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Error Analysis for Solving Elliptic Interface Problems with Randomized Neural Networks

  • Jun Hu
  • Sidi Wu
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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 C2-approximation bound valid for any hidden-parameter distribution supported on a cube with strictly positive density. We then establish a quantitative π’ͺ(nβˆ’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 L2 error estimate for the neural network solution in the presence of interface discontinuities. Numerical experiments are presented to validate the theoretical results.

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Cite

BibTeXRIS
BibTeX
@misc{hu2026error,
  title = {{Error Analysis for Solving Elliptic Interface Problems with Randomized Neural Networks}},
  author = {Jun Hu and Sidi Wu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.26534},
  eprint = {2609.26534},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.26534},
}
RIS
TY  - UNPB
TI  - Error Analysis for Solving Elliptic Interface Problems with Randomized Neural Networks
AU  - Jun Hu
AU  - Sidi Wu
PY  - 2026
DA  - 2026/09/22
PB  - arXiv
DO  - 10.48550/arxiv.2609.26534
UR  - https://arxiv.org/abs/2609.26534
AB  - 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 π’ͺ(n^βˆ’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.
ER  -