With the remarkable empirical success of neural networks across diverse scientific disciplines, rigorous error and convergence analysis are also being developed and enriched. However, there has been little theoretical work focusing on neural networks in solving interface problems. In this paper, we perform a convergence analysis of physics-informed neural networks (PINNs) for solving second-order elliptic interface problems. Specifically, we consider PINNs with domain decomposition technologies and introduce gradient-enhanced strategies on the interfaces to deal with boundary and interface jump conditions. It is shown that the neural network sequence obtained by minimizing a Lipschitz regularized loss function converges to the unique solution to the interface problem in $H^2$ as the number of samples increases. Numerical experiments are provided to demonstrate our theoretical analysis.

翻译：随着神经网络在众多科学领域中取得显著的经验成功，其严格的误差与收敛性分析也在不断发展与完善。然而，针对神经网络求解界面问题的理论研究仍较为匮乏。本文对用于求解二阶椭圆界面问题的物理信息神经网络（PINNs）进行了收敛性分析。具体而言，我们考虑采用区域分解技术的PINNs，并在界面上引入梯度增强策略以处理边界及界面跳跃条件。研究表明，通过最小化Lipschitz正则化损失函数获得的神经网络序列，在样本数量增加时，会在$H^2$空间内收敛至界面问题的唯一解。数值实验验证了我们的理论分析结果。