{"id":"https://openalex.org/W3135895479","doi":"https://doi.org/10.1090/mcom/3781","title":"A deep learning approach to Reduced Order Modelling of parameter dependent partial differential equations","display_name":"A deep learning approach to Reduced Order Modelling of parameter dependent partial differential equations","publication_year":2022,"publication_date":"2022-11-16","ids":{"openalex":"https://openalex.org/W3135895479","doi":"https://doi.org/10.1090/mcom/3781","mag":"3135895479"},"language":"en","primary_location":{"id":"doi:10.1090/mcom/3781","is_oa":false,"landing_page_url":"https://doi.org/10.1090/mcom/3781","pdf_url":null,"source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2103.06183","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5089119201","display_name":"Nicola Rares Franco","orcid":"https://orcid.org/0000-0002-4980-5606"},"institutions":[{"id":"https://openalex.org/I93860229","display_name":"Politecnico di Milano","ror":"https://ror.org/01nffqt88","country_code":"IT","type":"education","lineage":["https://openalex.org/I93860229"]}],"countries":["IT"],"is_corresponding":true,"raw_author_name":"Nicola Franco","raw_affiliation_strings":["MOX, Department of Mathematics, Politecnico di Milano, via Edoardo, Bonardi 9, 20133, Milano, Italy"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"MOX, Department of Mathematics, Politecnico di Milano, via Edoardo, Bonardi 9, 20133, Milano, Italy","institution_ids":["https://openalex.org/I93860229"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5082126056","display_name":"Andrea Manzoni","orcid":"https://orcid.org/0000-0001-8277-2802"},"institutions":[{"id":"https://openalex.org/I93860229","display_name":"Politecnico di Milano","ror":"https://ror.org/01nffqt88","country_code":"IT","type":"education","lineage":["https://openalex.org/I93860229"]}],"countries":["IT"],"is_corresponding":false,"raw_author_name":"Andrea Manzoni","raw_affiliation_strings":["MOX, Department of Mathematics, Politecnico di Milano, via Edoardo, Bonardi 9, 20133, Milano, Italy"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"MOX, Department of Mathematics, Politecnico di Milano, via Edoardo, Bonardi 9, 20133, Milano, Italy","institution_ids":["https://openalex.org/I93860229"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5038341430","display_name":"Paolo Zunino","orcid":"https://orcid.org/0000-0002-2470-0189"},"institutions":[{"id":"https://openalex.org/I93860229","display_name":"Politecnico di Milano","ror":"https://ror.org/01nffqt88","country_code":"IT","type":"education","lineage":["https://openalex.org/I93860229"]}],"countries":["IT"],"is_corresponding":false,"raw_author_name":"Paolo Zunino","raw_affiliation_strings":["MOX, Department of Mathematics, Politecnico di Milano, via Edoardo, Bonardi 9, 20133, Milano, Italy"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"MOX, Department of Mathematics, Politecnico di Milano, via Edoardo, Bonardi 9, 20133, Milano, Italy","institution_ids":["https://openalex.org/I93860229"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5089119201"],"corresponding_institution_ids":["https://openalex.org/I93860229"],"apc_list":null,"apc_paid":null,"fwci":4.8817,"has_fulltext":true,"cited_by_count":44,"citation_normalized_percentile":{"value":0.95887234,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":100},"biblio":{"volume":"92","issue":"340","first_page":"483","last_page":"524"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11206","display_name":"Model Reduction and Neural Networks","score":0.9998999834060669,"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":0.9998999834060669,"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/T10339","display_name":"Advanced Numerical Methods in Computational Mathematics","score":0.9876000285148621,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10928","display_name":"Probabilistic and Robust Engineering Design","score":0.987500011920929,"subfield":{"id":"https://openalex.org/subfields/1804","display_name":"Statistics, Probability and Uncertainty"},"field":{"id":"https://openalex.org/fields/18","display_name":"Decision Sciences"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.5801267623901367},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.5125732421875},{"id":"https://openalex.org/keywords/partial-differential-equation","display_name":"Partial differential equation","score":0.5001039505004883},{"id":"https://openalex.org/keywords/dimension","display_name":"Dimension (graph theory)","score":0.48755449056625366},{"id":"https://openalex.org/keywords/semantics","display_name":"Semantics (computer science)","score":0.45117276906967163},{"id":"https://openalex.org/keywords/deep-learning","display_name":"Deep learning","score":0.4432840943336487},{"id":"https://openalex.org/keywords/manifold","display_name":"Manifold (fluid mechanics)","score":0.4247722029685974},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.4194173514842987},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.38429075479507446},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.183068186044693},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.145806223154068}],"concepts":[{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.5801267623901367},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.5125732421875},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.5001039505004883},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.48755449056625366},{"id":"https://openalex.org/C184337299","wikidata":"https://www.wikidata.org/wiki/Q1437428","display_name":"Semantics (computer science)","level":2,"score":0.45117276906967163},{"id":"https://openalex.org/C108583219","wikidata":"https://www.wikidata.org/wiki/Q197536","display_name":"Deep learning","level":2,"score":0.4432840943336487},{"id":"https://openalex.org/C529865628","wikidata":"https://www.wikidata.org/wiki/Q1790740","display_name":"Manifold (fluid mechanics)","level":2,"score":0.4247722029685974},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.4194173514842987},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.38429075479507446},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.183068186044693},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.145806223154068},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","level":0,"score":0.0},{"id":"https://openalex.org/C78519656","wikidata":"https://www.wikidata.org/wiki/Q101333","display_name":"Mechanical engineering","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1090/mcom/3781","is_oa":false,"landing_page_url":"https://doi.org/10.1090/mcom/3781","pdf_url":null,"source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:2103.06183","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2103.06183","pdf_url":"https://arxiv.org/pdf/2103.06183","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:re.public.polimi.it:11311/1225297","is_oa":true,"landing_page_url":"https://hdl.handle.net/11311/1225297","pdf_url":"https://re.public.polimi.it/bitstream/11311/1225297/6/11311-1225297_Franco.pdf","source":{"id":"https://openalex.org/S4306400312","display_name":"Virtual Community of Pathological Anatomy (University of Castilla La Mancha)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I79189158","host_organization_name":"University of Castilla-La Mancha","host_organization_lineage":["https://openalex.org/I79189158"],"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":null,"raw_type":"info:eu-repo/semantics/article"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2103.06183","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2103.06183","pdf_url":"https://arxiv.org/pdf/2103.06183","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"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":86,"referenced_works":["https://openalex.org/W934230699","https://openalex.org/W1523918473","https://openalex.org/W1553600813","https://openalex.org/W1604809219","https://openalex.org/W1605528072","https://openalex.org/W1677182931","https://openalex.org/W1855169650","https://openalex.org/W1873968263","https://openalex.org/W1967011851","https://openalex.org/W1983156129","https://openalex.org/W1984337824","https://openalex.org/W1990261069","https://openalex.org/W2003207029","https://openalex.org/W2039055345","https://openalex.org/W2047591100","https://openalex.org/W2064543119","https://openalex.org/W2067981301","https://openalex.org/W2076063813","https://openalex.org/W2093472053","https://openalex.org/W2100495367","https://openalex.org/W2116341502","https://openalex.org/W2122494265","https://openalex.org/W2158581396","https://openalex.org/W2278536479","https://openalex.org/W2295099209","https://openalex.org/W2316762840","https://openalex.org/W2528305538","https://openalex.org/W2562390720","https://openalex.org/W2609668139","https://openalex.org/W2620903196","https://openalex.org/W2745324256","https://openalex.org/W2772709428","https://openalex.org/W2784733489","https://openalex.org/W2808746463","https://openalex.org/W2809491586","https://openalex.org/W2883486956","https://openalex.org/W2905589030","https://openalex.org/W2933916181","https://openalex.org/W2936544000","https://openalex.org/W2944394121","https://openalex.org/W2962761333","https://openalex.org/W2962949934","https://openalex.org/W2963146412","https://openalex.org/W2963982269","https://openalex.org/W2969750612","https://openalex.org/W2979786244","https://openalex.org/W2986795381","https://openalex.org/W3000293971","https://openalex.org/W3004086380","https://openalex.org/W3006108674","https://openalex.org/W3008349919","https://openalex.org/W3020191331","https://openalex.org/W3021748939","https://openalex.org/W3028938632","https://openalex.org/W3035087317","https://openalex.org/W3040277050","https://openalex.org/W3087255898","https://openalex.org/W3093786597","https://openalex.org/W3102139197","https://openalex.org/W3104577153","https://openalex.org/W3107277102","https://openalex.org/W3108091126","https://openalex.org/W3111771943","https://openalex.org/W3123208564","https://openalex.org/W3126658681","https://openalex.org/W3133338006","https://openalex.org/W3137474564","https://openalex.org/W3137611682","https://openalex.org/W3153519416","https://openalex.org/W3160447895","https://openalex.org/W3166995168","https://openalex.org/W3167031531","https://openalex.org/W3175496536","https://openalex.org/W3176116059","https://openalex.org/W3177136884","https://openalex.org/W3178968719","https://openalex.org/W3181924469","https://openalex.org/W3206057793","https://openalex.org/W4200096621","https://openalex.org/W4229054729","https://openalex.org/W4229801863","https://openalex.org/W4234524531","https://openalex.org/W4238472920","https://openalex.org/W4240395309","https://openalex.org/W4292079189","https://openalex.org/W6600281463"],"related_works":["https://openalex.org/W4375867731","https://openalex.org/W2611989081","https://openalex.org/W4230611425","https://openalex.org/W2731899572","https://openalex.org/W4304166257","https://openalex.org/W4294635752","https://openalex.org/W4383066092","https://openalex.org/W3215138031","https://openalex.org/W4327774331","https://openalex.org/W2952670135"],"abstract_inverted_index":{"Within":[0],"the":[1,21,25,33,42,55,75,92,99,110,129,139,145,149,166,170,174,189,211,214],"framework":[2],"of":[3,24,37,77,91,101,109,131,148,173,216],"parameter":[4],"dependent":[5],"Partial":[6],"Differential":[7],"Equations":[8],"(PDEs),":[9],"we":[10,81,104,127,153,187,203,209],"develop":[11],"a":[12,51,87,106,132],"constructive":[13],"approach":[14,191],"based":[15,73],"on":[16,74,160],"Deep":[17],"Neural":[18],"Networks":[19],"for":[20,83,98],"efficient":[22],"approximation":[23,90,171],"parameter-to-solution":[26],"map.":[27],"The":[28,177],"research":[29],"is":[30,72,141,180],"motivated":[31],"by":[32,183],"limitations":[34],"and":[35,85,152,169,208,222],"drawbacks":[36],"state-of-the-art":[38],"algorithms,":[39],"such":[40],"as":[41],"Reduced":[43],"Basis":[44],"method,":[45],"when":[46],"addressing":[47],"problems":[48],"that":[49,138],"show":[50,137],"slow":[52],"decay":[53],"in":[54,213],"Kolmogorov":[56,111],"<inline-formula":[57,112],"content-type=\"math/mathml\">":[58,113],"<mml:math":[59,114],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[60,115],"alttext=\"n\">":[61,116],"<mml:semantics>":[62,117],"<mml:mi>n</mml:mi>":[63,118],"<mml:annotation":[64,119],"encoding=\"application/x-tex\">n</mml:annotation>":[65,120],"</mml:semantics>":[66,121],"</mml:math>":[67,122],"</inline-formula>":[68,123],"-width.":[69],"Our":[70],"work":[71],"use":[76],"deep":[78,102],"autoencoders,":[79,103],"which":[80,126],"employ":[82],"encoding":[84],"decoding":[86],"high":[88],"fidelity":[89],"solution":[93,150],"manifold.":[94],"To":[95],"provide":[96,154],"guidelines":[97],"design":[100],"consider":[105,204],"nonlinear":[107],"version":[108],"-width":[124],"over":[125],"base":[128],"concept":[130],"minimal":[133,167],"latent":[134],"dimension.":[135],"We":[136],"latter":[140],"intimately":[142],"related":[143],"to":[144],"topological":[146],"properties":[147],"manifold,":[151],"theoretical":[155],"results":[156],"with":[157,192],"particular":[158],"emphasis":[159],"second":[161],"order":[162,199],"elliptic":[163],"PDEs,":[164,207],"characterizing":[165],"dimension":[168],"errors":[172],"proposed":[175,190],"approach.":[176],"theory":[178],"presented":[179],"further":[181],"supported":[182],"numerical":[184],"experiments,":[185],"where":[186],"compare":[188],"classical":[193],"Principal":[194],"Orthogonal":[195],"Decomposition":[196],"(POD)-Galerkin":[197],"reduced":[198],"models.":[200],"In":[201],"particular,":[202],"parametrized":[205],"advection-diffusion":[206],"test":[210],"methodology":[212],"presence":[215],"strong":[217],"transport":[218],"fields,":[219],"singular":[220],"terms":[221],"stochastic":[223],"coefficients.":[224]},"counts_by_year":[{"year":2026,"cited_by_count":4},{"year":2025,"cited_by_count":19},{"year":2024,"cited_by_count":9},{"year":2023,"cited_by_count":10},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":1}],"updated_date":"2026-04-23T06:14:38.165362","created_date":"2021-03-15T00:00:00"}
