{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,31]],"date-time":"2026-05-31T11:00:36Z","timestamp":1780225236430,"version":"3.54.0"},"reference-count":63,"publisher":"Elsevier BV","license":[{"start":{"date-parts":[[2026,10,1]],"date-time":"2026-10-01T00:00:00Z","timestamp":1790812800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/tdm\/userlicense\/1.0\/"},{"start":{"date-parts":[[2026,10,1]],"date-time":"2026-10-01T00:00:00Z","timestamp":1790812800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/legal\/tdmrep-license"},{"start":{"date-parts":[[2026,10,1]],"date-time":"2026-10-01T00:00:00Z","timestamp":1790812800000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-017"},{"start":{"date-parts":[[2026,10,1]],"date-time":"2026-10-01T00:00:00Z","timestamp":1790812800000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-037"},{"start":{"date-parts":[[2026,10,1]],"date-time":"2026-10-01T00:00:00Z","timestamp":1790812800000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-012"},{"start":{"date-parts":[[2026,10,1]],"date-time":"2026-10-01T00:00:00Z","timestamp":1790812800000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-029"},{"start":{"date-parts":[[2026,10,1]],"date-time":"2026-10-01T00:00:00Z","timestamp":1790812800000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-004"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["elsevier.com","sciencedirect.com"],"crossmark-restriction":true},"short-container-title":["Communications in Nonlinear Science and Numerical Simulation"],"published-print":{"date-parts":[[2026,10]]},"DOI":"10.1016\/j.cnsns.2026.110060","type":"journal-article","created":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T23:39:19Z","timestamp":1776814759000},"page":"110060","update-policy":"https:\/\/doi.org\/10.1016\/elsevier_cm_policy","source":"Crossref","is-referenced-by-count":0,"special_numbering":"P1","title":["Subspace method based on neural networks for solving eigenvalue problems"],"prefix":"10.1016","volume":"161","author":[{"given":"Xiaoying","family":"Dai","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yunying","family":"Fan","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zhiqiang","family":"Sheng","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"78","reference":[{"issue":"2","key":"10.1016\/j.cnsns.2026.110060_bib0001","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1137\/1026033","article-title":"Eigenvalues of the Laplacian in two dimensions","volume":"26","author":"Kuttler","year":"1984","journal-title":"SIAM Rev"},{"issue":"1","key":"10.1016\/j.cnsns.2026.110060_bib0002","doi-asserted-by":"crossref","first-page":"208","DOI":"10.1137\/090757046","article-title":"Finite volume discretizations for eigenvalue problems with applications to electronic structure calculations","volume":"9","author":"Dai","year":"2011","journal-title":"Multiscale Model Simul"},{"key":"10.1016\/j.cnsns.2026.110060_bib0003","doi-asserted-by":"crossref","first-page":"345","DOI":"10.1023\/A:1021946607960","article-title":"Finite volume methods for eigenvalue problems","volume":"41","author":"Liang","year":"2001","journal-title":"BIT"},{"key":"10.1016\/j.cnsns.2026.110060_bib0004","series-title":"Handbook of numerical analysis","doi-asserted-by":"crossref","first-page":"641","DOI":"10.1016\/S1570-8659(05)80042-0","article-title":"Eigenvalue problems","volume":"vol. 2","author":"Babu\u0161ka","year":"1991"},{"key":"10.1016\/j.cnsns.2026.110060_bib0005","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1017\/S0962492910000012","article-title":"Finite element approximation of eigenvalue problems","volume":"19","author":"Boffi","year":"2010","journal-title":"Acta Numer"},{"key":"10.1016\/j.cnsns.2026.110060_bib0006","series-title":"Finite element methods for eigenvalue problems","author":"Sun","year":"2016"},{"issue":"3","key":"10.1016\/j.cnsns.2026.110060_bib0007","doi-asserted-by":"crossref","first-page":"412","DOI":"10.1016\/0021-9991(82)90091-2","article-title":"Solution of the Schr\u00f6dinger equation by a spectral method","volume":"47","author":"Feit","year":"1982","journal-title":"J Comput Phys"},{"key":"10.1016\/j.cnsns.2026.110060_bib0008","doi-asserted-by":"crossref","first-page":"686","DOI":"10.1016\/j.jcp.2018.10.045","article-title":"Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations","volume":"378","author":"Raissi","year":"2019","journal-title":"J Comput Phys"},{"key":"10.1016\/j.cnsns.2026.110060_bib0009","doi-asserted-by":"crossref","first-page":"1339","DOI":"10.1016\/j.jcp.2018.08.029","article-title":"DGM: a deep learning algorithm for solving partial differential equations","volume":"375","author":"Sirignano","year":"2018","journal-title":"J Comput Phys"},{"issue":"1","key":"10.1016\/j.cnsns.2026.110060_bib0010","first-page":"1","article-title":"The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems","volume":"6","author":"Yu","year":"2018","journal-title":"Commun Math Stat"},{"key":"10.1016\/j.cnsns.2026.110060_bib0011","doi-asserted-by":"crossref","DOI":"10.1016\/j.jcp.2020.109409","article-title":"Weak adversarial networks for high-dimensional partial differential equations","volume":"411","author":"Zang","year":"2020","journal-title":"J Comput Phys"},{"key":"10.1016\/j.cnsns.2026.110060_bib0012","doi-asserted-by":"crossref","DOI":"10.1016\/j.cma.2023.116666","article-title":"Multi-level neural networks for accurate solutions of boundary-value problems","volume":"419","author":"Aldirany","year":"2024","journal-title":"Comput Methods Appl Mech Eng"},{"key":"10.1016\/j.cnsns.2026.110060_bib0013","doi-asserted-by":"crossref","first-page":"327","DOI":"10.1017\/S0962492921000052","article-title":"Neural network approximation","volume":"30","author":"DeVore","year":"2021","journal-title":"Acta Numer"},{"issue":"3","key":"10.1016\/j.cnsns.2026.110060_bib0014","doi-asserted-by":"crossref","first-page":"502","DOI":"10.4208\/jcm.1901-m2018-0160","article-title":"ReLU deep neural networks and linear finite elements","volume":"38","author":"He","year":"2020","journal-title":"J Comput Math"},{"key":"10.1016\/j.cnsns.2026.110060_bib0015","unstructured":"He J., Xu J.. Deep neural networks and finite elements of any order on arbitrary dimensions. 2023, arXiv: 231214276."},{"issue":"6","key":"10.1016\/j.cnsns.2026.110060_bib0016","doi-asserted-by":"crossref","first-page":"861","DOI":"10.1016\/S0893-6080(05)80131-5","article-title":"Multilayer feedforward networks with a nonpolynomial activation function can approximate any function","volume":"6","author":"Leshno","year":"1993","journal-title":"Neural Netw"},{"issue":"5","key":"10.1016\/j.cnsns.2026.110060_bib0017","doi-asserted-by":"crossref","first-page":"5465","DOI":"10.1137\/20M134695X","article-title":"Deep network approximation for smooth functions","volume":"53","author":"Lu","year":"2021","journal-title":"SIAM J Math Anal"},{"issue":"1","key":"10.1016\/j.cnsns.2026.110060_bib0018","doi-asserted-by":"crossref","first-page":"369","DOI":"10.1007\/s00365-021-09549-y","article-title":"The barron space and the flow-induced function spaces for neural network models","volume":"55","author":"Ma","year":"2022","journal-title":"Constr Approx"},{"key":"10.1016\/j.cnsns.2026.110060_bib0019","unstructured":"Shen Z., Yang H., Zhang S.. Deep network approximation characterized by number of neurons. 2019, arXiv: 190605497."},{"key":"10.1016\/j.cnsns.2026.110060_bib0020","doi-asserted-by":"crossref","first-page":"313","DOI":"10.1016\/j.neunet.2020.05.019","article-title":"Approximation rates for neural networks with general activation functions","volume":"128","author":"Siegel","year":"2020","journal-title":"Neural Netw"},{"issue":"2","key":"10.1016\/j.cnsns.2026.110060_bib0021","doi-asserted-by":"crossref","first-page":"481","DOI":"10.1007\/s10208-022-09595-3","article-title":"Sharp bounds on the approximation rates, metric entropy, and n-widths of shallow neural networks","volume":"24","author":"Siegel","year":"2024","journal-title":"Found Comput Math"},{"key":"10.1016\/j.cnsns.2026.110060_bib0022","doi-asserted-by":"crossref","DOI":"10.1016\/j.jcp.2019.108929","article-title":"Solving many-electron Schr\u00f6dinger equation using deep neural networks","volume":"399","author":"Han","year":"2019","journal-title":"J Comput Phys"},{"issue":"10","key":"10.1016\/j.cnsns.2026.110060_bib0023","doi-asserted-by":"crossref","first-page":"891","DOI":"10.1038\/s41557-020-0544-y","article-title":"Deep-neural-network solution of the electronic Schr\u00f6dinger equation","volume":"12","author":"Hermann","year":"2020","journal-title":"Nat Chem"},{"issue":"10","key":"10.1016\/j.cnsns.2026.110060_bib0024","doi-asserted-by":"crossref","first-page":"692","DOI":"10.1038\/s41570-023-00516-8","article-title":"Ab initio quantum chemistry with neural-network wavefunctions","volume":"7","author":"Hermann","year":"2023","journal-title":"Nat Rev Chem"},{"key":"10.1016\/j.cnsns.2026.110060_bib0025","article-title":"Neural-network-based multistate solver for a static Schr\u00f6dinger equation","volume":"103","author":"Li","year":"2021","journal-title":"Phys Rev A"},{"key":"10.1016\/j.cnsns.2026.110060_bib0026","doi-asserted-by":"crossref","DOI":"10.1016\/j.jcp.2020.109792","article-title":"Solving high-dimensional eigenvalue problems using deep neural networks: a diffusion Monte Carlo like approach","volume":"423","author":"Han","year":"2020","journal-title":"J Comput Phys"},{"key":"10.1016\/j.cnsns.2026.110060_bib0027","doi-asserted-by":"crossref","DOI":"10.1016\/j.jcp.2022.110939","article-title":"A semigroup method for high dimensional elliptic PDEs and eigenvalue problems based on neural networks","volume":"453","author":"Li","year":"2022","journal-title":"J Comput Phys"},{"key":"10.1016\/j.cnsns.2026.110060_bib0028","unstructured":"Li Y., Lin Z., Wang Y., Xie H.. Tensor neural network interpolation and its applications. 2024, arXiv: 240407805."},{"key":"10.1016\/j.cnsns.2026.110060_bib0029","doi-asserted-by":"crossref","unstructured":"Wang Y., Lin Z., Liao Y., Liu H., Xie H.. Solving high dimensional partial differential equations using tensor neural network and a posteriori error estimators. 2024, arXiv: 231102732.","DOI":"10.1007\/s10915-024-02700-4"},{"key":"10.1016\/j.cnsns.2026.110060_bib0030","unstructured":"Wang Y., Liao Y., Xie H.. Solving Schr\u00f6dinger equation using tensor neural network. 2022, arXiv: 220912572."},{"key":"10.1016\/j.cnsns.2026.110060_bib0031","doi-asserted-by":"crossref","DOI":"10.1016\/j.jcp.2024.112928","article-title":"Computing multi-eigenpairs of high-dimensional eigenvalue problems using tensor neural networks","volume":"506","author":"Wang","year":"2024","journal-title":"J Comput Phys"},{"key":"10.1016\/j.cnsns.2026.110060_bib0032","unstructured":"Yu H., Guo Y., Ming P.. Generalization error estimates of machine learning methods for solving high dimensional Schr\u00f6dinger eigenvalue problems. 2024, arXiv: 240813511."},{"issue":"1\u20133","key":"10.1016\/j.cnsns.2026.110060_bib0033","doi-asserted-by":"crossref","first-page":"489","DOI":"10.1016\/j.neucom.2005.12.126","article-title":"Extreme learning machine: theory and applications","volume":"70","author":"Huang","year":"2006","journal-title":"Neurocomputing"},{"issue":"3","key":"10.1016\/j.cnsns.2026.110060_bib0034","doi-asserted-by":"crossref","first-page":"268","DOI":"10.4208\/jml.220726","article-title":"Bridging traditional and machine learning-based algorithms for solving PDEs: the random feature method","volume":"1","author":"Chen","year":"2022","journal-title":"J Mach Learn"},{"key":"10.1016\/j.cnsns.2026.110060_bib0035","doi-asserted-by":"crossref","DOI":"10.1016\/j.cma.2023.116719","article-title":"The random feature method for solving interface problems","volume":"420","author":"Chi","year":"2024","journal-title":"Comput Methods Appl Mech Eng"},{"key":"10.1016\/j.cnsns.2026.110060_bib0036","doi-asserted-by":"crossref","DOI":"10.1016\/j.cma.2021.114129","article-title":"Local extreme learning machines and domain decomposition for solving linear and nonlinear partial differential equations","volume":"387","author":"Dong","year":"2021","journal-title":"Comput Methods Appl Mech Engrg"},{"key":"10.1016\/j.cnsns.2026.110060_bib0037","doi-asserted-by":"crossref","DOI":"10.1016\/j.cnsns.2023.107518","article-title":"Randomized neural network with Petrov-Galerkin methods for solving linear and nonlinear partial differential equations","volume":"127","author":"Shang","year":"2023","journal-title":"Commun Nonlinear Sci Numer Simul"},{"key":"10.1016\/j.cnsns.2026.110060_bib0038","doi-asserted-by":"crossref","DOI":"10.1016\/j.cam.2024.115830","article-title":"Local randomized neural networks with discontinuous Galerkin methods for partial differential equations","volume":"445","author":"Sun","year":"2024","journal-title":"J Comput Appl Math"},{"issue":"6","key":"10.1016\/j.cnsns.2026.110060_bib0039","doi-asserted-by":"crossref","first-page":"1281","DOI":"10.4208\/jcm.2205-m2021-0277","article-title":"A deep learning based discontinuous Galerkin method for hyperbolic equations with discontinuous solutions and random uncertainties","volume":"41","author":"Chen","year":"2023","journal-title":"J Comput Math"},{"key":"10.1016\/j.cnsns.2026.110060_bib0040","doi-asserted-by":"crossref","unstructured":"Li Y., Wang F.. Local randomized neural networks methods for interface problems. 2023, arXiv: 230803087.","DOI":"10.2139\/ssrn.4608002"},{"key":"10.1016\/j.cnsns.2026.110060_bib0041","doi-asserted-by":"crossref","DOI":"10.1016\/j.jcp.2023.112452","article-title":"Solving multiscale elliptic problems by sparse radial basis function neural networks","volume":"492","author":"Wang","year":"2023","journal-title":"J Comput Phys"},{"key":"10.1016\/j.cnsns.2026.110060_bib0042","doi-asserted-by":"crossref","DOI":"10.1016\/j.jcp.2021.110585","article-title":"A modified batch intrinsic plasticity method for pre-training the random coefficients of extreme learning machines","volume":"445","author":"Dong","year":"2021","journal-title":"J Comput Phys"},{"key":"10.1016\/j.cnsns.2026.110060_bib0043","doi-asserted-by":"crossref","DOI":"10.1016\/j.jcp.2022.111290","article-title":"On computing the hyperparameter of extreme learning machines: algorithm and application to computational PDEs, and comparison with classical and high-order finite elements","volume":"463","author":"Dong","year":"2022","journal-title":"J Comput Phys"},{"issue":"1","key":"10.1016\/j.cnsns.2026.110060_bib0044","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s10915-024-02463-y","article-title":"Transferable neural networks for partial differential equations","volume":"99","author":"Zhang","year":"2024","journal-title":"J Sci Comput"},{"issue":"4","key":"10.1016\/j.cnsns.2026.110060_bib0045","doi-asserted-by":"crossref","first-page":"A2474","DOI":"10.1137\/20M1366587","article-title":"Galerkin neural networks: a framework for approximating variational equations with error control","volume":"43","author":"Ainsworth","year":"2021","journal-title":"SIAM J Sci Comput"},{"key":"10.1016\/j.cnsns.2026.110060_bib0046","series-title":"Mathematical and scientific machine learning","first-page":"512","article-title":"Robust training and initialization of deep neural networks: an adaptive basis viewpoint","author":"Cyr","year":"2020"},{"key":"10.1016\/j.cnsns.2026.110060_bib0047","unstructured":"Xu Z., Sheng Z.. Subspace method based on neural networks for solving the partial differential equation. 2024, arXiv: 240408223."},{"key":"10.1016\/j.cnsns.2026.110060_bib0048","unstructured":"Liu P., Xu Z., Sheng Z.. Subspace method based on neural networks for solving the partial differential equation in weak form. 2024, arXiv: 240508513."},{"key":"10.1016\/j.cnsns.2026.110060_bib0049","series-title":"Sobolev spaces","author":"Adams","year":"2003"},{"issue":"4","key":"10.1016\/j.cnsns.2026.110060_bib0050","doi-asserted-by":"crossref","first-page":"581","DOI":"10.1137\/1034116","article-title":"Iterative methods by space decomposition and subspace correction","volume":"34","author":"Xu","year":"1992","journal-title":"SIAM Rev"},{"issue":"186","key":"10.1016\/j.cnsns.2026.110060_bib0051","doi-asserted-by":"crossref","first-page":"275","DOI":"10.1090\/S0025-5718-1989-0962210-8","article-title":"Finite element-Galerkin approximation of the eigenvalues and eigenvectors of selfadjoint problems","volume":"52","author":"Babu\u0161ka","year":"1989","journal-title":"Math Comp"},{"issue":"3","key":"10.1016\/j.cnsns.2026.110060_bib0052","doi-asserted-by":"crossref","first-page":"313","DOI":"10.1007\/s00211-008-0169-3","article-title":"Convergence and optimal complexity of adaptive finite element eigenvalue computations","volume":"110","author":"Dai","year":"2008","journal-title":"Numer Math"},{"key":"10.1016\/j.cnsns.2026.110060_bib0053","unstructured":"Kingma D.P.. Adam: a method for stochastic optimization. 2014, arXiv: 14126980."},{"key":"10.1016\/j.cnsns.2026.110060_bib0054","series-title":"Model order reduction: theory, research aspects and applications","first-page":"95","article-title":"Model reduction via proper orthogonal decomposition","author":"Pinnau","year":"2008"},{"key":"10.1016\/j.cnsns.2026.110060_bib0055","series-title":"Model reduction using proper orthogonal decomposition","author":"Volkwein","year":"2011"},{"key":"10.1016\/j.cnsns.2026.110060_bib0056","series-title":"Templates for the solution of algebraic eigenvalue problems: a practical guide","author":"Bai","year":"2000"},{"key":"10.1016\/j.cnsns.2026.110060_bib0057","series-title":"Numerical methods for large eigenvalue problems: revised edition","author":"Saad","year":"2011"},{"issue":"1","key":"10.1016\/j.cnsns.2026.110060_bib0058","doi-asserted-by":"crossref","first-page":"64","DOI":"10.1007\/BF01017247","article-title":"Difference analogs of a harmonic oscillator","volume":"85","author":"Atakishiev","year":"1990","journal-title":"Theor Math Phys"},{"issue":"2","key":"10.1016\/j.cnsns.2026.110060_bib0059","doi-asserted-by":"crossref","first-page":"1490","DOI":"10.1007\/s42967-024-00389-8","article-title":"Optimization of random feature method in the high-precision regime","volume":"6","author":"Chen","year":"2024","journal-title":"Commun Appl Math Comput"},{"key":"10.1016\/j.cnsns.2026.110060_bib0060","series-title":"Advances in neural information processing systems","article-title":"Fast randomized kernel ridge regression with statistical guarantees","volume":"vol. 28","author":"Alaoui","year":"2015"},{"key":"10.1016\/j.cnsns.2026.110060_bib0061","series-title":"Conference on learning theory","first-page":"185","article-title":"Sharp analysis of low-rank kernel matrix approximations","author":"Bach","year":"2013"},{"key":"10.1016\/j.cnsns.2026.110060_bib0062","series-title":"Advances in neural information processing systems","first-page":"1657","article-title":"Less is more: Nystr\u00f6m computational regularization","volume":"vol. 1","author":"Rudi","year":"2015"},{"key":"10.1016\/j.cnsns.2026.110060_bib0063","series-title":"Advances in neural information processing systems","first-page":"682","article-title":"Using the Nystr\u00f6m method to speed up kernel machines","volume":"vol. 13","author":"Williams","year":"2000"}],"container-title":["Communications in Nonlinear Science and Numerical Simulation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S1007570426004193?httpAccept=text\/xml","content-type":"text\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S1007570426004193?httpAccept=text\/plain","content-type":"text\/plain","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2026,5,31]],"date-time":"2026-05-31T10:03:40Z","timestamp":1780221820000},"score":1,"resource":{"primary":{"URL":"https:\/\/linkinghub.elsevier.com\/retrieve\/pii\/S1007570426004193"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,10]]},"references-count":63,"alternative-id":["S1007570426004193"],"URL":"https:\/\/doi.org\/10.1016\/j.cnsns.2026.110060","relation":{},"ISSN":["1007-5704"],"issn-type":[{"value":"1007-5704","type":"print"}],"subject":[],"published":{"date-parts":[[2026,10]]},"assertion":[{"value":"Elsevier","name":"publisher","label":"This article is maintained by"},{"value":"Subspace method based on neural networks for solving eigenvalue problems","name":"articletitle","label":"Article Title"},{"value":"Communications in Nonlinear Science and Numerical Simulation","name":"journaltitle","label":"Journal Title"},{"value":"https:\/\/doi.org\/10.1016\/j.cnsns.2026.110060","name":"articlelink","label":"CrossRef DOI link to publisher maintained version"},{"value":"article","name":"content_type","label":"Content Type"},{"value":"\u00a9 2026 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.","name":"copyright","label":"Copyright"}],"article-number":"110060"}}