文献类型：期刊文献

英文题名：Utilizing Correlations in Singular Vector Space for Nonlinear Partial Differential Equation Discovery

作者：Pan, Chunjian[1]; Jiang, Qingchao[2]; Yan, Xuefeng[2]

机构：[1] College of Automation Engineering, Shanghai University of Electric Power, Shanghai, 200090, China; [2] Key Laboratory of Smart Manufacturing in Energy Chemical Process, East China University of Science and Technology, Shanghai, 200237, China

年份：2023

外文期刊名：SSRN

收录：EI(收录号：20230403227)

语种：英文

外文关键词：Inverse problems - Learning algorithms - Machine learning - Nonlinear equations - Partial differential equations - Singular value decomposition

摘要：Machine learning algorithms have shown promising potentials in solving large-scale inverse problems for modeling modern complex systems. However, conventionally used l1-norm regulation method introduces bias on the discovered models, as it sacrifices some variance in order to achieve sparsity, which could disregard some true physics. Singular value decomposition gives the optimal low-rank representation of a data matrix if the noise is i.i.d. Gaussian. In this paper, using correlations in singular vector space to discover nonlinear partial differential equations (PDEs) from data, hence preventing such a bias, is proposed. An observable matrix is constructed using basis functions to lift the original data space into a higher dimensional space, where linear correlations among the basis functions are discovered by the coefficients of the singular vectors corresponding to zero singular values. The parsimony of the discovered model is achieved by applying Gauss-Jordan elimination to the discovered null subspace. The performance of the proposed approach is tested on some canonical nonlinear PDE problems, including Burger’s equation, the Korteweg-de Vries (KdV) equation, the Kuramoto-Sivashinsky (KS) equation and a two-dimensional Allen-Cahn equation. It was observed that only a small number of local spatial data is required to discover PDEs of constant parameters. ? 2023, The Authors. All rights reserved.