{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,22]],"date-time":"2026-08-22T07:33:05Z","timestamp":1787383985591,"version":"build-2736575974"},"reference-count":91,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-2012253"],"award-info":[{"award-number":["DMS-2012253"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-2012255"],"award-info":[{"award-number":["DMS-2012255"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["2012286"],"award-info":[{"award-number":["2012286"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000015","name":"U.S. Department of Energy","doi-asserted-by":"publisher","award":["DE-SC0019290"],"award-info":[{"award-number":["DE-SC0019290"]}],"id":[{"id":"10.13039\/100000015","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2023,6,30]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>We propose a new physics guided machine learning (PGML) paradigm that leverages the variational multiscale (VMS) framework and available data to dramatically increase the accuracy of reduced order models (ROMs) at a modest computational cost. The hierarchical structure of the ROM basis and the VMS framework enable a natural separation of the resolved and unresolved ROM spatial scales. Modern PGML algorithms are used to construct novel models for the interaction among the resolved and unresolved ROM scales. Specifically, the new framework builds ROM operators that are closest to the true interaction terms in the VMS framework. Finally, machine learning is used to reduce the projection error and further increase the ROM accuracy. Our numerical experiments for a two-dimensional vorticity transport problem show that the novel PGML-VMS-ROM paradigm maintains the low computational cost of current ROMs while significantly increasing the ROM accuracy.<\/jats:p>","DOI":"10.1137\/22m1496360","type":"journal-article","created":{"date-parts":[[2023,6,6]],"date-time":"2023-06-06T14:43:28Z","timestamp":1686062608000},"page":"B283-B313","source":"Crossref","is-referenced-by-count":12,"title":["Physics Guided Machine Learning for Variational Multiscale Reduced Order Modeling"],"prefix":"10.1137","volume":"45","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5548-0265","authenticated-orcid":true,"given":"Shady E.","family":"Ahmed","sequence":"first","affiliation":[{"name":"School of Mechanical & Aerospace Engineering, Oklahoma State University, Stillwater, OK 74078 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2241-4648","authenticated-orcid":true,"given":"Omer","family":"San","sequence":"additional","affiliation":[{"name":"School of Mechanical & Aerospace Engineering, Oklahoma State University, Stillwater, OK 74078 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Adil","family":"Rasheed","sequence":"additional","affiliation":[{"name":"Department of Engineering Cybernetics, Norwegian University of Science and Technology, N-7465, Trondheim, Norway."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Traian","family":"Iliescu","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Virginia Tech, Blacksburg, VA 24061 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8916-1190","authenticated-orcid":true,"given":"Alessandro","family":"Veneziani","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Department of Computer Science, Emory University, Atlanta, GA 30322 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2023,6,6]]},"reference":[{"key":"ref1","doi-asserted-by":"publisher","DOI":"10.2514\/6.2020-2946"},{"key":"ref2","doi-asserted-by":"publisher","DOI":"10.1063\/5.0061577"},{"key":"ref3","doi-asserted-by":"publisher","DOI":"10.3390\/fluids5010026"},{"key":"ref4","doi-asserted-by":"publisher","DOI":"10.1016\/j.physd.2020.132471"},{"key":"ref5","doi-asserted-by":"publisher","DOI":"10.1063\/5.0074310"},{"key":"ref6","doi-asserted-by":"crossref","unstructured":"S. E. Ahmed , \nO. San , \nA. Rasheed , \nT. Iliescu , and \nA. Veneziani , Physics Guided Machine Learning for Variational Multiscale Reduced Order Modeling: Python Codes, https:\/\/github.com\/Shady-Ahmed\/PGML-VMS-NLPOD, 2022.","DOI":"10.1137\/22M1496360"},{"key":"ref7","doi-asserted-by":"publisher","DOI":"10.1007\/s00162-009-0112-y"},{"key":"ref8","doi-asserted-by":"publisher","DOI":"10.1186\/s40323-016-0059-7"},{"key":"ref9","doi-asserted-by":"publisher","DOI":"10.1007\/s00158-014-1183-y"},{"key":"ref10","doi-asserted-by":"publisher","DOI":"10.1007\/s11071-012-0561-5"},{"key":"ref11","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2016.05.037"},{"key":"ref12","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-05083-6_19"},{"key":"ref13","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/30\/5\/055006"},{"key":"ref14","first-page":"59","volume-title":"Mechanic Futures","author":"Boschert S.","year":"2016"},{"key":"ref15","doi-asserted-by":"publisher","DOI":"10.2514\/1.13679"},{"key":"ref16","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2006.10.026"},{"key":"ref17","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2016.10.033"},{"key":"ref18","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4419-1002-8"},{"key":"ref19","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.97.7.2968"},{"key":"ref20","doi-asserted-by":"publisher","DOI":"10.1016\/S0167-2789(02)00446-3"},{"key":"ref21","doi-asserted-by":"publisher","DOI":"10.1002\/9781119176817.ecm2117"},{"key":"ref22","doi-asserted-by":"publisher","DOI":"10.1002\/fld.2089"},{"key":"ref23","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2016.07.002"},{"key":"ref24","doi-asserted-by":"publisher","DOI":"10.1137\/140969841"},{"key":"ref25","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2019.01.024"},{"key":"ref26","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4471-0219-9_20"},{"key":"ref27","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2021.110260"},{"key":"ref28","first-page":"20170385","volume":"473","author":"Gouasmi A.","year":"2017","journal-title":"Proc. R. Soc. A Math. Phys. Eng. Sci."},{"key":"ref29","doi-asserted-by":"publisher","DOI":"10.1098\/rspa.2020.1004"},{"key":"ref30","doi-asserted-by":"publisher","DOI":"10.1016\/j.mfglet.2018.02.006"},{"key":"ref31","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-75319-5_8"},{"key":"ref32","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2020.109513"},{"key":"ref33","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511919701"},{"key":"ref34","doi-asserted-by":"publisher","DOI":"10.1109\/MCOM.2019.1800155"},{"key":"ref35","doi-asserted-by":"publisher","DOI":"10.1016\/S0045-7825(98)00079-6"},{"key":"ref36","doi-asserted-by":"publisher","DOI":"10.1007\/s007910050051"},{"key":"ref37","doi-asserted-by":"publisher","DOI":"10.1063\/1.1367868"},{"key":"ref38","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-2013-02683-X"},{"key":"ref39","doi-asserted-by":"publisher","DOI":"10.1002\/num.21835"},{"key":"ref40","doi-asserted-by":"publisher","DOI":"10.1007\/s11071-019-05087-2"},{"key":"ref41","doi-asserted-by":"publisher","DOI":"10.1007\/s001620050119"},{"key":"ref42","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.1998.5943"},{"key":"ref43","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-45750-5"},{"key":"ref44","doi-asserted-by":"publisher","DOI":"10.1038\/s43588-021-00069-0"},{"key":"ref45","doi-asserted-by":"publisher","DOI":"10.1007\/s10915-022-02019-y"},{"key":"ref46","doi-asserted-by":"publisher","DOI":"10.1021\/ef800984v"},{"key":"ref47","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2019.108973"},{"key":"ref48","doi-asserted-by":"publisher","DOI":"10.1016\/j.paerosci.2003.12.001"},{"key":"ref49","doi-asserted-by":"publisher","DOI":"10.1143\/PTP.33.423"},{"key":"ref50","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2020.113470"},{"key":"ref51","doi-asserted-by":"crossref","unstructured":"C. Mou , \nE. Merzari , \nO. San , and \nT. Iliescu , An Energy-Based Lengthscale for Reduced Order Models of Turbulent Flows, preprint, https:\/\/arxiv.org\/abs\/2211.04404, 2022.","DOI":"10.1016\/j.nucengdes.2023.112454"},{"key":"ref52","doi-asserted-by":"publisher","DOI":"10.1017\/S0022112003006694"},{"key":"ref53","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-7091-0758-4"},{"key":"ref54","doi-asserted-by":"publisher","DOI":"10.1017\/S0022112004002149"},{"key":"ref55","doi-asserted-by":"publisher","DOI":"10.1137\/18M1177263"},{"key":"ref56","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2020.112990"},{"key":"ref57","doi-asserted-by":"publisher","DOI":"10.1063\/5.0038929"},{"key":"ref58","doi-asserted-by":"publisher","DOI":"10.1063\/5.0053349"},{"key":"ref59","doi-asserted-by":"crossref","unstructured":"S. Pawar , \nO. San , \nP. Vedula , \nA. Rasheed , and \nT. Kvamsdal , Multi-Fidelity Information Fusion with Concatenated Neural Networks, preprint, https:\/\/arxiv.org\/abs\/2110.04170, 2021.","DOI":"10.1038\/s41598-022-09938-8"},{"key":"ref60","doi-asserted-by":"publisher","DOI":"10.1090\/noti2475"},{"key":"ref61","doi-asserted-by":"publisher","DOI":"10.1137\/16M1082469"},{"key":"ref62","doi-asserted-by":"publisher","DOI":"10.1063\/1.5093355"},{"key":"ref63","doi-asserted-by":"publisher","DOI":"10.1109\/ACCESS.2020.2970143"},{"key":"ref64","doi-asserted-by":"publisher","DOI":"10.1023\/A:1011184714898"},{"key":"ref65","doi-asserted-by":"publisher","DOI":"10.1146\/annurev-fluid-010816-060042"},{"key":"ref66","doi-asserted-by":"publisher","DOI":"10.1002\/gamm.201010015"},{"key":"ref67","first-page":"217","volume":"5","author":"San O.","year":"2014","journal-title":"Int. J. Numer. Anal. Model. Ser. B"},{"key":"ref68","doi-asserted-by":"publisher","DOI":"10.1007\/s10444-015-9417-0"},{"key":"ref69","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.97.042322"},{"key":"ref70","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2012.09.005"},{"key":"ref71","doi-asserted-by":"publisher","DOI":"10.1007\/s00332-013-9178-1"},{"key":"ref72","doi-asserted-by":"crossref","unstructured":"S. Siami-Namini , \nN. Tavakoli , and \nA. S. Namin , The performance of LSTM and BiLSTM in forecasting time series, in 2019 IEEE International Conference on Big Data (Big Data), IEEE, 2019, pp. 3285\u20133292.","DOI":"10.1109\/BigData47090.2019.9005997"},{"key":"ref73","doi-asserted-by":"publisher","DOI":"10.1137\/120886947"},{"key":"ref74","doi-asserted-by":"publisher","DOI":"10.1090\/qam\/910462"},{"key":"ref75","doi-asserted-by":"publisher","DOI":"10.1090\/qam\/910463"},{"key":"ref76","doi-asserted-by":"publisher","DOI":"10.1090\/qam\/910464"},{"key":"ref77","doi-asserted-by":"publisher","DOI":"10.1002\/fld.3946"},{"key":"ref78","doi-asserted-by":"publisher","DOI":"10.1098\/rspa.2014.0446"},{"key":"ref79","doi-asserted-by":"publisher","DOI":"10.1016\/j.compfluid.2018.07.021"},{"key":"ref80","doi-asserted-by":"publisher","DOI":"10.2514\/1.J056060"},{"key":"ref81","doi-asserted-by":"publisher","DOI":"10.2514\/1.J058462"},{"key":"ref82","doi-asserted-by":"publisher","DOI":"10.1109\/TII.2018.2873186"},{"key":"ref83","doi-asserted-by":"publisher","DOI":"10.1016\/j.ymeth.2020.01.011"},{"key":"ref84","series-title":"Lecture Notes","volume-title":"Proper Orthogonal Decomposition: Theory and Reduced-Order Modelling","author":"Volkwein S.","year":"2013"},{"key":"ref85","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2020.109402"},{"key":"ref86","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2012.04.015"},{"key":"ref87","doi-asserted-by":"publisher","DOI":"10.1137\/17M1145136"},{"key":"ref88","doi-asserted-by":"publisher","DOI":"10.1016\/j.apnum.2017.01.006"},{"key":"ref89","doi-asserted-by":"publisher","DOI":"10.1002\/nme.4770"},{"key":"ref90","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0025619"},{"key":"ref91","doi-asserted-by":"publisher","DOI":"10.1093\/oso\/9780195140187.001.0001"}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/22M1496360","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:52:20Z","timestamp":1787334740000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/22M1496360"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,6,6]]},"references-count":91,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2023,6,30]]}},"alternative-id":["10.1137\/22M1496360"],"URL":"https:\/\/doi.org\/10.1137\/22m1496360","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,6,6]]}}}