Error Analysis for Solving Elliptic Interface Problems with Randomized Neural Networks

Abstract

This paper presents a unified error analysis of randomized neural networks (RaNNs) for elliptic interface problems. We first derive an integral representation of Barron functions using the standard $\tanh$ activation, which yields a $C^2$-approximation bound valid for any hidden-parameter distribution supported on a cube with strictly positive density. We then establish a quantitative $\mathcal{O}(n^{-1})$ generalization bound for the composite loss that enforces the partial differential equation, boundary, and interface conditions simultaneously, where $n$ denotes the number of training samples. Combining these results with the optimization guarantees for RaNNs, we obtain an a priori $L^2$ error estimate for the neural network solution in the presence of interface discontinuities. Numerical experiments are presented to validate the theoretical results.