  In this paper we show how to augment classical methods for inverse problems
with artificial neural networks. The neural network acts as a prior for the
coefficient to be estimated from noisy data. Neural networks are global, smooth
function approximators and as such they do not require explicit regularization
of the error functional to recover smooth solutions and coefficients. We give
detailed examples using the Poisson equation in 1, 2, and 3 space dimensions
and show that the neural network augmentation is robust with respect to noisy
and incomplete data, mesh, and geometry.
