  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.
