Point Source Identification Using Singularity Enriched Neural Networks
This addresses a gap in neural network methods for point source identification, which is important in applied inverse problems, though it appears incremental as it builds on existing techniques.
The paper tackled the inverse problem of recovering point sources by developing a neural network-based algorithm with singularity enrichment, demonstrating effectiveness in challenging experiments.
The inverse problem of recovering point sources represents an important class of applied inverse problems. However, there is still a lack of neural network-based methods for point source identification, mainly due to the inherent solution singularity. In this work, we develop a novel algorithm to identify point sources, utilizing a neural network combined with a 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 the unknown point sources, as well as 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.