{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,15]],"date-time":"2026-10-15T23:21:37Z","timestamp":1792106497795,"version":"4.4.0"},"reference-count":27,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","funder":[{"DOI":"10.13039\/100000001","name":"NSF of China","doi-asserted-by":"publisher","award":["12171085"],"award-info":[{"award-number":["12171085"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012166","name":"National Key R&D Program of China","doi-asserted-by":"crossref","award":["2020YFA0712000"],"award-info":[{"award-number":["2020YFA0712000"]}],"id":[{"id":"10.13039\/501100012166","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/100000001","name":"NSF of China","doi-asserted-by":"publisher","award":["12288201"],"award-info":[{"award-number":["12288201"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Strategic Priority Research Program of the Chinese Academy of Sciences","award":["XDA25000404"],"award-info":[{"award-number":["XDA25000404"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2023,8,31]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>Physics-informed neural networks (PINNs) have emerged as an effective technique for solving PDEs in a wide range of domains. It is noticed, however, that the performance of PINNs can vary dramatically with different sampling procedures. For instance, a fixed set of (prior chosen) training points may fail to capture the effective solution region (especially for problems with singularities). To overcome this issue, we present in this work an adaptive strategy, termed failure-informed PINNs (FI-PINNs), which is inspired by the viewpoint of reliability analysis. The key idea is to define an effective failure probability based on the residual, and then, with the aim of placing more samples in the failure region, the FI-PINNs employs a failure-informed enrichment technique to adaptively add new collocation points to the training set, such that the numerical accuracy is dramatically improved. In short, similar to adaptive finite element methods, the proposed FI-PINNs adopt the failure probability as the posterior error indicator to generate new training points. We prove rigorous error bounds of FI-PINNs and illustrate their performance through several problems.<\/jats:p>","DOI":"10.1137\/22m1527763","type":"journal-article","created":{"date-parts":[[2023,7,27]],"date-time":"2023-07-27T05:30:53Z","timestamp":1690435853000},"page":"A1971-A1994","source":"Crossref","is-referenced-by-count":178,"title":["Failure-Informed Adaptive Sampling for PINNs"],"prefix":"10.1137","volume":"45","author":[{"given":"Zhiwei","family":"Gao","sequence":"first","affiliation":[{"name":"School of Mathematics, Southeast University, Nanjing 210096, China."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4515-9125","authenticated-orcid":true,"given":"Liang","family":"Yan","sequence":"additional","affiliation":[{"name":"School of Mathematics, Southeast University, Nanjing 210096, and Nanjing Center for Applied Mathematics, Nanjing 211135, China."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Tao","family":"Zhou","sequence":"additional","affiliation":[{"name":"LSEC, Institute of Computational Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2023,7,27]]},"reference":[{"key":"ref1","series-title":"Inform. 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Yan , and \nT. Zhou , Failure-Informed Adaptive Sampling for PINNs, Part II: Combining with Re-sampling and Subset Simulation, https:\/\/arxiv.org\/abs\/2302.01529, 2023.","DOI":"10.1007\/s42967-023-00312-7"},{"key":"ref9","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2022.115523"},{"key":"ref10","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2022.111202"},{"key":"ref11","doi-asserted-by":"publisher","DOI":"10.4208\/cicp.OA-2021-0176"},{"key":"ref12","doi-asserted-by":"crossref","unstructured":"A. D. Jagtap  and \nG. E. Karniadakis , Extended physics-informed neural networks (xPINNs): A generalized space-time domain decomposition based deep learning framework for nonlinear partial differential equations, in Proceedings of AAAI-MLPS, 2021.","DOI":"10.4208\/cicp.OA-2020-0164"},{"key":"ref13","doi-asserted-by":"publisher","DOI":"10.1038\/s42254-021-00314-5"},{"key":"ref14","unstructured":"D. P. Kingma  and \nJ. Ba , Adam: A Method for Stochastic Optimization, preprint, https:\/\/arxiv.org\/abs\/1412.6980, 2014."},{"key":"ref15","first-page":"26548","volume":"34","author":"Krishnapriyan A.","year":"2021","journal-title":"Adv. Neur. Inf. Process. Syst."},{"key":"ref16","doi-asserted-by":"publisher","DOI":"10.1137\/19M1274067"},{"key":"ref17","unstructured":"L. McClenny  and \nU. Braga-Neto , Self-Adaptive Physics-Informed Neural Networks Using a Soft Attention Mechanism, preprint, https:\/\/arxiv.org\/abs\/2009.04544, 2020."},{"key":"ref18","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2018.10.045"},{"key":"ref19","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2018.08.029"},{"key":"ref20","doi-asserted-by":"crossref","unstructured":"S. Subramanian , \nR. M. Kirby , \nM. W. Mahoney , and \nA. Gholami , Adaptive Self-Supervision Algorithms for Physics-Informed Neural Networks, preprint, https:\/\/arxiv.org\/abs\/2207.04084, 2022.","DOI":"10.3233\/FAIA230521"},{"key":"ref21","unstructured":"K. Tang , \nX. Wan , and \nC. Yang , DAS: A Deep Adaptive Sampling Method for Solving Partial Differential Equations, preprint, https:\/\/arxiv.org\/abs\/2112.14038, 2021."},{"key":"ref22","unstructured":"S. Wang , \nS. Sankaran , and \nP. Perdikaris , Respecting Causality Is All You Need for Training Physics-Informed Neural Networks, preprint, https:\/\/arxiv.org\/abs\/2203.07404, 2022."},{"key":"ref23","doi-asserted-by":"publisher","DOI":"10.1137\/20M1318043"},{"key":"ref24","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2021.110768"},{"key":"ref25","doi-asserted-by":"crossref","unstructured":"C. Wu , \nM. Zhu , \nQ. Tan , \nY. Kartha , and \nL. Lu , A Comprehensive Study of Non-adaptive and Residual-Based Adaptive Sampling for Physics-Informed Neural Networks, preprint, https:\/\/arxiv.org\/abs\/2207.10289, 2022.","DOI":"10.1016\/j.cma.2022.115671"},{"key":"ref26","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2022.114823"},{"key":"ref27","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2020.109409"}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/22M1527763","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T15:57:18Z","timestamp":1787327838000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/22M1527763"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,7,27]]},"references-count":27,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2023,8,31]]}},"alternative-id":["10.1137\/22M1527763"],"URL":"https:\/\/doi.org\/10.1137\/22m1527763","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,7,27]]}}}