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

Andrés Felipe Lerma Pineda, Philipp Christian Petersen

Published 2022-06-02Version 1

We study the problem of reconstructing solutions of inverse problems with neural networks when only noisy data is available. We assume 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 function. 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.