Xi‐Yao Li

Bangti Jin, Xiyao Li, Xiliang Lu

Conductivity imaging represents one of the most important tasks in medical imaging. In this work we develop a neural network based reconstruction technique for imaging the conductivity from the magnitude of the internal current density. It is achieved by formulating the problem as a relaxed weighted least-gradient problem, and then approximating its minimizer by standard fully connected feedforward neural networks. We derive bounds on two components of the generalization error, i.e., approximation error and statistical error, explicitly in terms of properties of the neural networks (e.g., depth, total number of parameters, and the bound of the network parameters). We illustrate the performance and distinct features of the approach on several numerical experiments. Numerically, it is observed that the approach enjoys remarkable robustness with respect to the presence of data noise.

Bangti Jin, Xiyao Li, Qimeng Quan et al.

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.