Semantic Autoencoder for PDE models

This semantic autoencoder is used to solve the following inverse problem: Given (perhaps noisy) observations of some phenomenon which is modeled as a solution to a known partial differential equation (PDE) with unknown parameters theta, can we infer theta?

The semantic autoencoder merges Tensorflow 2 and FEniCS, a finite element method package. Tensorflow encodes the observations to a latent space that represents theta, then FEniCS decodes theta to reproduce the observations according to the PDE model. Unsupervised training is performed by comparing the observations with those produced by the network, thereby training the network to learn theta.

Adjoint Poisson's Equation Gradient Descent Usage

See this notebook for a thorough demonstration.

Supervised Encoding Usage

Requirements: Tensorflow 2.2 and scikit-learn 0.23.1

See this notebook for a demonstration, and compare it to the analytic autoencoder version.

The equations/ directory contains some example config files. There are four requirements for the config files:

1. theta_names is a list of strings representing names for each parameter in theta
2. domain() is a function that returns the domain for the PDE
3. solution(theta) is a function that calculates the analytic solution to the PDE as it depends on theta.
4. theta(replicates) is a function that simulates theta from a prior distribution.

With a config file containing these, such as equations/poisson.py, one simply needs to run

python train_encoder.py --trainon poisson

and observe the encoder will be trained. You will find plots of the predicted vs true theta values in logs/fit/{datetime}/ where datetime is the date and time at which the encoder was trained.

A more detailed example of training using poisson.py is worked through in poisson_encoder_example.ipynb.