  This paper introduces a new approach for solving electrical impedance
tomography (EIT) problems using deep neural networks. The mathematical problem
of EIT is to invert the electrical conductivity from the Dirichlet-to-Neumann
(DtN) map. Both the forward map from the electrical conductivity to the DtN map
and the inverse map are high-dimensional and nonlinear. Motivated by the linear
perturbative analysis of the forward map and based on a numerically low-rank
property, we propose compact neural network architectures for the forward and
inverse maps for both 2D and 3D problems. Numerical results demonstrate the
efficiency of the proposed neural networks.
