Imaging Anisotropic Conductivity from Internal Measurements with Mixed Least-Squares Deep Neural Networks

TL;DR

This paper introduces MLS-DNN, a deep learning-based method for reconstructing anisotropic conductivity tensors from internal measurements, demonstrating high accuracy in 2D and 3D with noise robustness.

Contribution

The paper proposes a novel mixed least-squares deep neural network approach for anisotropic conductivity imaging, with theoretical error bounds and comprehensive numerical validation.

Findings

Accurately recovers anisotropic conductivity in 2D and 3D.

Performs well with up to 10% noise in data.

Outperforms standard finite element and physics-informed neural network methods.

Abstract

In this work we develop a novel algorithm, termed as mixed least-squares deep neural network (MLS-DNN), to recover an anisotropic conductivity tensor from the internal measurements of the solutions. It is based on applying the least-squares formulation to the mixed form of the elliptic problem, and approximating the internal flux and conductivity tensor simultaneously using deep neural networks. We provide error bounds on the approximations obtained via both population and empirical losses. The analysis relies on the canonical source condition, approximation theory of deep neural networks and statistical learning theory. We also present multiple numerical experiments to illustrate the performance of the method, and conduct a comparative study with the standard Galerkin finite element method and physics informed neural network. The results indicate that the method can accurately recover the anisotropic conductivity in both two- and three-dimensional cases, up to 10\% noise in the data.