{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,8]],"date-time":"2026-04-08T16:47:24Z","timestamp":1775666844454,"version":"3.50.1"},"reference-count":73,"publisher":"IOP Publishing","issue":"2","license":[{"start":{"date-parts":[[2024,6,11]],"date-time":"2024-06-11T00:00:00Z","timestamp":1718064000000},"content-version":"vor","delay-in-days":10,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2024,6,11]],"date-time":"2024-06-11T00:00:00Z","timestamp":1718064000000},"content-version":"tdm","delay-in-days":10,"URL":"https:\/\/iopscience.iop.org\/info\/page\/text-and-data-mining"}],"content-domain":{"domain":["iopscience.iop.org"],"crossmark-restriction":false},"short-container-title":["Mach. Learn.: Sci. Technol."],"published-print":{"date-parts":[[2024,6,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>Physics Informed Neural Networks (PINNs) have been achieving ever newer feats of solving complicated Partial Differential Equations (PDEs) numerically while offering an attractive trade-off between accuracy and speed of inference. A particularly challenging aspect of PDEs is that there exist simple PDEs which can evolve into singular solutions in finite time starting from smooth initial conditions. In recent times some striking experiments have suggested that PINNs might be good at even detecting such finite-time blow-ups. In this work, we embark on a program to investigate this stability of PINNs from a rigorous theoretical viewpoint. Firstly, we derive error bounds for PINNs for Burgers\u2019 PDE, in arbitrary dimensions, under conditions that allow for a finite-time blow-up. Our bounds give a theoretical justification for the functional regularization terms that have been reported to be useful for training PINNs near finite-time blow-up. Then we demonstrate via experiments that our bounds are significantly correlated to the <jats:inline-formula>\n                     <jats:tex-math\/>\n                     <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" overflow=\"scroll\">\n                        <mml:mrow>\n                           <mml:msub>\n                              <mml:mi>\u2113<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                           <\/mml:msub>\n                        <\/mml:mrow>\n                     <\/mml:math>\n                  <\/jats:inline-formula>-distance of the neurally found surrogate from the true blow-up solution, when computed on sequences of PDEs that are getting increasingly close to a blow-up.<\/jats:p>","DOI":"10.1088\/2632-2153\/ad51cd","type":"journal-article","created":{"date-parts":[[2024,5,29]],"date-time":"2024-05-29T22:30:23Z","timestamp":1717021823000},"page":"025063","update-policy":"https:\/\/doi.org\/10.1088\/crossmark-policy","source":"Crossref","is-referenced-by-count":1,"title":["Investigating the ability of PINNs to solve Burgers\u2019 PDE near finite-time blowup"],"prefix":"10.1088","volume":"5","author":[{"ORCID":"https:\/\/orcid.org\/0009-0009-7864-5865","authenticated-orcid":true,"given":"Dibyakanti","family":"Kumar","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5189-8939","authenticated-orcid":true,"given":"Anirbit","family":"Mukherjee","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"266","published-online":{"date-parts":[[2024,6,11]]},"reference":[{"key":"mlstad51cdbib1","first-page":"pp 2002","article-title":"Using computational fluid dynamics for aerodynamics\u2013a critical assessment","author":"Jameson","year":"2002"},{"key":"mlstad51cdbib2","doi-asserted-by":"publisher","first-page":"987","DOI":"10.1109\/72.712178","volume":"9","author":"Lagaris","year":"1998","journal-title":"IEEE Trans. Neural Netw."},{"key":"mlstad51cdbib3","article-title":"Royal signals and radar establishment malvern (United Kingdom) RSRE-MEMO-4148","author":"Broomhead","year":"1988"},{"key":"mlstad51cdbib4","doi-asserted-by":"publisher","first-page":"422","DOI":"10.1038\/s42254-021-00314-5","volume":"3","author":"Karniadakis","year":"2021","journal-title":"Nat. Rev. Phys."},{"key":"mlstad51cdbib5","doi-asserted-by":"publisher","first-page":"278","DOI":"10.1088\/1361-6544\/ac337f","volume":"35","author":"Weinan","year":"2021","journal-title":"Nonlinearity"},{"key":"mlstad51cdbib6","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":"mlstad51cdbib7","doi-asserted-by":"publisher","first-page":"140","DOI":"10.3390\/bdcc6040140","volume":"6","author":"Lawal","year":"2022","journal-title":"Big Data Cogn. Comput."},{"key":"mlstad51cdbib8","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s40304-018-0127-z","volume":"6","author":"Yu","year":"2018","journal-title":"Commun. Math. Stat."},{"key":"mlstad51cdbib9","doi-asserted-by":"publisher","first-page":"1339","DOI":"10.1016\/j.jcp.2018.08.029","volume":"375","author":"Sirignano","year":"2018","journal-title":"J. Comput. Phys."},{"key":"mlstad51cdbib10","doi-asserted-by":"crossref","DOI":"10.1088\/2632-2153\/abf0f5","article-title":"Data-driven discovery of koopman eigenfunctions for control","author":"Kaiser","year":"2021"},{"key":"mlstad51cdbib11","article-title":"Physics-informed autoencoders for lyapunov-stable fluid flow prediction","author":"Erichson","year":"2019"},{"key":"mlstad51cdbib12","doi-asserted-by":"publisher","DOI":"10.1063\/5.0047428","volume":"33","author":"Wandel","year":"2021","journal-title":"Phys. Fluids"},{"key":"mlstad51cdbib13","article-title":"Learning dissipative dynamics in chaotic systems","author":"Li","year":"2022"},{"key":"mlstad51cdbib14","article-title":"Neural stochastic pdes: Resolution-invariant learning of continuous spatiotemporal dynamics","author":"Salvi","year":"2022"},{"key":"mlstad51cdbib15","doi-asserted-by":"publisher","first-page":"218","DOI":"10.1038\/s42256-021-00302-5","volume":"3","author":"Lu","year":"2021","journal-title":"Nat. Mach. Intell."},{"key":"mlstad51cdbib16","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2022.114778","volume":"393","author":"Lu","year":"2022","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"mlstad51cdbib17","doi-asserted-by":"publisher","first-page":"eabi8605","DOI":"10.1126\/sciadv.abi8605","volume":"7","author":"Wang","year":"2021","journal-title":"Sci. Adv."},{"key":"mlstad51cdbib18","author":"Raoni\u0107","year":"2023"},{"key":"mlstad51cdbib19","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2021.110364","volume":"438","author":"Arthurs","year":"2021","journal-title":"J. Comput. Phys."},{"key":"mlstad51cdbib20","doi-asserted-by":"crossref","DOI":"10.1145\/3394486.3403198","article-title":"Towards physics-informed deep learning for turbulent flow prediction","author":"Wang","year":"2020"},{"key":"mlstad51cdbib21","doi-asserted-by":"publisher","DOI":"10.1063\/5.0095270","volume":"34","author":"Eivazi","year":"2022","journal-title":"Phys. Fluids"},{"key":"mlstad51cdbib22","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.130.244002","volume":"130","author":"Wang","year":"2023","journal-title":"Phys. Rev. Lett."},{"key":"mlstad51cdbib23","doi-asserted-by":"publisher","first-page":"A3158","DOI":"10.1137\/21M1447039","volume":"44","author":"Hu","year":"2022","journal-title":"SIAM J. Sci. Comput."},{"key":"mlstad51cdbib24","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1093\/imanum\/drab093","volume":"43","author":"Mishra","year":"2022","journal-title":"IMA J. Numer. Anal."},{"key":"mlstad51cdbib25","author":"De Ryck","year":"2022"},{"key":"mlstad51cdbib26","first-page":"pp 26548","volume":"vol 34","author":"Krishnapriyan","year":"2021"},{"key":"mlstad51cdbib27","doi-asserted-by":"publisher","first-page":"A3055","DOI":"10.1137\/20M1318043","volume":"43","author":"Wang","year":"2021","journal-title":"SIAM J. Sci. Comput."},{"key":"mlstad51cdbib28","author":"Rohrhofer","year":"2022"},{"key":"mlstad51cdbib29","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1109\/TAI.2022.3192362","volume":"5","author":"Cheng Wong","year":"2022","journal-title":"IEEE Trans. Artif. Intell."},{"key":"mlstad51cdbib30","author":"Wang","year":"2022"},{"key":"mlstad51cdbib31","doi-asserted-by":"crossref","first-page":"277","DOI":"10.2307\/2371729","volume":"67","author":"Wintner","year":"1945","journal-title":"Am. J. Math."},{"key":"mlstad51cdbib32","doi-asserted-by":"publisher","first-page":"301","DOI":"10.1007\/BF02854201","volume":"4","author":"Cooke","year":"1955","journal-title":"Rend. Circ. Mat. Palermo"},{"key":"mlstad51cdbib33","first-page":"pp 183","author":"Pazy","year":"1983"},{"key":"mlstad51cdbib34","doi-asserted-by":"publisher","first-page":"124","DOI":"10.1137\/S0363012993259981","volume":"34","author":"Lin","year":"1996","journal-title":"SIAM J. Control Optim."},{"key":"mlstad51cdbib35","doi-asserted-by":"publisher","first-page":"47","DOI":"10.1017\/S095679250000005X","volume":"1","author":"Stuart","year":"1990","journal-title":"Eur. J. Appl. Math."},{"key":"mlstad51cdbib36","article-title":"On the blowing up of solutions of the cauchy problem for u1+\u03b4u+u1+\u03b1","author":"Fujita","year":"1966"},{"key":"mlstad51cdbib37","doi-asserted-by":"publisher","first-page":"132","DOI":"10.1090\/S0002-9904-1969-12175-0","volume":"75","author":"Fujita","year":"1969","journal-title":"Bull. Am. Math. Soc."},{"key":"mlstad51cdbib38","first-page":"633","volume":"24","author":"Herrero","year":"1997","journal-title":"Ann. Scuola. Norm.-Sci."},{"key":"mlstad51cdbib39","doi-asserted-by":"publisher","first-page":"951","DOI":"10.1007\/s00205-018-01336-7","volume":"232","author":"He","year":"2019","journal-title":"Arch. Ration. Mech. Anal."},{"key":"mlstad51cdbib40","doi-asserted-by":"publisher","first-page":"24","DOI":"10.1007\/s40818-022-00140-7","volume":"8","author":"Chen","year":"2022","journal-title":"Ann. PDE"},{"key":"mlstad51cdbib41","author":"Tanaka","year":"2023"},{"key":"mlstad51cdbib42","doi-asserted-by":"publisher","first-page":"476","DOI":"10.1137\/0140040","volume":"40","author":"Bebernes","year":"1981","journal-title":"SIAM J. Appl. Math."},{"key":"mlstad51cdbib43","doi-asserted-by":"publisher","first-page":"1350","DOI":"10.1137\/0143090","volume":"43","author":"Lacey","year":"1983","journal-title":"SIAM J. Appl. Math."},{"key":"mlstad51cdbib44","doi-asserted-by":"publisher","first-page":"521","DOI":"10.1098\/rspa.1991.0063","volume":"433","author":"Dold","year":"1991","journal-title":"Proc. R. Soc. A"},{"key":"mlstad51cdbib45","doi-asserted-by":"publisher","first-page":"321","DOI":"10.1007\/BF02764836","volume":"81","author":"Herrero","year":"1993","journal-title":"Isr. J. Math."},{"key":"mlstad51cdbib46","doi-asserted-by":"publisher","first-page":"201","DOI":"10.1017\/S0956792500001807","volume":"6","author":"Lacey","year":"1995","journal-title":"Eur. J. Appl. Math."},{"key":"mlstad51cdbib47","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s40818-016-0019-z","volume":"2","author":"Tao","year":"2016","journal-title":"Ann. PDE"},{"key":"mlstad51cdbib48","doi-asserted-by":"publisher","first-page":"601","DOI":"10.1090\/jams\/838","volume":"29","author":"Tao","year":"2016","journal-title":"J. Am. Math. Soc."},{"key":"mlstad51cdbib49","first-page":"1","volume":"18","author":"Baydin","year":"2018","journal-title":"J. March. Learn. Res."},{"key":"mlstad51cdbib50","first-page":"pp 8278","volume":"vol 35","author":"Wang","year":"2022"},{"key":"mlstad51cdbib51","doi-asserted-by":"publisher","DOI":"10.1088\/1402-4896\/ace21f","volume":"98","author":"Sultan","year":"2023","journal-title":"Phys. Scr."},{"key":"mlstad51cdbib52","doi-asserted-by":"publisher","first-page":"651","DOI":"10.1002\/fld.5259","volume":"96","author":"Sultan","year":"2024","journal-title":"Int. J. Numer. Methods Fluids"},{"key":"mlstad51cdbib53","author":"Dziugaite","year":"2017"},{"key":"mlstad51cdbib54","author":"Neyshabur","year":"2017"},{"key":"mlstad51cdbib55","article-title":"A study of the mathematics of deep learning","author":"Mukherjee","year":"2020"},{"key":"mlstad51cdbib56","author":"Neyshabur","year":"2018"},{"key":"mlstad51cdbib57","first-page":"pp 254","article-title":"Stronger generalization bounds for deep nets via a compression approach","author":"Arora","year":"2018"},{"key":"mlstad51cdbib58","first-page":"pp 5311","article-title":"Sparsity-aware generalization theory for deep neural networks","author":"Muthukumar","year":"2023"},{"key":"mlstad51cdbib59","doi-asserted-by":"publisher","first-page":"1394","DOI":"10.1016\/j.mcm.2008.12.006","volume":"49","author":"Biazar","year":"2009","journal-title":"Math. Comput. Model."},{"key":"mlstad51cdbib60","first-page":"pp 7264","article-title":"Mitigating propagation failures in physics-informed neural networks using retain-resample-release (R3) sampling","author":"Daw","year":"2023"},{"key":"mlstad51cdbib61","author":"Daw","year":"2022"},{"key":"mlstad51cdbib62","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2021.110768","volume":"449","author":"Wang","year":"2022","journal-title":"J. Comput. Phys."},{"key":"mlstad51cdbib63","first-page":"pp 5301","article-title":"On the spectral bias of neural networks","author":"Rahaman","year":"2019"},{"key":"mlstad51cdbib64","first-page":"pp 1","article-title":"Adversarial multi-task learning enhanced physics-informed neural networks for solving partial differential equations","author":"Thanasutives","year":"2021"},{"key":"mlstad51cdbib65","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2022.111722","volume":"474","author":"McClenny","year":"2023","journal-title":"J. Comput. Phys."},{"key":"mlstad51cdbib66","doi-asserted-by":"publisher","first-page":"427","DOI":"10.1090\/S0025-5718-1987-0906180-5","volume":"49","author":"Johnson","year":"1987","journal-title":"Math. Comput."},{"key":"mlstad51cdbib67","doi-asserted-by":"publisher","first-page":"891","DOI":"10.1137\/0728048","volume":"28","author":"Tadmor","year":"1991","journal-title":"SIAM J. Numer. Anal."},{"key":"mlstad51cdbib68","doi-asserted-by":"publisher","first-page":"1505","DOI":"10.1137\/0729087","volume":"29","author":"Nessyahu","year":"1992","journal-title":"SIAM J. Numer. Anal."},{"key":"mlstad51cdbib69","doi-asserted-by":"publisher","first-page":"12968","DOI":"10.1073\/pnas.1405238111","volume":"111","author":"Luo","year":"2014","journal-title":"Proc. Natl Acad. Sci."},{"key":"mlstad51cdbib70","doi-asserted-by":"publisher","first-page":"647","DOI":"10.4007\/annals.2021.194.3.2","volume":"194","author":"Elgindi","year":"2021","journal-title":"Ann. Math."},{"key":"mlstad51cdbib71","doi-asserted-by":"publisher","first-page":"1559","DOI":"10.1007\/s00220-021-04067-1","volume":"383","author":"Chen","year":"2021","journal-title":"Commun. Math. Phys."},{"key":"mlstad51cdbib72","doi-asserted-by":"publisher","first-page":"1722","DOI":"10.1137\/140966411","volume":"12","author":"Luo","year":"2014","journal-title":"Multiscale Model. Simul."},{"key":"mlstad51cdbib73","doi-asserted-by":"publisher","first-page":"400","DOI":"10.1137\/0721030","volume":"21","author":"DeVore","year":"1984","journal-title":"SIAM J. Numer. Anal."}],"container-title":["Machine Learning: Science and Technology"],"original-title":[],"link":[{"URL":"https:\/\/iopscience.iop.org\/article\/10.1088\/2632-2153\/ad51cd","content-type":"text\/html","content-version":"am","intended-application":"text-mining"},{"URL":"https:\/\/iopscience.iop.org\/article\/10.1088\/2632-2153\/ad51cd\/pdf","content-type":"application\/pdf","content-version":"am","intended-application":"text-mining"},{"URL":"https:\/\/iopscience.iop.org\/article\/10.1088\/2632-2153\/ad51cd","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/iopscience.iop.org\/article\/10.1088\/2632-2153\/ad51cd\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/iopscience.iop.org\/article\/10.1088\/2632-2153\/ad51cd\/pdf","content-type":"application\/pdf","content-version":"am","intended-application":"syndication"},{"URL":"https:\/\/iopscience.iop.org\/article\/10.1088\/2632-2153\/ad51cd\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/iopscience.iop.org\/article\/10.1088\/2632-2153\/ad51cd\/pdf","content-type":"application\/pdf","content-version":"am","intended-application":"similarity-checking"},{"URL":"https:\/\/iopscience.iop.org\/article\/10.1088\/2632-2153\/ad51cd\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,6,11]],"date-time":"2024-06-11T09:30:08Z","timestamp":1718098208000},"score":1,"resource":{"primary":{"URL":"https:\/\/iopscience.iop.org\/article\/10.1088\/2632-2153\/ad51cd"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,6,1]]},"references-count":73,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2024,6,11]]},"published-print":{"date-parts":[[2024,6,1]]}},"URL":"https:\/\/doi.org\/10.1088\/2632-2153\/ad51cd","relation":{},"ISSN":["2632-2153"],"issn-type":[{"value":"2632-2153","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,6,1]]},"assertion":[{"value":"Investigating the ability of PINNs to solve Burgers\u2019 PDE near finite-time blowup","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":"2024-02-14","name":"date_received","label":"Date Received","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2024-05-29","name":"date_accepted","label":"Date Accepted","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2024-06-11","name":"date_epub","label":"Online publication date","group":{"name":"publication_dates","label":"Publication dates"}}]}}