On convergence of neural network methods for solving elliptic interface problems
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With the remarkable empirical success of neural networks across diverse scientific disciplines, rigorous error and convergence analysis are also being developed and enriched. However, most existing theoretical works fail to explain the performance of neural networks in solving interface problems. In this paper, we perform a convergence analysis of the neural network-based method for solving second-order elliptic interface problems. Here, we consider the domain decomposition method with gradient-enhanced on the boundary and interface and Lipschitz regularized loss. It is shown that the neural network sequence obtained by minimizing the regularization loss function converges to the unique solution to the interface problem in H^2 as the number of samples increases. This result serves the theoretical understanding of deep learning-based solvers for interface problems.
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