Deep Domain Decomposition Methods: Helmholtz Equation

doi:10.4208/aamm.OA-2021-0305

CSpace

	Deep Domain Decomposition Methods: Helmholtz Equation
	Li, Wuyang 1,3; Wang, Ziming 2,4; Cui, Tao 2,4; Xu, Yingxiang 1; Xiang, Xueshuang 3
	2023-02-01
发表期刊	ADVANCES IN APPLIED MATHEMATICS AND MECHANICS
ISSN	2070-0733
卷号	15 期号:1 页码:118-138
摘要	This paper proposes a deep-learning-based Robin-Robin domain decom-position 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 differ-ence methods (FDM). On this basis, we use PWNN to discretize the subproblems di-vided by domain decomposition methods (DDM), which is the main idea of Deep-DDM. 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 num-ber 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.
关键词	Helmholtz equation deep learning domain decomposition method plane wave method
DOI	10.4208/aamm.OA-2021-0305
收录类别	SCI
语种	英语
资助项目	National Key R&D Program of China[2019YFA0709600] ; National Key R&D Program of China[2019YFA0709602] ; China NSF[11831016] ; China NSF[12171468] ; China NSF[11771440] ; China NSF[12071069] ; Fundamental Research Funds for the Central Universities[JGPY202101] ; Innovation Foundation of Qian Xuesen Laboratory of Space Technology
WOS研究方向	Mathematics ; Mechanics
WOS类目	Mathematics, Applied ; Mechanics
WOS记录号	WOS:000880390000004
出版者	GLOBAL SCIENCE PRESS
引用统计
文献类型	期刊论文
条目标识符	http://ir.amss.ac.cn/handle/2S8OKBNM/60662
专题	中国科学院数学与系统科学研究院
通讯作者	Xiang, Xueshuang
作者单位	1.Northeast Normal Univ, Jilin Natl Appl Math Ctr NENU, Sch Math & Stat, Changchun 130024, Jilin, Peoples R China 2.Chinese Acad Sci, Acad Math & Syst Sci, NCMIS, LSEC, Beijing 100190, Peoples R China 3.China Acad Space Technol, Qian Xuesen Lab Space Technol, Beijing 100094, Peoples R China 4.Univ Chinese Acad Sci, Sch Math Sci, Beijing 100049, Peoples R China
推荐引用方式 GB/T 7714	Li, Wuyang,Wang, Ziming,Cui, Tao,et al. Deep Domain Decomposition Methods: Helmholtz Equation[J]. ADVANCES IN APPLIED MATHEMATICS AND MECHANICS,2023,15(1):118-138.
APA	Li, Wuyang,Wang, Ziming,Cui, Tao,Xu, Yingxiang,&Xiang, Xueshuang.(2023).Deep Domain Decomposition Methods: Helmholtz Equation.ADVANCES IN APPLIED MATHEMATICS AND MECHANICS,15(1),118-138.
MLA	Li, Wuyang,et al."Deep Domain Decomposition Methods: Helmholtz Equation".ADVANCES IN APPLIED MATHEMATICS AND MECHANICS 15.1(2023):118-138.