  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.
