Deep neural networks can stably solve high-dimensional, noisy, non-linear inverse problems

TL;DR

This paper shows that deep neural networks can reliably reconstruct solutions to high-dimensional, noisy, non-linear inverse problems by approximating the inverse operator in a stable manner, supported by theoretical and numerical evidence.

Contribution

It introduces a framework for using neural networks to stably approximate inverse operators in high-dimensional inverse problems, even with noisy data.

Findings

Neural networks can approximate inverse operators with stability to noise.

The approach is applicable to a wide range of practical inverse problems.

Numerical experiments support the theoretical stability and robustness of the method.

Abstract

We study the problem of reconstructing solutions of inverse problems when only noisy measurements are available. We assume that the problem can be modeled with an infinite-dimensional forward operator that is not continuously invertible. Then, we restrict this forward operator to finite-dimensional spaces so that the inverse is Lipschitz continuous. For the inverse operator, we demonstrate that there exists a neural network which is a robust-to-noise approximation of the operator. In addition, we show that these neural networks can be learned from appropriately perturbed training data. We demonstrate the admissibility of this approach to a wide range of inverse problems of practical interest. Numerical examples are given that support the theoretical findings.