{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,17]],"date-time":"2026-04-17T16:06:07Z","timestamp":1776441967015,"version":"3.51.2"},"reference-count":27,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2023,4,3]],"date-time":"2023-04-03T00:00:00Z","timestamp":1680480000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Science Foundation","award":["DMS-1555072"],"award-info":[{"award-number":["DMS-1555072"]}]},{"name":"National Science Foundation","award":["DMS-2053746"],"award-info":[{"award-number":["DMS-2053746"]}]},{"name":"National Science Foundation","award":["DMS-2134209"],"award-info":[{"award-number":["DMS-2134209"]}]},{"name":"National Science Foundation","award":["382247"],"award-info":[{"award-number":["382247"]}]},{"name":"National Science Foundation","award":["DE-SC0021142"],"award-info":[{"award-number":["DE-SC0021142"]}]},{"name":"National Science Foundation","award":["DE-SC0023161"],"award-info":[{"award-number":["DE-SC0023161"]}]},{"DOI":"10.13039\/100006231","name":"Brookhaven National Laboratory","doi-asserted-by":"publisher","award":["DMS-1555072"],"award-info":[{"award-number":["DMS-1555072"]}],"id":[{"id":"10.13039\/100006231","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100006231","name":"Brookhaven National Laboratory","doi-asserted-by":"publisher","award":["DMS-2053746"],"award-info":[{"award-number":["DMS-2053746"]}],"id":[{"id":"10.13039\/100006231","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100006231","name":"Brookhaven National Laboratory","doi-asserted-by":"publisher","award":["DMS-2134209"],"award-info":[{"award-number":["DMS-2134209"]}],"id":[{"id":"10.13039\/100006231","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100006231","name":"Brookhaven National Laboratory","doi-asserted-by":"publisher","award":["382247"],"award-info":[{"award-number":["382247"]}],"id":[{"id":"10.13039\/100006231","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100006231","name":"Brookhaven National Laboratory","doi-asserted-by":"publisher","award":["DE-SC0021142"],"award-info":[{"award-number":["DE-SC0021142"]}],"id":[{"id":"10.13039\/100006231","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100006231","name":"Brookhaven National Laboratory","doi-asserted-by":"publisher","award":["DE-SC0023161"],"award-info":[{"award-number":["DE-SC0023161"]}],"id":[{"id":"10.13039\/100006231","id-type":"DOI","asserted-by":"publisher"}]},{"name":"U.S. Department of Energy","award":["DMS-1555072"],"award-info":[{"award-number":["DMS-1555072"]}]},{"name":"U.S. Department of Energy","award":["DMS-2053746"],"award-info":[{"award-number":["DMS-2053746"]}]},{"name":"U.S. Department of Energy","award":["DMS-2134209"],"award-info":[{"award-number":["DMS-2134209"]}]},{"name":"U.S. Department of Energy","award":["382247"],"award-info":[{"award-number":["382247"]}]},{"name":"U.S. Department of Energy","award":["DE-SC0021142"],"award-info":[{"award-number":["DE-SC0021142"]}]},{"name":"U.S. Department of Energy","award":["DE-SC0023161"],"award-info":[{"award-number":["DE-SC0023161"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>This paper presents NSGA-PINN, a multi-objective optimization framework for the effective training of physics-informed neural networks (PINNs). The proposed framework uses the non-dominated sorting genetic algorithm (NSGA-II) to enable traditional stochastic gradient optimization algorithms (e.g., ADAM) to escape local minima effectively. Additionally, the NSGA-II algorithm enables satisfying the initial and boundary conditions encoded into the loss function during physics-informed training precisely. We demonstrate the effectiveness of our framework by applying NSGA-PINN to several ordinary and partial differential equation problems. In particular, we show that the proposed framework can handle challenging inverse problems with noisy data.<\/jats:p>","DOI":"10.3390\/a16040194","type":"journal-article","created":{"date-parts":[[2023,4,3]],"date-time":"2023-04-03T02:32:27Z","timestamp":1680489147000},"page":"194","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":22,"title":["NSGA-PINN: A Multi-Objective Optimization Method for Physics-Informed Neural Network Training"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0009-0001-6001-6632","authenticated-orcid":false,"given":"Binghang","family":"Lu","sequence":"first","affiliation":[{"name":"Department of Computer Science, Purdue University, West Lafayette, IN 47906, USA"}]},{"given":"Christian","family":"Moya","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Purdue University, West Lafayette, IN 47906, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0976-1987","authenticated-orcid":false,"given":"Guang","family":"Lin","sequence":"additional","affiliation":[{"name":"Department of Mathematics and School of Mechanical Engineering, Purdue University, West Lafayette, IN 47906, USA"}]}],"member":"1968","published-online":{"date-parts":[[2023,4,3]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1026","DOI":"10.1126\/science.aaw4741","article-title":"Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizations","volume":"367","author":"Raissi","year":"2020","journal-title":"Science"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"422","DOI":"10.1038\/s42254-021-00314-5","article-title":"Physics-informed machine learning","volume":"3","author":"Karniadakis","year":"2021","journal-title":"Nat. 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