Add technical implementation in jax documentation

VERSION

@@ -1 +1 @@
-0.0.12
+0.0.13

docs/api.md

@@ -3,6 +3,7 @@
 ::: driftbench.data_generation.sample
 ::: driftbench.data_generation.drifts
 ::: driftbench.data_generation.solvers
+::: driftbench.data_generation.latent_information
 
 # Drift detection
 

docs/how_it_works.md

@@ -2,3 +2,34 @@
 
 
 # Technical implementation in `jax`
+This package uses [JAX](JAX (https://github.com/jax-ml/jax) as its backend for generating synthetic curves.
+In particular, `jax` is used for:
+
+- Solving non-linear optimization problems.
+- Automatic differentiation for calculating partial derivates of arbitrary functions.
+- Just-in-time (JIT)-compilation with XLA for performance optimization.
+
+For more detailed information regarding the XLA-compilation, please see the 
+[offical JAX documentation](https://docs.jax.dev/en/latest/index.html)
+or [XLA-documentation](https://openxla.org/xla/tf2xla?hl=en).
+
+These three points ensure an efficient generation of curves, while being
+able to control the latent information used and the behaviour of drifts applied
+on the curves.
+
+The method used to solve optimization problems is the 
+[LBFGS](https://en.wikipedia.org/wiki/Limited-memory_BFGS) algorithm, which is supported by `jax`.
+The corresponding functions in order to compute the error term, running a iteration of the
+optimization solving problem, and computing the gradients is all done in functions which are 
+compiled just-in-time and can be run on a GPU.
+The procedure can be described as follows:
+
+- Choose a function $f(w(t), x)$, which describes the shape of the curves to generate with 
+inital internal parameters $w_0(t)$.
+- Provide problem constraints encoded in
+[`LatentInformation`][driftbench.data_generation.latent_information.LatentInformation]
+objects.
+- Compute the partial derivates of $f(w(t), x)$ with respect to $x_i$.
+- For each latent information object, compute $w(t)$ according to the LBFGS-algorithm.
+- Return all computed solutions $w(t)$ for each curve.
+

driftbench/data_generation/latent_information.py

@@ -19,8 +19,8 @@ class LatentInformation:
         y2 (list-like): The y-values of the derivative of a function.
         x2 (list-like): The x-values of the second derivative of a function.
         Hence, no duplicates are allowed.
-
     """
+
     y0: np.ndarray
     x0: np.ndarray
     y1: np.ndarray
@@ -35,11 +35,17 @@ def __post_init__(self):
 
     def _validate_matching_shapes(self):
         if self.y0.shape != self.x0.shape:
-            raise ValueError("Features y0 and x0 are not allowed to have different shape")
+            raise ValueError(
+                "Features y0 and x0 are not allowed to have different shape"
+            )
         if self.y1.shape != self.x1.shape:
-            raise ValueError("Features y1 and x1 are not allowed to have different shape")
+            raise ValueError(
+                "Features y1 and x1 are not allowed to have different shape"
+            )
         if self.y2.shape != self.x2.shape:
-            raise ValueError("Features y2 and x2 are not allowed to have different shape")
+            raise ValueError(
+                "Features y2 and x2 are not allowed to have different shape"
+            )
 
     def _validate_1d_array(self):