Neural operators for solving nonlinear inverse problems

TL;DR

This paper analyzes the use of neural operators as surrogates in Tikhonov regularization for solving infinite dimensional, ill-posed operator equations, extending their approximation capabilities and discussing network training.

Contribution

It provides a theoretical analysis of neural operators in regularization, extending their approximation to Sobolev and Lebesgue spaces, and discusses network structure selection.

Findings

Neural operators can effectively approximate infinite dimensional operators.

The analysis balances approximation errors, regularization, and noise.

Numerical experiments demonstrate the approach's viability.

Abstract

We consider solving a probably infinite dimensional operator equation, where the operator is not modeled by physical laws but is specified indirectly via training pairs of the input-output relation of the operator. Neural operators have proven to be efficient to approximate infinite dimensional operators. In this paper we analyze Tikhonov regularization with neural operators as surrogates for solving ill-posed operator equations. The analysis is based on balancing approximation errors of neural operators, regularization parameters, and noise. Moreover, we extend the approximation properties of neural operators from sets of continuous functions to Sobolev and Lebesgue spaces, which is crucial for solving inverse problems and we discuss the problem of finding an appropriate network structure of neural operators (training). Finally, we present some numerical experiments.