Deep Learning and Bayesian inference for Inverse Problems

TL;DR

This paper explores how Bayesian Deep Learning can be applied to inverse problems, offering computationally efficient solutions by leveraging neural network surrogates and uncertainty quantification.

Contribution

It provides a detailed analysis of Bayesian DL methods for inverse problems, including approximate computations and applications to physics-constrained and data-driven models.

Findings

Bayesian DL effectively quantifies uncertainty in inverse problems.

Surrogate neural networks reduce computational costs significantly.

Bayesian methods improve solution robustness in complex inverse scenarios.

Abstract

Inverse problems arise anywhere we have indirect measurement. As, in general they are ill-posed, to obtain satisfactory solutions for them needs prior knowledge. Classically, different regularization methods and Bayesian inference based methods have been proposed. As these methods need a great number of forward and backward computations, they become costly in computation, in particular, when the forward or generative models are complex and the evaluation of the likelihood becomes very costly. Using Deep Neural Network surrogate models and approximate computation can become very helpful. However, accounting for the uncertainties, we need first understand the Bayesian Deep Learning and then, we can see how we can use them for inverse problems. In this work, we focus on NN, DL and more specifically the Bayesian DL particularly adapted for inverse problems. We first give details of Bayesian DL approximate computations with exponential families, then we will see how we can use them for inverse problems. We consider two cases: First the case where the forward operator is known and used as physics constraint, the second more general data driven DL methods. keyword: Neural Network, Variational Bayesian inference, Bayesian Deep Learning (DL), Inverse problems, Physics based DL.