Point source identification using singularity-enriched neural networks

Abstract Neural network-based methods have shown great promise in stably solving ill-posed inverse problems. In this work we focus on the inverse problem of recovering point sources, an important class of applied inverse problems. Despite their potential neural network-based methods for identifying point sources remain underdeveloped, primarily due to the inherent singularity of the solution. To address this challenge we develop a novel neural algorithm for identifying point sources, utilizing the singularity enrichment technique. We employ the fundamental solution and neural networks to represent the singular and regular parts, respectively, and then minimize an empirical loss involving the intensities and locations of unknown point sources and the parameters of the neural network. Moreover, by combining the conditional stability argument of the inverse problem with the generalization error of the empirical loss we conduct a rigorous error analysis of the algorithm. We demonstrate the effectiveness of the method with several challenging experiments.