  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.
