Recovering the Source Term in Elliptic Equation via Deep Learning: Method and Convergence Analysis

Abstract

In this paper, we present a deep learning approach to tackle elliptic inverse source problems. Our method combines Tikhonov regularization with physics-informed neural networks, utilizing separate neural networks to approximate the source term and solution. Firstly, we construct a population loss and derive stability estimates. Furthermore, we conduct a convergence analysis of the empirical risk minimization estimator. This analysis yields a prior rule for selecting regularization parameters, determining the number of observations, and choosing the size of neural networks. Finally, we validate our proposed method through numerical experiments. These experiments also demonstrate the remarkable robustness of our approach against data noise, even at high levels of up to 50%.