Functional normalizing flow for statistical inverse problems of partial differential equations

TL;DR

This paper introduces a novel infinite-dimensional variational inference method using normalizing flows for solving large-scale inverse problems in PDEs, ensuring discretization invariance and computational efficiency.

Contribution

It proposes a new NF-iVI framework with concrete transformations and a conditional variant CNF-iVI for efficient, discretization-invariant Bayesian inference in PDE inverse problems.

Findings

Algorithms effectively recover posterior distributions.

Method demonstrates discretization-invariance across problems.

Numerical results confirm efficiency and theoretical properties.

Abstract

Inverse problems of partial differential equations are ubiquitous across various scientific disciplines and can be formulated as statistical inference problems using Bayes' theorem. To address large-scale problems, it is crucial to develop discretization-invariant algorithms, which can be achieved by formulating methods directly in infinite-dimensional space. We propose a novel normalizing flow based infinite-dimensional variational inference method (NF-iVI) to extract posterior information efficiently. Specifically, by introducing well-defined transformations, the prior in Bayes' formula is transformed into post-transformed measures that approximate the posterior. To circumvent the issue of mutually singular probability measures, we formulate general conditions for the employed transformations. As guiding principles, these conditions yield four concrete transformations. Additionally, to minimize computational demands, we have developed a conditional normalizing flow variant, termed CNF-iVI, which is adapt at processing measurement data of varying dimensions while requiring minimal computational resources. We apply the proposed algorithms to three typical inverse problems governed by the simple smooth equation, the steady-state Darcy flow equation, and the electric impedance tomography. Numerical results confirm our theoretical findings, illustrate the efficiency of our algorithms, and verify the discretization-invariant property.