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

• URL: http://arxiv.org/abs/2206.00934v5
• Date: Fri, 20 Oct 2023 14:06:30 GMT
• Title: Deep neural networks can stably solve high-dimensional, noisy, non-linear inverse problems
• Authors: Andr\'es Felipe Lerma Pineda and Philipp Christian Petersen
• Abstract summary: We study the problem of reconstructing solutions of inverse problems when only noisy measurements are available. For the inverse operator, we demonstrate that there exists a neural network which is a robust-to-noise approximation of the operator.
• Score: 2.6651200086513107
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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