{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,23]],"date-time":"2026-10-23T23:43:37Z","timestamp":1792799017711,"version":"4.5.0"},"reference-count":58,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","funder":[{"DOI":"10.13039\/100006831","name":"U.S. Air Force","doi-asserted-by":"publisher","award":["FA9550-17-1-0195"],"award-info":[{"award-number":["FA9550-17-1-0195"]}],"id":[{"id":"10.13039\/100006831","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000015","name":"U.S. Department of Energy","doi-asserted-by":"publisher","award":["DE-SC0019303"],"award-info":[{"award-number":["DE-SC0019303"]}],"id":[{"id":"10.13039\/100000015","id-type":"DOI","asserted-by":"publisher"}]},{"name":"US Department of Energy National Nuclear Security Administration","award":["DE-NA0003969"],"award-info":[{"award-number":["DE-NA0003969"]}],"id":[{"id":"https:\/\/ror.org\/03sk1we31","id-type":"ROR","asserted-by":"crossref"}]}],"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>This work formulates a new approach to reduced modeling of parameterized, time-dependent partial differential equations (PDEs). The method employs Operator Inference, a scientific machine learning framework combining data-driven learning and physics-based modeling. The parametric structure of the governing equations is embedded directly into the reduced-order model, and parameterized reduced-order operators are learned via a data-driven linear regression problem. The result is a reduced-order model that can be solved rapidly to map parameter values to approximate PDE solutions. Such parameterized reduced-order models may be used as physics-based surrogates for uncertainty quantification and inverse problems that require many forward solves of parametric PDEs. Numerical issues such as well-posedness and the need for appropriate regularization in the learning problem are considered, and an algorithm for hyperparameter selection is presented. The method is illustrated for a parametric heat equation and demonstrated for the FitzHugh\u2013Nagumo neuron model.<\/jats:p>","DOI":"10.1137\/21m1452810","type":"journal-article","created":{"date-parts":[[2023,7,25]],"date-time":"2023-07-25T03:57:41Z","timestamp":1690257461000},"page":"A1917-A1946","source":"Crossref","is-referenced-by-count":56,"title":["Nonintrusive Reduced-Order Models for Parametric Partial Differential Equations via Data-Driven Operator Inference"],"prefix":"10.1137","volume":"45","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0231-5359","authenticated-orcid":true,"given":"Shane A.","family":"McQuarrie","sequence":"first","affiliation":[{"name":"Oden Institute for Computational Engineering and Sciences, University of Texas at Austin, Austin, TX 78712 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8807-8192","authenticated-orcid":true,"given":"Parisa","family":"Khodabakhshi","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering and Mechanics, Lehigh University, Bethlehem, PA 18015 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2156-9338","authenticated-orcid":true,"given":"Karen E.","family":"Willcox","sequence":"additional","affiliation":[{"name":"Oden Institute for Computational Engineering and Sciences, University of Texas at Austin, Austin, TX 78712 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2023,7,25]]},"reference":[{"key":"ref1","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.1969.1054286"},{"key":"ref2","doi-asserted-by":"publisher","DOI":"10.1093\/imamci\/3.2-3.61"},{"key":"ref3","doi-asserted-by":"publisher","DOI":"10.1137\/15M1041432"},{"key":"ref4","doi-asserted-by":"publisher","DOI":"10.1007\/s11837-020-04399-8"},{"key":"ref5","doi-asserted-by":"publisher","DOI":"10.1137\/0146047"},{"key":"ref6","doi-asserted-by":"publisher","DOI":"10.1016\/j.crma.2004.08.006"},{"key":"ref7","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2020.113433"},{"key":"ref8","doi-asserted-by":"publisher","DOI":"10.1137\/130932715"},{"key":"ref9","doi-asserted-by":"publisher","DOI":"10.1146\/annurev.fl.25.010193.002543"},{"key":"ref10","doi-asserted-by":"publisher","DOI":"10.5802\/smai-jcm.74"},{"key":"ref11","doi-asserted-by":"crossref","unstructured":"\u00c5. Bj\u00f6rck , Numerical Methods for Least Squares Problems, SIAM, Philadelphia, 1996.","DOI":"10.1137\/1.9781611971484"},{"key":"ref12","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.1517384113"},{"key":"ref13","doi-asserted-by":"publisher","DOI":"10.3934\/jcd.2015002"},{"key":"ref14","doi-asserted-by":"publisher","DOI":"10.1137\/090766498"},{"key":"ref15","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.38.3.235"},{"key":"ref16","first-page":"20210904","volume":"478","author":"Fasel U.","year":"2022","journal-title":"Proc. R. Soc. A Math. Phys. Eng. Sci."},{"key":"ref17","doi-asserted-by":"publisher","DOI":"10.1016\/S0006-3495(61)86902-6"},{"key":"ref18","doi-asserted-by":"publisher","DOI":"10.1017\/S0962492921000064"},{"key":"ref19","doi-asserted-by":"publisher","DOI":"10.56021\/9781421407944"},{"key":"ref20","doi-asserted-by":"publisher","DOI":"10.1002\/(SICI)1097-0207(19990310)44:7<945::AID-NME537>3.0.CO;2-F"},{"key":"ref21","doi-asserted-by":"publisher","DOI":"10.1051\/m2an:2005006"},{"key":"ref22","doi-asserted-by":"publisher","DOI":"10.1137\/1034115"},{"key":"ref23","doi-asserted-by":"publisher","DOI":"10.1137\/130914619"},{"key":"ref24","doi-asserted-by":"crossref","unstructured":"P. Jain , \nS. McQuarrie , and \nB. Kramer , Performance comparison of data-driven reduced models for a single-injector combustion process, in Proceedings of the AIAA Propulsion and Energy 2021 Forum, Virtual Event, 2021, Paper AIAA-2021-3633.","DOI":"10.2514\/6.2021-3633"},{"key":"ref25","first-page":"167","volume":"30","author":"Khatri C.","year":"1968","journal-title":"Sankhy\u0101 Indian J. Stat. Series A"},{"key":"ref26","doi-asserted-by":"crossref","unstructured":"P. Khodabakhshi  and \nK. E. Willcox , Non-intrusive Data-Driven Model Reduction for Differential Algebraic Equations Derived from Lifting Transformations, Oden Institute Report 21-08, University of Texas at Austin, 2021.","DOI":"10.1016\/j.cma.2021.114296"},{"key":"ref27","doi-asserted-by":"publisher","DOI":"10.1137\/07070111X"},{"key":"ref28","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2019.108973"},{"key":"ref29","doi-asserted-by":"publisher","DOI":"10.1063\/1.5126869"},{"key":"ref30","doi-asserted-by":"publisher","DOI":"10.1137\/090775622"},{"key":"ref31","unstructured":"J. Lumley , The structures of inhomogeneous turbulent flow, in Atmospheric Turbulence and Radio Wave Propagation, Nauka, Moscow, 1967, pp. 166\u2013178."},{"key":"ref32","doi-asserted-by":"publisher","DOI":"10.1080\/03036758.2020.1863237"},{"key":"ref33","doi-asserted-by":"publisher","DOI":"10.1109\/JRPROC.1962.288235"},{"key":"ref34","doi-asserted-by":"publisher","DOI":"10.1093\/comjnl\/7.4.308"},{"key":"ref35","doi-asserted-by":"publisher","DOI":"10.1137\/20M1323448"},{"key":"ref36","doi-asserted-by":"publisher","DOI":"10.1137\/19M1292448"},{"key":"ref37","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2016.03.025"},{"key":"ref38","doi-asserted-by":"publisher","DOI":"10.1137\/21M1393972"},{"key":"ref39","doi-asserted-by":"publisher","DOI":"10.1016\/j.physd.2020.132401"},{"key":"ref40","doi-asserted-by":"publisher","DOI":"10.1063\/1.5027470"},{"key":"ref41","doi-asserted-by":"publisher","DOI":"10.1007\/s11831-008-9019-9"},{"key":"ref42","doi-asserted-by":"publisher","DOI":"10.1137\/18M1191944"},{"key":"ref43","doi-asserted-by":"publisher","DOI":"10.1098\/rspa.2016.0446"},{"key":"ref44","doi-asserted-by":"publisher","DOI":"10.1017\/S0022112010001217"},{"key":"ref45","doi-asserted-by":"publisher","DOI":"10.1016\/j.physd.2021.133122"},{"key":"ref46","doi-asserted-by":"publisher","DOI":"10.1090\/qam\/910462"},{"key":"ref47","doi-asserted-by":"publisher","DOI":"10.1023\/A:1021160023092"},{"key":"ref48","doi-asserted-by":"publisher","DOI":"10.2514\/1.J058943"},{"key":"ref49","doi-asserted-by":"publisher","DOI":"10.1186\/s40323-020-00177-y"},{"key":"ref50","volume-title":"Solutions of Ill-Posed Inverse Problems","author":"Tikhonov A. N.","year":"1977"},{"key":"ref51","doi-asserted-by":"crossref","unstructured":"L. N. Trefethen  and \nD. Bau III , Numerical Linear Algebra, SIAM, Philadelphia, 1997.","DOI":"10.1137\/1.9780898719574"},{"key":"ref52","doi-asserted-by":"publisher","DOI":"10.1051\/m2an\/2021010"},{"key":"ref53","doi-asserted-by":"publisher","DOI":"10.1016\/S0377-0427(00)00393-9"},{"key":"ref54","doi-asserted-by":"publisher","DOI":"10.1002\/fld.867"},{"key":"ref55","doi-asserted-by":"crossref","unstructured":"K. Veroy , \nC. Prud\u2019homme , \nD. Rovas , and \nA. Patera , A posteriori error bounds for reduced-basis approximation of parametrized noncoercive and nonlinear elliptic partial differential equations, in Proceedings of the 16th AIAA Computational Fluid Dynamics Conference, Orlando, FL, 2003, Paper AIAA-2003-3847.","DOI":"10.2514\/6.2003-3847"},{"key":"ref56","doi-asserted-by":"publisher","DOI":"10.1038\/s41592-019-0686-2"},{"key":"ref57","volume-title":"Solving Ordinary Differential Equations II: Stiff and Differential-algebraic Problems","volume":"375","author":"Wanner G.","year":"1996"},{"key":"ref58","doi-asserted-by":"publisher","DOI":"10.1002\/fld.4998"}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/21M1452810","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T15:57:04Z","timestamp":1787327824000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/21M1452810"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,7,25]]},"references-count":58,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2023,8,31]]}},"alternative-id":["10.1137\/21M1452810"],"URL":"https:\/\/doi.org\/10.1137\/21m1452810","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,7,25]]}}}