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arXiv · 2410.18439

Multiscale Neural Networks for Approximating Green's Functions

Wenrui Hao·Rui Peng Li·Yuanzhe Xi·Tianshi Xu·Yahong Yang
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Abstract

Neural networks (NNs) have been widely used to solve partial differential equations (PDEs) in the applications of physics, biology, and engineering. One effective approach for solving PDEs with a fixed differential operator is learning Green's functions. However, Green's functions are notoriously difficult to learn due to their poor regularity, which typically requires larger NNs and longer training times. In this paper, we address these challenges by leveraging multiscale NNs to learn Green's functions. Through theoretical analysis using multiscale Barron space methods and experimental validation, we show that the multiscale approach significantly reduces the necessary NN size and accelerates training.

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Wenrui Hao, Rui Peng Li, Yuanzhe Xi, Tianshi Xu, Yahong Yang. 2024-10-24. Multiscale Neural Networks for Approximating Green's Functions. https://arxiv.org/abs/2410.18439

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