On the recovery of two function-valued coefficients in the Helmholtz equation for inverse scattering problems via neural networks

TL;DR

This paper explores the use of combined deep neural networks to recover two function-valued coefficients in the Helmholtz equation for inverse scattering, analyzing their approximation and generalization capabilities with promising numerical results.

Contribution

It introduces novel combined DNN architectures for reconstructing two coefficients in inverse scattering and analyzes their approximation and generalization properties.

Findings

Neural networks can effectively approximate the inverse process with sufficient data.

Proposed neural networks successfully recover isotropic inhomogeneous media.

Networks can reconstruct isotropic representations of certain anisotropic media.

Abstract

Recently, deep neural networks (DNNs) have become powerful tools for solving inverse scattering problems. However, the approximation and generalization rates of DNNs for solving these problems remain largely under-explored. In this work, we introduce two types of combined DNNs (uncompressed and compressed) to reconstruct {two function-valued coefficients} in the Helmholtz equation for inverse scattering problems from the scattering data at two different frequencies. An analysis of the approximation and generalization capabilities of the proposed neural networks for simulating the regularized pseudo-inverses of the linearized forward operators in direct scattering problems is provided. The results show that, with sufficient training data and parameters, the proposed neural networks can effectively approximate the inverse process with desirable generalization. Preliminary numerical results show the feasibility of the proposed neural networks for recovering two types of isotropic inhomogeneous media. Furthermore, the trained neural network is capable of reconstructing the isotropic representation of certain types of anisotropic media.