Convergence of Physics-Informed Neural Networks Applied to Linear Second-Order Elliptic Interface Problems (Communications in Computational Physics, 2023)
With the remarkable empirical success of neural networks across diversescientific 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 convergenceanalysis of physics-informed neural networks (PINNs) for solving second-order elliptic interface problems. Specifically, we consider PINNs with domain decompositiontechnologies and introduce gradient-enhanced strategies on the interfaces to deal withboundary and interface jump conditions. It is shown that the neural network sequenceobtained by minimizing a Lipschitz regularized loss function converges to the uniquesolution to the interface problem in $H^2$ as the number of samples increases. Numericalexperiments are provided to demonstrate our theoretical analysis.
Keywords: Elliptic interface problems; generalization errors; convergence analysis; neural networks.