An artificial neural network approximation for Cauchy inverse problems

TL;DR

This paper introduces a neural network approach for solving Cauchy inverse problems, demonstrating improved accuracy and stability across various complex scenarios, including high dimensions and noisy data.

Contribution

It proposes a flexible neural network framework with multiple hidden layers, establishing theoretical existence and convergence results for the inverse problem solution.

Findings

Neural networks with wider and deeper layers improve approximation accuracy.

The method is effective for high-dimensional and noisy data scenarios.

Theoretical proof of well-posedness supports the approach's reliability.

Abstract

A novel artificial neural network method is proposed for solving Cauchy inverse problems. It allows multiple hidden layers with arbitrary width and depth, which theoretically yields better approximations to the inverse problems. In this research, the existence and convergence are shown to establish the well-posedness of neural network method for Cauchy inverse problems, and various numerical examples are presented to illustrate its accuracy and stability. The numerical examples are from different points of view, including time-dependent and time-independent cases, high spatial dimension cases up to 8D, and cases with noisy boundary data and singular computational domain. Moreover, numerical results also show that neural networks with wider and deeper hidden layers could lead to better approximation for Cauchy inverse problems.