Markov chain generative adversarial neural networks for solving Bayesian inverse problems in physics applications

Abstract

In the context of solving inverse problems for physics applications within a Bayesian framework, we present a new approach, the Markov Chain Generative Adversarial Neural Network (MCGAN), to alleviate the computational costs associated with solving the Bayesian inference problem. GANs pose a very suitable framework to aid in the solution of Bayesian inference problems, as they are designed to generate samples from complicated high-dimensional distributions. By training a GAN to sample from a low-dimensional latent space and then embedding it in a Markov Chain Monte Carlo method, we can highly efficiently sample from the posterior, by replacing both the high-dimensional prior and the expensive forward map. This comes at the cost of a potentially expensive offline stage in which training data must be simulated or gathered and the GAN has to be trained. We prove that the proposed methodology converges to the true posterior in the Wasserstein-1 distance and that sampling from the latent space is equivalent to sampling in the high-dimensional space in a weak sense. The method is showcased in two test cases where we perform both state and parameter estimation simultaneously and it is compared with two conventional approaches, polynomial chaos expansion and ensemble Kalman filter, and a deep learning-based approach, deep Bayesian inversion. The method is shown to be more accurate than alternative approaches while also being computationally faster, in multiple test cases, including the important engineering setting of detecting leaks in pipelines.