This repository supplements H108-0002 - Efficient Data Assimilation with Latent-Space Representations for Subsurface Flow Systems and Mohd-Razak et al (SPE Reservoir Simulation Conference, 2021).
The dataset used in this demo repository is the digit-MNIST images, X, with the forward model being a linear operator G and the resulting simulated responses denoted as Y. The physical system is represented as Y=G(X) or D=G(M). More description on the dataset is available here (in readme) and here is a demo on using convolutional autoencoders for dimensionality reduction of the digit-MNIST images.
We are interested in learning the inverse mapping M=G'(D) which is not trivial if the M is non-Gaussian (which is the case with the digit-MNIST images) and G is nonlinear (in this demo we assume a linear operator). Such complex mapping (also known as history-matching) may result in solutions that are non-unique with features that may not be consistent. Multiple history-matching approaches exist, such as gradient-based, direct-inversion and ensemble-based methods. In ensemble-based methods, we start with an initial ensemble of prior models that are gradually updated, based on the mismatch between data simulated from each of the prior models, to the observed data. Once the iterative update steps are done, the information within the observed data has been assimilated into the ensemble of prior models, and they become posterior models that can reproduce the observed data. The problem with this approach is that the forward model used as the simulator (i.e. G) can be computationally prohibitive especially when the models M are of high-fidelity and the size of the prior ensemble is large.
To address these challenges, LSDA performs simultaneous dimensionality reduction (by extracting salient spatial features from M and temporal features from D) and forward mapping (by mapping the salient features in M to D, i.e. latent spaces z_m and z_d). The architecture is composed of dual autoencoders connected with a regression model as shown below and is trained jointly. See this Jupyter Notebook to train/load/QC the architecture. Use this Jupyter Notebook to train/load/QC the architecture without the variational losses (i.e. no Gaussian prior constraint on the latent variables in variational autoencoders). Typically when the architecture has enough capacity to represent the model and data, the latent variables naturally become Gaussian and the variational loss becomes unnecessary. Note that the constraint on the latent variables can also reduce the performance of LSDA, so evaluate your dataset and check if the constraint is really necessary!
Once the architecture is trained, the low-dimensional vectors z_m represent the high-fidelity models M and z_d represent the simulated data D. The (potentially) computationally expensive forward model G is now represented by the regression model that maps z_m to z_d, as an efficient proxy model. Given an observation vector d_obs, the ensemble of priors z_m is iteratively assimilated using the following ESMDA equation, where each corresponding z_d is obtained from the proxy model:
The LSDA workflow is illustrated here:
The pseudocode is described here:
Note that when the variational loss is omitted, all the loss functions simply become squared-error terms (i.e. similar to Latent-Space-Inversion, LSI).
The figures in this demo can be reproduced using this notebook. In this demonstration, assume the following reference case of digit zero (the same reference case in LSI) and no Gaussian prior constraint on the latent variables z_m and z_d. The first left image in the figure below shows the reference m and the second shows its reconstruction from the model autoencoder. The third plot is the d_obs with its reconstruction from the data autoencoder and the fourth plot shows the prediction/forecast from the proxy model.
The histograms below show z_d (simulated and reduced from the entire initial prior ensemble) and the red line represents the latent variables z_d_obs. The black unfilled bars denoted as "Update" represent the z_d from the ensemble of posteriors, and they agree with z_d_obs.
Similarly, the histograms below show z_m (reduced from the entire initial prior ensemble) and the red line represents the latent variables z_m_ref of the reference m (assume known). The black unfilled bars denoted as "Update" represent the assimilated z_m (i.e. posteriors).
In the figure below, sample prior-posterior pairs from the ensemble are shown:
The mean and variance maps from the ensemble of priors and posteriors show that the information in d_obs has been assimilated into the ensemble of prior models.
The posteriors are fed to the operator G (i.e. D=G(M)) to validate if the solutions can sufficiently reproduce the d_obs. In the figure below, gray points/lines represent the entire testing dataset (priors) and the orange points/lines represent the predicted (by proxy, second plot) / simulated (third plot) D from the posteriors, where a significant reduction in uncertainty is obtained.
In practical applications, d_obs can be noisy and LSDA helps us to quickly obtain the ensemble of posteriors that can be accepted within the noise level, as well as understand the variations of spatial features within the posteriors, to improve the predictive power of the calibrated/assimilated models.