Year: 2023
Author: Wuyang Li, Ziming Wang, Tao Cui, Yingxiang Xu, Xueshuang Xiang
Advances in Applied Mathematics and Mechanics, Vol. 15 (2023), Iss. 1 : pp. 118–138
Abstract
This paper proposes a deep-learning-based Robin-Robin domain decomposition method (DeepDDM) for Helmholtz equations. We first present the plane wave activation-based neural network (PWNN), which is more efficient for solving Helmholtz equations with constant coefficients and wavenumber $k$ than finite difference methods (FDM). On this basis, we use PWNN to discretize the subproblems divided by domain decomposition methods (DDM), which is the main idea of DeepDDM. This paper will investigate the number of iterations of using DeepDDM for continuous and discontinuous Helmholtz equations. The results demonstrate that: DeepDDM exhibits behaviors consistent with conventional robust FDM-based domain decomposition method (FDM-DDM) under the same Robin parameters, i.e., the number of iterations by DeepDDM is almost the same as that of FDM-DDM. By choosing suitable Robin parameters on different subdomains, the convergence rate is almost constant with the rise of wavenumber in both continuous and discontinuous cases. The performance of DeepDDM on Helmholtz equations may provide new insights for improving the PDE solver by deep learning.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/aamm.OA-2021-0305
Advances in Applied Mathematics and Mechanics, Vol. 15 (2023), Iss. 1 : pp. 118–138
Published online: 2023-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 21
Keywords: Helmholtz equation deep learning domain decomposition method plane wave method.
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