Solving PDEs on unknown manifolds with machine learning

doi:10.1016/j.acha.2024.101652

KMS > 数学科学研究所

	Solving PDEs on unknown manifolds with machine learning
	Liang, Senwei 1; Jiang, Shixiao W.2 ; Harlim, John 3,4; Yang, Haizhao 5,6
	2024-07
发表期刊	APPLIED AND COMPUTATIONAL HARMONIC ANALYSIS (IF:2.6[JCR-2023],2.5[5-Year])
ISSN	1063-5203
EISSN	1096-603X
卷号	71
发表状态	已发表
DOI	10.1016/j.acha.2024.101652
摘要	This paper proposes a mesh-free computational framework and machine learning theory for solving elliptic PDEs on unknown manifolds, identified with point clouds, based on diffusion maps (DM) and deep learning. The PDE solver is formulated as a supervised learning task to solve a least-squares regression problem that imposes an algebraic equation approximating a PDE (and boundary conditions if applicable). This algebraic equation involves a graph-Laplacian type matrix obtained via DM asymptotic expansion, which is a consistent estimator of second-order elliptic differential operators. The resulting numerical method is to solve a highly non-convex empirical risk minimization problem subjected to a solution from a hypothesis space of neural networks (NNs). In a well-posed elliptic PDE setting, when the hypothesis space consists of neural networks with either infinite width or depth, we show that the global minimizer of the empirical loss function is a consistent solution in the limit of large training data. When the hypothesis space is a two-layer neural network, we show that for a sufficiently large width, gradient descent can identify a global minimizer of the empirical loss function. Supporting numerical examples demonstrate the convergence of the solutions, ranging from simple manifolds with low and high co-dimensions, to rough surfaces with and without boundaries. We also show that the proposed NN solver can robustly generalize the PDE solution on new data points with generalization errors that are almost identical to the training errors, superseding a Nyström-based interpolation method. © 2024 Elsevier Inc.
关键词	Algebra Computation theory Deep neural networks Gradient methods Learning systems Mathematical operators Multilayer neural networks Network layers Convergence analysis Diffusion maps High-dimensional High-dimensional PDE Higher-dimensional Hypothesis space Least squares minimization Manifold Neural-networks Point-clouds
URL	查看原文
收录类别	EI ; SCI
语种	英语
资助项目	NSF[DMS-1854299] ; ONR[N00014-22-1-2193] ; NSFC[12101408] ; US National Science Foundation["DMS-2206333","N00014-23-1-2007"] ; null[DMS-2244988]
WOS研究方向	Mathematics
WOS类目	Mathematics, Applied
WOS记录号	WOS:001208694100001
出版者	Academic Press Inc.
EI入藏号	20241015672645
EI主题词	Numerical methods
EI分类号	461.4 Ergonomics and Human Factors Engineering ; 721.1 Computer Theory, Includes Formal Logic, Automata Theory, Switching Theory, Programming Theory ; 723 Computer Software, Data Handling and Applications ; 921.1 Algebra ; 921.6 Numerical Methods
原始文献类型	Journal article (JA)
文献类型	期刊论文
条目标识符	https://kms.shanghaitech.edu.cn/handle/2MSLDSTB/352546
专题	数学科学研究所数学科学研究所_PI研究组(P)_蒋诗晓组
通讯作者	Yang, Haizhao
作者单位	1.Department of Mathematics, Purdue University, IN; 47907, United States; 2.Institute of Mathematical Sciences, ShanghaiTech University, Shanghai; 201210, China; 3.Department of Mathematics, Institute for Computational and Data Sciences, The Pennsylvania State University, University Park; PA; 16802, United States; 4.Department of Meteorology and Atmospheric Science, Institute for Computational and Data Sciences, The Pennsylvania State University, University Park; PA; 16802, United States; 5.Department of Mathematics, University of Maryland, College Park; MD; 20742, United States; 6.Department of Computer Science, University of Maryland, College Park; MD; 20742, United States
推荐引用方式 GB/T 7714	Liang, Senwei,Jiang, Shixiao W.,Harlim, John,et al. Solving PDEs on unknown manifolds with machine learning[J]. APPLIED AND COMPUTATIONAL HARMONIC ANALYSIS,2024,71.
APA	Liang, Senwei,Jiang, Shixiao W.,Harlim, John,&Yang, Haizhao.(2024).Solving PDEs on unknown manifolds with machine learning.APPLIED AND COMPUTATIONAL HARMONIC ANALYSIS,71.
MLA	Liang, Senwei,et al."Solving PDEs on unknown manifolds with machine learning".APPLIED AND COMPUTATIONAL HARMONIC ANALYSIS 71(2024).