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

In this work, we develop a novel approach using deep neural networks (DNNs) to reconstruct the conductivity distribution in elliptic problems from one measurement of the solution over the whole domain. The approach is based on a mixed reformulation of the governing equation and utilizes the standard least-squares objective, with DNNs as ansatz functions to approximate the conductivity and flux simultaneously. We provide a thorough analysis of the DNN approximations of the conductivity for both continuous and empirical losses, including rigorous error estimates that are explicit in terms of the noise level, various penalty parameters, and neural network architectural parameters (depth, width, and parameter bounds). We also provide multiple numerical experiments in two dimensions and multidimensions to illustrate distinct features of the approach, e.g., excellent stability with respect to data noise and capability of solving high-dimensional problems.