论文标题
适用于线性二阶椭圆界面问题的物理信息神经网络的收敛性
Convergence of physics-informed neural networks applied to linear second-order elliptic interface problems
论文作者
论文摘要
随着神经网络在各种科学学科中的显着经验成功，严格的错误和收敛分析也得到了发展和丰富。但是，在解决界面问题时，几乎没有理论工作的重点是神经网络。在本文中，我们对物理信息的神经网络（PINN）进行收敛分析，以解决二阶椭圆界面问题。具体而言，我们考虑具有域分解技术的PINN，并在接口上引入梯度增强策略，以处理边界和接口跳跃条件。结果表明，随着样品数量的增加，通过最小化Lipschitz正则损耗函数来获得的神经网络序列会收敛到$ H^2 $的界面问题的唯一解决方案。提供数值实验以证明我们的理论分析。
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.
