{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T08:28:45Z","timestamp":1760171325601,"version":"3.37.3"},"reference-count":70,"publisher":"IOP Publishing","issue":"3","license":[{"start":{"date-parts":[[2024,8,14]],"date-time":"2024-08-14T00:00:00Z","timestamp":1723593600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"},{"start":{"date-parts":[[2024,8,14]],"date-time":"2024-08-14T00:00:00Z","timestamp":1723593600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/iopscience.iop.org\/info\/page\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100001314","name":"Durham University","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100001314","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/100010665","name":"H2020 Marie Sk\u0142odowska-Curie Actions","doi-asserted-by":"crossref","award":["872172"],"award-info":[{"award-number":["872172"]}],"id":[{"id":"10.13039\/100010665","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["iopscience.iop.org"],"crossmark-restriction":false},"short-container-title":["Mach. Learn.: Sci. Technol."],"published-print":{"date-parts":[[2024,9,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>Identifying partial differential equations (PDEs) from data is crucial for understanding the governing mechanisms of natural phenomena, yet it remains a challenging task. We present an extension to the ARGOS framework, ARGOS-RAL, which leverages sparse regression with the recurrent adaptive lasso to identify PDEs from limited prior knowledge automatically. Our method automates calculating partial derivatives, constructing a candidate library, and estimating a sparse model. We rigorously evaluate the performance of ARGOS-RAL in identifying canonical PDEs under various noise levels and sample sizes, demonstrating its robustness in handling noisy and non-uniformly distributed data. We also test the algorithm\u2019s performance on datasets consisting solely of random noise to simulate scenarios with severely compromised data quality. Our results show that ARGOS-RAL effectively and reliably identifies the underlying PDEs from data, outperforming the sequential threshold ridge regression method in most cases. We highlight the potential of combining statistical methods, machine learning, and dynamical systems theory to automatically discover governing equations from collected data, streamlining the scientific modeling process.<\/jats:p>","DOI":"10.1088\/2632-2153\/ad682f","type":"journal-article","created":{"date-parts":[[2024,7,26]],"date-time":"2024-07-26T22:57:20Z","timestamp":1722034640000},"page":"035046","update-policy":"https:\/\/doi.org\/10.1088\/crossmark-policy","source":"Crossref","is-referenced-by-count":2,"title":["Automating the discovery of partial differential equations in dynamical systems"],"prefix":"10.1088","volume":"5","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9368-236X","authenticated-orcid":true,"given":"Weizhen","family":"Li","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3279-4218","authenticated-orcid":true,"given":"Rui","family":"Carvalho","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"266","published-online":{"date-parts":[[2024,8,14]]},"reference":[{"key":"mlstad682fbib1","doi-asserted-by":"publisher","first-page":"81","DOI":"10.1126\/science.1165893","volume":"324","author":"Schmidt","year":"2009","journal-title":"Science"},{"key":"mlstad682fbib2","doi-asserted-by":"publisher","first-page":"1009","DOI":"10.1080\/01621459.2013.794730","volume":"108","author":"Xun","year":"2013","journal-title":"J. Am. Stat. Assoc."},{"key":"mlstad682fbib3","doi-asserted-by":"publisher","first-page":"3932","DOI":"10.1073\/pnas.1517384113","volume":"113","author":"Brunton","year":"2016","journal-title":"Proc. Natl Acad. Sci."},{"key":"mlstad682fbib4","doi-asserted-by":"publisher","DOI":"10.1098\/rspa.2016.0446","volume":"473","author":"Schaeffer","year":"2017","journal-title":"Proc. R. Soc. A"},{"key":"mlstad682fbib5","doi-asserted-by":"publisher","DOI":"10.1126\/sciadv.1602614","volume":"3","author":"Rudy","year":"2017","journal-title":"Sci. Adv."},{"key":"mlstad682fbib6","doi-asserted-by":"publisher","first-page":"686","DOI":"10.1016\/j.jcp.2018.10.045","volume":"378","author":"Raissi","year":"2019","journal-title":"J. Comput. Phys."},{"key":"mlstad682fbib7","doi-asserted-by":"publisher","first-page":"148","DOI":"10.3390\/universe9030148","volume":"9","author":"Meng","year":"2023","journal-title":"Universe"},{"key":"mlstad682fbib8","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2019.108925","volume":"399","author":"Long","year":"2019","journal-title":"J. Comput. Phys."},{"key":"mlstad682fbib9","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2020.109985","volume":"428","author":"Both","year":"2021","journal-title":"J. Comput. Phys."},{"key":"mlstad682fbib10","doi-asserted-by":"publisher","first-page":"3279","DOI":"10.1137\/18M116798X","volume":"78","author":"Schaeffer","year":"2018","journal-title":"SIAM J. Appl. Math."},{"key":"mlstad682fbib11","doi-asserted-by":"publisher","first-page":"eaav6971","DOI":"10.1126\/sciadv.aav6971","volume":"6","author":"Guimer\u00e1","year":"2020","journal-title":"Sci. Adv."},{"key":"mlstad682fbib12","doi-asserted-by":"publisher","first-page":"206","DOI":"10.1038\/s42005-022-00987-z","volume":"5","author":"Lu","year":"2022","journal-title":"Commun. Phys."},{"key":"mlstad682fbib13","doi-asserted-by":"publisher","first-page":"eabk0644","DOI":"10.1126\/sciadv.abk0644","volume":"8","author":"Zhang","year":"2022","journal-title":"Sci. Adv."},{"key":"mlstad682fbib14","doi-asserted-by":"publisher","first-page":"1043","DOI":"10.1038\/s41467-023-36657-z","volume":"14","author":"Fajardo-Fontiveros","year":"2023","journal-title":"Nat. Commun."},{"key":"mlstad682fbib15","doi-asserted-by":"publisher","first-page":"2775","DOI":"10.1007\/s11071-021-06707-6","volume":"105","author":"Jiang","year":"2021","journal-title":"Nonlinear Dyn."},{"key":"mlstad682fbib16","doi-asserted-by":"publisher","DOI":"10.1098\/rspa.2021.0916","volume":"478","author":"Maddu","year":"2022","journal-title":"Proc. R. Soc. A"},{"key":"mlstad682fbib17","doi-asserted-by":"publisher","first-page":"2085","DOI":"10.1109\/TPWRS.2022.3189602","volume":"38","author":"Cai","year":"2023","journal-title":"IEEE Trans. Power Syst."},{"key":"mlstad682fbib18","doi-asserted-by":"publisher","DOI":"10.1016\/j.ijmecsci.2023.108905","volume":"265","author":"Sun","year":"2024","journal-title":"Int. J. Mech. Sci."},{"key":"mlstad682fbib19","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1038\/s42005-023-01516-2","volume":"7","author":"Egan","year":"2024","journal-title":"Commun. Phys."},{"key":"mlstad682fbib20","doi-asserted-by":"publisher","first-page":"28","DOI":"10.1137\/0903003","volume":"3","author":"Varah","year":"1982","journal-title":"SIAM J. Sci. Stat. Comput."},{"key":"mlstad682fbib21","first-page":"417","volume":"1","author":"Crutchfield","year":"1987","journal-title":"Complex Syst."},{"key":"mlstad682fbib22","doi-asserted-by":"publisher","first-page":"337","DOI":"10.1103\/PhysRevE.59.337","volume":"59","author":"B\u00e4r","year":"1999","journal-title":"Phys. Rev. E"},{"key":"mlstad682fbib23","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/S0167-2789(02)00546-8","volume":"171","author":"M\u00fcller","year":"2002","journal-title":"Physica D"},{"key":"mlstad682fbib24","doi-asserted-by":"publisher","first-page":"1570","DOI":"10.1198\/016214508000000797","volume":"103","author":"Liang","year":"2008","journal-title":"J. Am. Stat. Assoc."},{"key":"mlstad682fbib25","doi-asserted-by":"publisher","first-page":"2463","DOI":"10.1002\/(SICI)1097-0258(19981115)17:213.0.CO;2-A","volume":"17","author":"Wu","year":"1998","journal-title":"Stat. Med."},{"key":"mlstad682fbib26","doi-asserted-by":"publisher","first-page":"410","DOI":"10.1111\/j.0006-341X.1999.00410.x","volume":"55","author":"Wu","year":"1999","journal-title":"Biometrics"},{"key":"mlstad682fbib27","doi-asserted-by":"publisher","first-page":"2199","DOI":"10.1002\/sim.1211","volume":"21","author":"Putter","year":"2002","journal-title":"Stat. Med."},{"key":"mlstad682fbib28","doi-asserted-by":"publisher","first-page":"9943","DOI":"10.1073\/pnas.0609476104","volume":"104","author":"Bongard","year":"2007","journal-title":"Proc. Natl Acad. Sci."},{"key":"mlstad682fbib29","doi-asserted-by":"publisher","first-page":"eaay2631","DOI":"10.1126\/sciadv.aay2631","volume":"6","author":"Udrescu","year":"2020","journal-title":"Sci. Adv."},{"key":"mlstad682fbib30","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevResearch.3.033270","volume":"3","author":"Xu","year":"2021","journal-title":"Phys. Rev. Res."},{"key":"mlstad682fbib31","doi-asserted-by":"publisher","first-page":"643","DOI":"10.1137\/18M1191944","volume":"18","author":"Rudy","year":"2019","journal-title":"SIAM J. Appl. Dyn. Syst."},{"key":"mlstad682fbib32","doi-asserted-by":"publisher","DOI":"10.1098\/rspa.2020.0279","volume":"476","author":"Kaheman","year":"2020","journal-title":"Proc. R. Soc. A"},{"key":"mlstad682fbib33","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2021.110525","volume":"443","author":"Messenger","year":"2021","journal-title":"J. Comput. Phys."},{"key":"mlstad682fbib34","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2020.113620","volume":"376","author":"Cortiella","year":"2021","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"mlstad682fbib35","doi-asserted-by":"publisher","DOI":"10.1098\/rspa.2021.0904","volume":"478","author":"Fasel","year":"2022","journal-title":"Proc. R. Soc. A"},{"key":"mlstad682fbib36","doi-asserted-by":"publisher","DOI":"10.1088\/2632-2153\/ac567a","volume":"3","author":"Kaheman","year":"2022","journal-title":"Mach. Learn.: Sci. Technol."},{"key":"mlstad682fbib37","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevResearch.5.023126","volume":"5","author":"Li","year":"2023","journal-title":"Phys. Rev. Res."},{"key":"mlstad682fbib38","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2023.116096","volume":"413","author":"Wentz","year":"2023","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"mlstad682fbib39","doi-asserted-by":"publisher","first-page":"459","DOI":"10.1017\/jfm.2018.147","volume":"844","author":"Loiseau","year":"2018","journal-title":"J. Fluid Mech."},{"key":"mlstad682fbib40","doi-asserted-by":"publisher","first-page":"357","DOI":"10.1146\/annurev-fluid-010518-040547","volume":"51","author":"Duraisamy","year":"2019","journal-title":"Annu. Rev. Fluid Mech."},{"key":"mlstad682fbib41","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.100.022220","volume":"100","author":"Li","year":"2019","journal-title":"Phys. Rev. E"},{"key":"mlstad682fbib42","doi-asserted-by":"publisher","first-page":"52","DOI":"10.1109\/TMBMC.2016.2633265","volume":"2","author":"Mangan","year":"2016","journal-title":"IEEE Trans. Mol. Biol. Multi-Scale Commun."},{"key":"mlstad682fbib43","doi-asserted-by":"publisher","DOI":"10.1063\/1.5066099","volume":"150","author":"Hoffmann","year":"2019","journal-title":"J. Chem. Phys."},{"key":"mlstad682fbib44","doi-asserted-by":"publisher","DOI":"10.1098\/rspa.2019.0800","volume":"476","author":"Lagergren","year":"2020","journal-title":"Proc. R. Soc. A"},{"key":"mlstad682fbib45","doi-asserted-by":"publisher","first-page":"6136","DOI":"10.1038\/s41467-021-26434-1","volume":"12","author":"Chen","year":"2021","journal-title":"Nat. Commun."},{"key":"mlstad682fbib46","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2021.110657","volume":"446","author":"Zhang","year":"2021","journal-title":"J. Comput. Phys."},{"key":"mlstad682fbib47","doi-asserted-by":"publisher","DOI":"10.1088\/2632-2153\/acb1f0","volume":"4","author":"Thanasutives","year":"2023","journal-title":"Mach. Learn.: Sci. Technol."},{"key":"mlstad682fbib48","doi-asserted-by":"publisher","DOI":"10.1088\/2632-2153\/acffa4","volume":"4","author":"Jia","year":"2023","journal-title":"Mach. Learn.: Sci. Technol."},{"key":"mlstad682fbib49","doi-asserted-by":"publisher","first-page":"1627","DOI":"10.1021\/ac60214a047","volume":"36","author":"Savitzky","year":"1964","journal-title":"Anal. Chem."},{"key":"mlstad682fbib50","doi-asserted-by":"publisher","first-page":"196865","DOI":"10.1109\/ACCESS.2020.3034077","volume":"8","author":"Breugel","year":"2020","journal-title":"IEEE Access"},{"key":"mlstad682fbib51","doi-asserted-by":"publisher","first-page":"111","DOI":"10.1109\/MSP.2011.941097","volume":"28","author":"Schafer","year":"2011","journal-title":"IEEE Signal Process. Mag."},{"key":"mlstad682fbib52","doi-asserted-by":"publisher","first-page":"1418","DOI":"10.1198\/016214506000000735","volume":"101","author":"Zou","year":"2006","journal-title":"J. Am. Stat. Assoc."},{"key":"mlstad682fbib53","doi-asserted-by":"publisher","first-page":"1","DOI":"10.18637\/jss.v033.i01","volume":"33","author":"Friedman","year":"2010","journal-title":"J. Stat. Softw."},{"key":"mlstad682fbib54","doi-asserted-by":"publisher","first-page":"561","DOI":"10.1137\/1034115","volume":"34","author":"Hansen","year":"1992","journal-title":"SIAM Rev."},{"key":"mlstad682fbib55","doi-asserted-by":"publisher","first-page":"1095","DOI":"10.1016\/j.apm.2011.07.055","volume":"36","author":"Nasehi Tehrani","year":"2012","journal-title":"Appl. Math. Model."},{"key":"mlstad682fbib56","doi-asserted-by":"publisher","DOI":"10.1088\/2633-1357\/abad0d","volume":"1","author":"Cultrera","year":"2020","journal-title":"IOP SciNotes"},{"key":"mlstad682fbib57","doi-asserted-by":"publisher","first-page":"291","DOI":"10.1017\/CBO9780511804441","article-title":"Approximation and fitting","author":"Boyd","year":"2004"},{"key":"mlstad682fbib58","doi-asserted-by":"publisher","first-page":"7","DOI":"10.1007\/978-3-642-20192-9","article-title":"Lasso for linear models","author":"B\u00fchlmann","year":"2011"},{"key":"mlstad682fbib59","doi-asserted-by":"publisher","first-page":"267","DOI":"10.1111\/j.2517-6161.1996.tb02080.x","volume":"58","author":"Tibshirani","year":"1996","journal-title":"J. R. Stat. B"},{"key":"mlstad682fbib60","doi-asserted-by":"publisher","first-page":"937","DOI":"10.1093\/biomet\/92.4.937","volume":"92","author":"Yang","year":"2005","journal-title":"Biometrika"},{"key":"mlstad682fbib61","doi-asserted-by":"publisher","first-page":"631","DOI":"10.1890\/13-1452.1","volume":"95","author":"Aho","year":"2014","journal-title":"Ecology"},{"key":"mlstad682fbib62","first-page":"119","article-title":"Intensity transformations and spatial filtering","author":"Gonzalez","year":"2018","edition":"4th edn"},{"key":"mlstad682fbib63","doi-asserted-by":"publisher","first-page":"483","DOI":"10.1016\/j.jcp.2019.06.056","volume":"396","author":"Rudy","year":"2019","journal-title":"J. Comput. Phys."},{"key":"mlstad682fbib64","doi-asserted-by":"publisher","first-page":"2119","DOI":"10.1109\/LCOMM.2020.3001253","volume":"24","author":"Modonesi","year":"2020","journal-title":"IEEE Commun. Lett."},{"key":"mlstad682fbib65","doi-asserted-by":"publisher","first-page":"433","DOI":"10.1038\/s43588-022-00281-6","volume":"2","author":"Chen","year":"2022","journal-title":"Nat. Comput. Sci."},{"key":"mlstad682fbib66","doi-asserted-by":"publisher","first-page":"851","DOI":"10.1103\/RevModPhys.65.851","volume":"65","author":"Cross","year":"1993","journal-title":"Rev. Mod. Phys."},{"key":"mlstad682fbib67","doi-asserted-by":"publisher","first-page":"53","DOI":"10.1017\/9781009089517","article-title":"Fourier and wavelet transforms","author":"Brunton","year":"2022","edition":"2nd edn"},{"key":"mlstad682fbib68","doi-asserted-by":"publisher","first-page":"2118","DOI":"10.1016\/j.jcp.2007.03.005","volume":"225","author":"Taira","year":"2007","journal-title":"J. Comput. Phys."},{"key":"mlstad682fbib69","doi-asserted-by":"publisher","first-page":"2131","DOI":"10.1016\/j.cma.2007.08.014","volume":"197","author":"Colonius","year":"2008","journal-title":"Immersed Bound. Method Ext."},{"key":"mlstad682fbib70","doi-asserted-by":"publisher","first-page":"857","DOI":"10.1201\/b11412","article-title":"Third-order equations","author":"Polyanin","year":"2012","edition":"2nd edn"}],"container-title":["Machine Learning: Science and Technology"],"original-title":[],"link":[{"URL":"https:\/\/iopscience.iop.org\/article\/10.1088\/2632-2153\/ad682f","content-type":"text\/html","content-version":"am","intended-application":"text-mining"},{"URL":"https:\/\/iopscience.iop.org\/article\/10.1088\/2632-2153\/ad682f\/pdf","content-type":"application\/pdf","content-version":"am","intended-application":"text-mining"},{"URL":"https:\/\/iopscience.iop.org\/article\/10.1088\/2632-2153\/ad682f","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/iopscience.iop.org\/article\/10.1088\/2632-2153\/ad682f\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/iopscience.iop.org\/article\/10.1088\/2632-2153\/ad682f\/pdf","content-type":"application\/pdf","content-version":"am","intended-application":"syndication"},{"URL":"https:\/\/iopscience.iop.org\/article\/10.1088\/2632-2153\/ad682f\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/iopscience.iop.org\/article\/10.1088\/2632-2153\/ad682f\/pdf","content-type":"application\/pdf","content-version":"am","intended-application":"similarity-checking"},{"URL":"https:\/\/iopscience.iop.org\/article\/10.1088\/2632-2153\/ad682f\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,8,14]],"date-time":"2024-08-14T08:52:37Z","timestamp":1723625557000},"score":1,"resource":{"primary":{"URL":"https:\/\/iopscience.iop.org\/article\/10.1088\/2632-2153\/ad682f"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,8,14]]},"references-count":70,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2024,8,14]]},"published-print":{"date-parts":[[2024,9,1]]}},"URL":"https:\/\/doi.org\/10.1088\/2632-2153\/ad682f","relation":{},"ISSN":["2632-2153"],"issn-type":[{"type":"electronic","value":"2632-2153"}],"subject":[],"published":{"date-parts":[[2024,8,14]]},"assertion":[{"value":"Automating the discovery of partial differential equations in dynamical systems","name":"article_title","label":"Article Title"},{"value":"Machine Learning: Science and Technology","name":"journal_title","label":"Journal Title"},{"value":"paper","name":"article_type","label":"Article Type"},{"value":"\u00a9 2024 The Author(s). Published by IOP Publishing Ltd","name":"copyright_information","label":"Copyright Information"},{"value":"2024-05-03","name":"date_received","label":"Date Received","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2024-07-26","name":"date_accepted","label":"Date Accepted","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2024-08-14","name":"date_epub","label":"Online publication date","group":{"name":"publication_dates","label":"Publication dates"}}]}}