An efficient neural network method with plane wave activation functions for solving Helmholtz equation

doi:10.1016/j.camwa.2022.02.004

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	An efficient neural network method with plane wave activation functions for solving Helmholtz equation
	Cui, Tao 1,2; Wang, Ziming 1,2; Xiang, Xueshuang 3
	2022-04-01
发表期刊	COMPUTERS & MATHEMATICS WITH APPLICATIONS
ISSN	0898-1221
卷号	111 页码:34-49
摘要	In this paper, we propose a plane wave activation based neural network (PWNN) to solve the Helmholtz equation with constant coefficients and relatively large wave number kappa efficiently. Since the complex activation function ??(& igrave;& nbsp;->& nbsp;chi) is introduced to be the activation function of the neural network, PWNN significantly improves the computational speed and accuracy as compared to traditional activation based neural networks (TANN) and the finite difference method (FDM), for relatively large wave number problems. We establish a new upper bound for error estimates for homogeneous Helmholtz solutions in two dimensions by plane waves, which is different from the previous estimates by generalized harmonic polynomials. Based on the new error estimates, the theoretical guidance is given for choosing the number of neural network's neurons and the initial value to accelerate network training. The analyses of computational complexity related to the wave number kappa are given for PWNN with two layers, TANN, the plane wave partition of unity method (PWPUM) and FDM. Numerical experiments in 2D and 3D are performed to demonstrate the efficiency and accuracy of PWNN. Especially for large wave number problems, like kappa = 500, PWNN can get the solution with relative error less than 10-4 in less than 20 seconds, which is more efficient than other methods.
关键词	Helmholtz equation Deep learning Finite difference method Plane wave method
DOI	10.1016/j.camwa.2022.02.004
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[11771440] ; China NSF[12171468] ; Innovation Foundation of Qian Xuesen Laboratory of Space Technology
WOS研究方向	Mathematics
WOS类目	Mathematics, Applied
WOS记录号	WOS:000789919800003
出版者	PERGAMON-ELSEVIER SCIENCE LTD
引用统计
文献类型	期刊论文
条目标识符	http://ir.amss.ac.cn/handle/2S8OKBNM/61277
专题	中国科学院数学与系统科学研究院
通讯作者	Wang, Ziming; Xiang, Xueshuang
作者单位	1.Chinese Acad Sci, Acad Math & Syst Sci, NCMIS, LSEC, Beijing 100190, Peoples R China 2.Univ Chinese Acad Sci, Sch Math Sci, Beijing 100049, Peoples R China 3.China Acad Space Technol, Qian Xuesen Lab Space Technol, Beijing 100094, Peoples R China
推荐引用方式 GB/T 7714	Cui, Tao,Wang, Ziming,Xiang, Xueshuang. An efficient neural network method with plane wave activation functions for solving Helmholtz equation[J]. COMPUTERS & MATHEMATICS WITH APPLICATIONS,2022,111:34-49.
APA	Cui, Tao,Wang, Ziming,&Xiang, Xueshuang.(2022).An efficient neural network method with plane wave activation functions for solving Helmholtz equation.COMPUTERS & MATHEMATICS WITH APPLICATIONS,111,34-49.
MLA	Cui, Tao,et al."An efficient neural network method with plane wave activation functions for solving Helmholtz equation".COMPUTERS & MATHEMATICS WITH APPLICATIONS 111(2022):34-49.