The learned range test method for the inverse inclusion problem

TL;DR

This paper introduces a neural network-based learned range test method for reconstructing inclusions in a domain from boundary data, demonstrating improved accuracy and stability over traditional methods and pure deep learning approaches.

Contribution

It develops a novel approach combining a neural network architecture with the range test method for inverse problems, trained via supervised learning to enhance reconstruction quality.

Findings

The learned range test outperforms standard range test in accuracy.

The method provides stable reconstructions of polygonal inclusions.

It surpasses purely data-driven neural networks in performance.

Abstract

We consider the inverse problem consisting of the reconstruction of an inclusion $B$ contained in a bounded domain $\Omega\subset\mathbb{R}^d$ from a single pair of Cauchy data $(u|_{\partial\Omega},\partial_\nu u|_{\partial\Omega})$, where $\Delta u=0$ in $\Omega\setminus\overline B$ and $u=0$ on $\partial B$. We show that the reconstruction algorithm based on the range test, a domain sampling method, can be written as a neural network with a specific architecture. We propose to learn the weights of this network in the framework of supervised learning, and to combine it with a pre-trained classifier, with the purpose of distinguishing the inclusions based on their distance from the boundary. The numerical simulations show that this learned range test method provides accurate and stable reconstructions of polygonal inclusions. Furthermore, the results are superior to those obtained with the standard range test method (without learning) and with an end-to-end fully connected deep neural network, a purely data-driven method.