{"id":"https://openalex.org/W4225984980","doi":"https://doi.org/10.1186/s40323-023-00242-2","title":"POD-Galerkin reduced order models and physics-informed neural networks for solving inverse problems for the Navier\u2013Stokes equations","display_name":"POD-Galerkin reduced order models and physics-informed neural networks for solving inverse problems for the Navier\u2013Stokes equations","publication_year":2023,"publication_date":"2023-03-18","ids":{"openalex":"https://openalex.org/W4225984980","doi":"https://doi.org/10.1186/s40323-023-00242-2"},"language":"en","primary_location":{"id":"doi:10.1186/s40323-023-00242-2","is_oa":true,"landing_page_url":"https://doi.org/10.1186/s40323-023-00242-2","pdf_url":"https://amses-journal.springeropen.com/counter/pdf/10.1186/s40323-023-00242-2","source":{"id":"https://openalex.org/S3034486266","display_name":"Advanced Modeling and Simulation in Engineering Sciences","issn_l":"2213-7467","issn":["2213-7467"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Advanced Modeling and Simulation in Engineering Sciences","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref","datacite","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://amses-journal.springeropen.com/counter/pdf/10.1186/s40323-023-00242-2","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5088023955","display_name":"Saddam Hijazi","orcid":"https://orcid.org/0000-0002-9114-1102"},"institutions":[{"id":"https://openalex.org/I176453806","display_name":"University of Potsdam","ror":"https://ror.org/03bnmw459","country_code":"DE","type":"education","lineage":["https://openalex.org/I176453806"]}],"countries":["DE"],"is_corresponding":true,"raw_author_name":"Saddam Hijazi","raw_affiliation_strings":["Institute of Mathematics, University of Potsdam, Karl-Liebknecht-Str. 24-25, 14476, Potsdam, Germany"],"raw_orcid":"https://orcid.org/0000-0002-9114-1102","affiliations":[{"raw_affiliation_string":"Institute of Mathematics, University of Potsdam, Karl-Liebknecht-Str. 24-25, 14476, Potsdam, Germany","institution_ids":["https://openalex.org/I176453806"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5054331885","display_name":"Melina A. Freitag","orcid":"https://orcid.org/0000-0002-4539-2162"},"institutions":[{"id":"https://openalex.org/I176453806","display_name":"University of Potsdam","ror":"https://ror.org/03bnmw459","country_code":"DE","type":"education","lineage":["https://openalex.org/I176453806"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Melina Freitag","raw_affiliation_strings":["Institute of Mathematics, University of Potsdam, Karl-Liebknecht-Str. 24-25, 14476, Potsdam, Germany"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institute of Mathematics, University of Potsdam, Karl-Liebknecht-Str. 24-25, 14476, Potsdam, Germany","institution_ids":["https://openalex.org/I176453806"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5023752093","display_name":"Niels Landwehr","orcid":"https://orcid.org/0000-0001-5019-7620"},"institutions":[{"id":"https://openalex.org/I155765044","display_name":"University of Hildesheim","ror":"https://ror.org/02f9det96","country_code":"DE","type":"education","lineage":["https://openalex.org/I155765044"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Niels Landwehr","raw_affiliation_strings":["Institute of Computer Science, University of Hildesheim, Universit\u00e4tsplatz 1, 31141, Hildesheim, Germany"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institute of Computer Science, University of Hildesheim, Universit\u00e4tsplatz 1, 31141, Hildesheim, Germany","institution_ids":["https://openalex.org/I155765044"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5088023955"],"corresponding_institution_ids":["https://openalex.org/I176453806"],"apc_list":{"value":990,"currency":"GBP","value_usd":1214},"apc_paid":{"value":990,"currency":"GBP","value_usd":1214},"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.00412007,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"10","issue":"1","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11206","display_name":"Model Reduction and Neural Networks","score":0.9911999702453613,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11206","display_name":"Model Reduction and Neural Networks","score":0.9911999702453613,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11948","display_name":"Machine Learning in Materials Science","score":0.0013000000035390258,"subfield":{"id":"https://openalex.org/subfields/2505","display_name":"Materials Chemistry"},"field":{"id":"https://openalex.org/fields/25","display_name":"Materials Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12611","display_name":"Neural Networks and Reservoir Computing","score":0.0008999999845400453,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/galerkin-method","display_name":"Galerkin method","score":0.8146151304244995},{"id":"https://openalex.org/keywords/point-of-delivery","display_name":"Point of delivery","score":0.6333743929862976},{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.5978571176528931},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.5089907646179199},{"id":"https://openalex.org/keywords/turbulence","display_name":"Turbulence","score":0.491074800491333},{"id":"https://openalex.org/keywords/inverse","display_name":"Inverse","score":0.4908493161201477},{"id":"https://openalex.org/keywords/navier\u2013stokes-equations","display_name":"Navier\u2013Stokes equations","score":0.4831787645816803},{"id":"https://openalex.org/keywords/flow","display_name":"Flow (mathematics)","score":0.47787371277809143},{"id":"https://openalex.org/keywords/basis-function","display_name":"Basis function","score":0.4414742887020111},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.43393978476524353},{"id":"https://openalex.org/keywords/discontinuous-galerkin-method","display_name":"Discontinuous Galerkin method","score":0.4286280870437622},{"id":"https://openalex.org/keywords/inverse-problem","display_name":"Inverse problem","score":0.4212888479232788},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.4021804630756378},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.36838775873184204},{"id":"https://openalex.org/keywords/finite-element-method","display_name":"Finite element method","score":0.24102851748466492},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.20478275418281555},{"id":"https://openalex.org/keywords/mechanics","display_name":"Mechanics","score":0.12127992510795593},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.10020655393600464},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.09985458850860596}],"concepts":[{"id":"https://openalex.org/C186899397","wikidata":"https://www.wikidata.org/wiki/Q1491980","display_name":"Galerkin method","level":3,"score":0.8146151304244995},{"id":"https://openalex.org/C137776501","wikidata":"https://www.wikidata.org/wiki/Q7208270","display_name":"Point of delivery","level":2,"score":0.6333743929862976},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.5978571176528931},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5089907646179199},{"id":"https://openalex.org/C196558001","wikidata":"https://www.wikidata.org/wiki/Q190132","display_name":"Turbulence","level":2,"score":0.491074800491333},{"id":"https://openalex.org/C207467116","wikidata":"https://www.wikidata.org/wiki/Q4385666","display_name":"Inverse","level":2,"score":0.4908493161201477},{"id":"https://openalex.org/C2781278361","wikidata":"https://www.wikidata.org/wiki/Q201321","display_name":"Navier\u2013Stokes equations","level":3,"score":0.4831787645816803},{"id":"https://openalex.org/C38349280","wikidata":"https://www.wikidata.org/wiki/Q1434290","display_name":"Flow (mathematics)","level":2,"score":0.47787371277809143},{"id":"https://openalex.org/C5917680","wikidata":"https://www.wikidata.org/wiki/Q2621825","display_name":"Basis function","level":2,"score":0.4414742887020111},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.43393978476524353},{"id":"https://openalex.org/C92244383","wikidata":"https://www.wikidata.org/wiki/Q428273","display_name":"Discontinuous Galerkin method","level":3,"score":0.4286280870437622},{"id":"https://openalex.org/C135252773","wikidata":"https://www.wikidata.org/wiki/Q1567213","display_name":"Inverse problem","level":2,"score":0.4212888479232788},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.4021804630756378},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.36838775873184204},{"id":"https://openalex.org/C135628077","wikidata":"https://www.wikidata.org/wiki/Q220184","display_name":"Finite element method","level":2,"score":0.24102851748466492},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.20478275418281555},{"id":"https://openalex.org/C57879066","wikidata":"https://www.wikidata.org/wiki/Q41217","display_name":"Mechanics","level":1,"score":0.12127992510795593},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.10020655393600464},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.09985458850860596},{"id":"https://openalex.org/C6557445","wikidata":"https://www.wikidata.org/wiki/Q173113","display_name":"Agronomy","level":1,"score":0.0},{"id":"https://openalex.org/C84655787","wikidata":"https://www.wikidata.org/wiki/Q8067817","display_name":"Compressibility","level":2,"score":0.0},{"id":"https://openalex.org/C97355855","wikidata":"https://www.wikidata.org/wiki/Q11473","display_name":"Thermodynamics","level":1,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0}],"mesh":[],"locations_count":7,"locations":[{"id":"doi:10.1186/s40323-023-00242-2","is_oa":true,"landing_page_url":"https://doi.org/10.1186/s40323-023-00242-2","pdf_url":"https://amses-journal.springeropen.com/counter/pdf/10.1186/s40323-023-00242-2","source":{"id":"https://openalex.org/S3034486266","display_name":"Advanced Modeling and Simulation in Engineering Sciences","issn_l":"2213-7467","issn":["2213-7467"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Advanced Modeling and Simulation in Engineering Sciences","raw_type":"journal-article"},{"id":"doi:10.21203/rs.3.rs-1975535/v1","is_oa":true,"landing_page_url":"https://doi.org/10.21203/rs.3.rs-1975535/v1","pdf_url":"https://www.researchsquare.com/article/rs-1975535/latest.pdf","source":{"id":"https://openalex.org/S4306525896","display_name":"Research Square","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"acceptedVersion","is_accepted":true,"is_published":false,"raw_source_name":"Research Square","raw_type":"posted-content"},{"id":"pmh:oai:arXiv.org:2112.11950","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2112.11950","pdf_url":"https://arxiv.org/pdf/2112.11950","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"pmh:oai:HAL:hal-04513967v1","is_oa":true,"landing_page_url":"https://hal.science/hal-04513967","pdf_url":"https://hal.science/hal-04513967/document","source":{"id":"https://openalex.org/S4306402512","display_name":"HAL (Le Centre pour la Communication Scientifique Directe)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1294671590","host_organization_name":"Centre National de la Recherche Scientifique","host_organization_lineage":["https://openalex.org/I1294671590"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"https://link.springer.com/article/10.1186/s40323-023-00242-2/fulltext.html","raw_type":"info:eu-repo/semantics/article"},{"id":"pmh:oai:doaj.org/article:f3c02454dcd54e2c81ed3145c95fb1f5","is_oa":true,"landing_page_url":"https://doaj.org/article/f3c02454dcd54e2c81ed3145c95fb1f5","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Advanced Modeling and Simulation in Engineering Sciences, Vol 10, Iss 1, Pp 1-38 (2023)","raw_type":"article"},{"id":"pmh:oai:kobv.de-opus4-uni-potsdam:64312","is_oa":true,"landing_page_url":"https://publishup.uni-potsdam.de/frontdoor/index/index/docId/64312","pdf_url":null,"source":{"id":"https://openalex.org/S4306400594","display_name":"publish.UP (University of Potsdam)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I176453806","host_organization_name":"University of Potsdam","host_organization_lineage":["https://openalex.org/I176453806"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"article"},{"id":"doi:10.48550/arxiv.2112.11950","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2112.11950","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article-journal"}],"best_oa_location":{"id":"doi:10.1186/s40323-023-00242-2","is_oa":true,"landing_page_url":"https://doi.org/10.1186/s40323-023-00242-2","pdf_url":"https://amses-journal.springeropen.com/counter/pdf/10.1186/s40323-023-00242-2","source":{"id":"https://openalex.org/S3034486266","display_name":"Advanced Modeling and Simulation in Engineering Sciences","issn_l":"2213-7467","issn":["2213-7467"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Advanced Modeling and Simulation in Engineering Sciences","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G4092581361","display_name":"SFB 1294: Datenassimilation \u2013 Die nahtlose Verschmelzung von Daten und Modellen","funder_award_id":"318763901","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"},{"id":"https://openalex.org/G4686563519","display_name":null,"funder_award_id":"SFB1294","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"}],"funders":[{"id":"https://openalex.org/F4320320879","display_name":"Deutsche Forschungsgemeinschaft","ror":"https://ror.org/018mejw64"},{"id":"https://openalex.org/F4320322480","display_name":"Universit\u00e4t Potsdam","ror":"https://ror.org/03bnmw459"}],"has_content":{"grobid_xml":false,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4225984980.pdf"},"referenced_works_count":106,"referenced_works":["https://openalex.org/W37939860","https://openalex.org/W86368505","https://openalex.org/W581692639","https://openalex.org/W1483870296","https://openalex.org/W1489519217","https://openalex.org/W1531455566","https://openalex.org/W1613909081","https://openalex.org/W1853291804","https://openalex.org/W1934412359","https://openalex.org/W1973136150","https://openalex.org/W1974097079","https://openalex.org/W1978075626","https://openalex.org/W1981347630","https://openalex.org/W1984516949","https://openalex.org/W1984961557","https://openalex.org/W1985570191","https://openalex.org/W1988068819","https://openalex.org/W2006254762","https://openalex.org/W2007161305","https://openalex.org/W2011121645","https://openalex.org/W2014356541","https://openalex.org/W2017247230","https://openalex.org/W2017503182","https://openalex.org/W2017745440","https://openalex.org/W2026123674","https://openalex.org/W2029016890","https://openalex.org/W2039055345","https://openalex.org/W2047591100","https://openalex.org/W2063819119","https://openalex.org/W2068991306","https://openalex.org/W2074447412","https://openalex.org/W2074686342","https://openalex.org/W2085767064","https://openalex.org/W2088638050","https://openalex.org/W2089614907","https://openalex.org/W2096345667","https://openalex.org/W2097652335","https://openalex.org/W2109188552","https://openalex.org/W2110187357","https://openalex.org/W2110917150","https://openalex.org/W2126042578","https://openalex.org/W2128598780","https://openalex.org/W2130889131","https://openalex.org/W2133041109","https://openalex.org/W2134692705","https://openalex.org/W2134887646","https://openalex.org/W2137199868","https://openalex.org/W2142703030","https://openalex.org/W2149498546","https://openalex.org/W2151146539","https://openalex.org/W2152896489","https://openalex.org/W2153235652","https://openalex.org/W2158985775","https://openalex.org/W2159271996","https://openalex.org/W2163662580","https://openalex.org/W2166614820","https://openalex.org/W2168795429","https://openalex.org/W2169133564","https://openalex.org/W2277401480","https://openalex.org/W2319329301","https://openalex.org/W2325909323","https://openalex.org/W2353000616","https://openalex.org/W2496483732","https://openalex.org/W2517689716","https://openalex.org/W2561770886","https://openalex.org/W2563263390","https://openalex.org/W2695202536","https://openalex.org/W2749028154","https://openalex.org/W2753365189","https://openalex.org/W2766298346","https://openalex.org/W2766387390","https://openalex.org/W2772709428","https://openalex.org/W2778400616","https://openalex.org/W2884141227","https://openalex.org/W2885469054","https://openalex.org/W2889522259","https://openalex.org/W2899283552","https://openalex.org/W2909097035","https://openalex.org/W2945109751","https://openalex.org/W2963070178","https://openalex.org/W2963217000","https://openalex.org/W2986795381","https://openalex.org/W2987245967","https://openalex.org/W3006507139","https://openalex.org/W3021544678","https://openalex.org/W3037995832","https://openalex.org/W3098678593","https://openalex.org/W3101260193","https://openalex.org/W3105648287","https://openalex.org/W3121471846","https://openalex.org/W3137611682","https://openalex.org/W3141460013","https://openalex.org/W3149489117","https://openalex.org/W3153519416","https://openalex.org/W3185095713","https://openalex.org/W3206057793","https://openalex.org/W4221156222","https://openalex.org/W4231746175","https://openalex.org/W4232947228","https://openalex.org/W4240395309","https://openalex.org/W4241149657","https://openalex.org/W4241888870","https://openalex.org/W4242259929","https://openalex.org/W4245232001","https://openalex.org/W4293409666","https://openalex.org/W4300516407"],"related_works":["https://openalex.org/W8800753","https://openalex.org/W2149950","https://openalex.org/W89426","https://openalex.org/W12446239","https://openalex.org/W31993690","https://openalex.org/W32538687","https://openalex.org/W25865952","https://openalex.org/W24060463","https://openalex.org/W19391052","https://openalex.org/W24227501"],"abstract_inverted_index":{"<title>Abstract</title>":[0],"We":[1],"present":[2],"a":[3,62,169,175,184],"Reduced":[4],"Order":[5],"Model":[6],"(ROM)":[7],"which":[8,94,163],"exploits":[9],"recent":[10],"developments":[11],"in":[12,72],"Physics":[13],"Informed":[14],"Neural":[15],"Networks":[16],"(PINNs)":[17],"for":[18,22,36],"solving":[19],"inverse":[20],"problems":[21],"the":[23,28,31,37,53,57,66,69,78,96,100,107,113,126,132,146,152,155,165,172,179],"Navier\u2013Stokes":[24],"equations":[25,71,115],"(NSE).":[26],"In":[27],"proposed":[29,156],"approach,":[30,131],"presence":[32],"of":[33,56,65,74,106,125,151,154],"simulated":[34],"data":[35],"fluid":[38,58],"dynamics":[39],"fields":[40,59],"is":[41,46,86,135,158],"assumed.":[42],"A":[43,82,149],"POD-Galerkin":[44,83,117],"ROM":[45,85,157],"then":[47,87],"constructed":[48],"by":[49,89,160],"applying":[50],"POD":[51,79],"on":[52],"snapshots":[54],"matrices":[55],"and":[60,178],"performing":[61],"Galerkin":[63],"projection":[64],"NSE":[67],"(or":[68],"modified":[70],"case":[73],"turbulence":[75],"modeling)":[76],"onto":[77],"reduced":[80,97,118,133],"basis.":[81],"PINN":[84],"derived":[88],"introducing":[90],"deep":[91],"neural":[92,110],"networks":[93,111],"approximate":[95,138],"outputs":[98],"with":[99],"input":[101],"being":[102],"time":[103],"and/or":[104],"parameters":[105,140],"model.":[108],"The":[109],"incorporate":[112],"physical":[114,143],"(the":[116],"equations)":[119],"into":[120],"their":[121],"structure":[122],"as":[123,142],"part":[124],"loss":[127],"function.":[128],"Using":[129],"this":[130],"model":[134],"able":[136],"to":[137],"unknown":[139],"such":[141],"constants":[144],"or":[145],"boundary":[147],"conditions.":[148],"demonstration":[150],"applicability":[153],"illustrated":[159],"three":[161],"cases":[162],"are":[164],"steady":[166],"flow":[167,173,182],"around":[168,174,183],"backward":[170],"step,":[171],"circular":[176],"cylinder":[177],"unsteady":[180],"turbulent":[181],"surface":[185],"mounted":[186],"cubic":[187],"obstacle.":[188]},"counts_by_year":[],"updated_date":"2026-07-15T18:14:33.161393","created_date":"2022-05-05T00:00:00"}