Neural networks as smooth priors for inverse problems for PDEs

In this paper we discuss the potential of using artificial neural networks as smooth priors in classical methods for inverse problems for PDEs. Exploring that neural networks are global and smooth function approximators, the idea is that neural networks could act as attractive priors for the coefficients to be estimated from noisy data. We illustrate the capabilities of neural networks in the context of the Poisson equation and we show that the neural network approach show robustness with respect to noisy and incomplete data and with respect to mesh and geometry