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DOE OSTI · 3374095

Multiscale Neural Networks for Approximating Green’s Functions

Hao, Wenrui [Pennsylvania State Univ., University Park, PA (United States)] (ORCID:0000000269257424)·Li, Rui Peng [Lawrence Livermore National Laboratory (LLNL), Livermore, CA (United States). Center for Applied Scientific Computing] (ORCID:0000000328025763)·Xi, Yuanzhe [Emory Univ., Atlanta, GA (United States)] (ORCID:0000000303610931)·Xu, Tianshi [Emory Univ., Atlanta, GA (United States)] (ORCID:0000000331191957)·Yang, Yahong [Pennsylvania State Univ., University Park, PA (United States)] (ORCID:0000000297212362)
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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 work, 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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97 MATHEMATICS AND COMPUTINGBarron spaceGreen’s functionPDEsdomain decompositionfast solvermultiscale neural networks

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Hao, Wenrui [Pennsylvania State Univ., University Park, PA (United States)] (ORCID:0000000269257424), Li, Rui Peng [Lawrence Livermore National Laboratory (LLNL), Livermore, CA (United States). Center for Applied Scientific Computing] (ORCID:0000000328025763), Xi, Yuanzhe [Emory Univ., Atlanta, GA (United States)] (ORCID:0000000303610931), Xu, Tianshi [Emory Univ., Atlanta, GA (United States)] (ORCID:0000000331191957), Yang, Yahong [Pennsylvania State Univ., University Park, PA (United States)] (ORCID:0000000297212362). 2026-04-06. Multiscale Neural Networks for Approximating Green’s Functions. https://doi.org/10.1137/24m1709601

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