Conductivity Imaging from Internal Measurements with Mixed Least-Squares Deep Neural Networks
This work addresses conductivity imaging for applications like medical or material science, but it is incremental as it adapts existing neural network methods to a specific inverse problem.
The authors tackled the problem of reconstructing conductivity distributions in elliptic equations from a single measurement by proposing a deep neural network approach based on a mixed reformulation and least-squares objective, achieving excellent stability against data noise and capability in high-dimensional settings as demonstrated in numerical experiments.
In this work we develop a novel approach using deep neural networks 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 deep neural networks as ansatz functions to approximate the conductivity and flux simultaneously. We provide a thorough analysis of the deep neural network 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 bound). We also provide multiple numerical experiments in two- and multi-dimensions to illustrate distinct features of the approach, e.g., excellent stability with respect to data noise and capability of solving high-dimensional problems.