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.

翻译：点源反演问题是一类重要的应用反演问题。然而，目前仍缺乏基于神经网络的点源识别方法，这主要源于解固有的奇异性。本工作提出了一种识别点源的新算法，该算法结合了神经网络与奇异性增强技术。我们分别利用基本解和神经网络表示奇异部分与正则部分，进而最小化一个包含未知点源强度与位置以及神经网络参数的经验损失函数。此外，通过结合反问题的条件稳定性论证与经验损失函数的泛化误差，我们对算法进行了严格的误差分析。我们通过多个具有挑战性的实验验证了该方法的有效性。