The Calder\'on's problem via DeepONets

TL;DR

This paper demonstrates that DeepONets can rigorously approximate the Dirichlet-to-Neumann map and Calderón's inverse problem mappings, advancing the application of deep learning in inverse boundary value problems.

Contribution

The paper introduces a theoretical framework showing DeepONets can accurately approximate Calderón's mappings, bridging deep learning and inverse PDE problems.

Findings

DeepONets can approximate the Dirichlet-to-Neumann map.

The approach provides a rigorous foundation for deep learning in inverse problems.

The method applies to smooth bounded domains with positive isotropic conductivities.

Abstract

We consider the Dirichlet-to-Neumann operator and the direct and inverse Calder\'on's mappings appearing in the Inverse Problem of recovering a smooth bounded and positive isotropic conductivity of a material filling a smooth bounded domain in space. Using deep learning techniques, we prove that these mappings are rigorously approximated by DeepONets, infinite-dimensional counterparts of standard artificial neural networks.