{"id":"https://openalex.org/W2937868358","doi":"https://doi.org/10.48550/arxiv.1904.04685","title":"On the approximation of the solution of partial differential equations by artificial neural networks trained by a multilevel Levenberg-Marquardt method","display_name":"On the approximation of the solution of partial differential equations by artificial neural networks trained by a multilevel Levenberg-Marquardt method","publication_year":2019,"publication_date":"2019-04-09","ids":{"openalex":"https://openalex.org/W2937868358","doi":"https://doi.org/10.48550/arxiv.1904.04685","mag":"2937868358"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:1904.04685","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/1904.04685","pdf_url":"https://arxiv.org/pdf/1904.04685","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/1904.04685","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5107845515","display_name":"Henri Calandra","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I103084370","display_name":"Total (France)","ror":"https://ror.org/04sk34n56","country_code":"FR","type":"company","lineage":["https://openalex.org/I103084370"]}],"countries":["FR"],"is_corresponding":false,"raw_author_name":"Calandra, Henri","raw_affiliation_strings":["TOTAL , Centre Scientifique et Technique Jean F\u00e9ger , avenue de Larribau F-64000 Pau , France","Total E&P (France)"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"TOTAL , Centre Scientifique et Technique Jean F\u00e9ger , avenue de Larribau F-64000 Pau , France","institution_ids":["https://openalex.org/I103084370"]},{"raw_affiliation_string":"Total E&P (France)","institution_ids":["https://openalex.org/I103084370"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5036298459","display_name":"Serge Gratton","orcid":"https://orcid.org/0000-0002-5021-2357","observed_orcids":["https://orcid.org/0000-0002-5021-2357"]},"institutions":[{"id":"https://openalex.org/I205747304","display_name":"Institut National Polytechnique de Toulouse","ror":"https://ror.org/033p9g875","country_code":"FR","type":"education","lineage":["https://openalex.org/I205747304"]},{"id":"https://openalex.org/I4210119061","display_name":"Institut de Recherche en Informatique de Toulouse","ror":"https://ror.org/01rx4qw44","country_code":"FR","type":"facility","lineage":["https://openalex.org/I1294671590","https://openalex.org/I205747304","https://openalex.org/I4210119061","https://openalex.org/I4210152422","https://openalex.org/I4405258862","https://openalex.org/I4405259414"]},{"id":"https://openalex.org/I4387153255","display_name":"\u00c9cole Nationale Sup\u00e9rieure d'\u00c9lectrotechnique, d'\u00c9lectronique, d'Informatique, d'Hydraulique et des T\u00e9l\u00e9communications","ror":"https://ror.org/05wfw4946","country_code":"FR","type":"education","lineage":["https://openalex.org/I205747304","https://openalex.org/I4387153255"]},{"id":"https://openalex.org/I4405258862","display_name":"Universit\u00e9 de Toulouse","ror":"https://ror.org/01ahyrz84","country_code":"FR","type":"education","lineage":["https://openalex.org/I4405258862"]}],"countries":["FR"],"is_corresponding":false,"raw_author_name":"Gratton, Serge","raw_affiliation_strings":["INPT-IRIT , University of Toulouse and ENSEEIHT , 2 Rue Camichel , BP 7122 , F-31071 Toulouse Cedex 7 , France","IRIT-APO - Algorithmes Parall\u00e8les et Optimisation (Institut de recherche en informatique de Toulouse - IRIT\r\n2 rue Charles Camichel \r\n31071 Toulouse Cedex 7 \r\n - France)","Toulouse INP - Institut National Polytechnique (Toulouse) (France)"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"INPT-IRIT , University of Toulouse and ENSEEIHT , 2 Rue Camichel , BP 7122 , F-31071 Toulouse Cedex 7 , France","institution_ids":["https://openalex.org/I205747304","https://openalex.org/I4210119061","https://openalex.org/I4387153255","https://openalex.org/I4405258862"]},{"raw_affiliation_string":"IRIT-APO - Algorithmes Parall\u00e8les et Optimisation (Institut de recherche en informatique de Toulouse - IRIT\r\n2 rue Charles Camichel \r\n31071 Toulouse Cedex 7 \r\n - France)","institution_ids":["https://openalex.org/I4210119061"]},{"raw_affiliation_string":"Toulouse INP - Institut National Polytechnique (Toulouse) (France)","institution_ids":["https://openalex.org/I205747304"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5024258728","display_name":"Elisa Riccietti","orcid":"https://orcid.org/0000-0003-0864-6116","observed_orcids":["https://orcid.org/0000-0003-0864-6116"]},"institutions":[{"id":"https://openalex.org/I113428412","display_name":"\u00c9cole Normale Sup\u00e9rieure de Lyon","ror":"https://ror.org/04zmssz18","country_code":"FR","type":"education","lineage":["https://openalex.org/I113428412","https://openalex.org/I203339264"]},{"id":"https://openalex.org/I205747304","display_name":"Institut National Polytechnique de Toulouse","ror":"https://ror.org/033p9g875","country_code":"FR","type":"education","lineage":["https://openalex.org/I205747304"]},{"id":"https://openalex.org/I4210119061","display_name":"Institut de Recherche en Informatique de Toulouse","ror":"https://ror.org/01rx4qw44","country_code":"FR","type":"facility","lineage":["https://openalex.org/I1294671590","https://openalex.org/I205747304","https://openalex.org/I4210119061","https://openalex.org/I4210152422","https://openalex.org/I4405258862","https://openalex.org/I4405259414"]},{"id":"https://openalex.org/I4387153255","display_name":"\u00c9cole Nationale Sup\u00e9rieure d'\u00c9lectrotechnique, d'\u00c9lectronique, d'Informatique, d'Hydraulique et des T\u00e9l\u00e9communications","ror":"https://ror.org/05wfw4946","country_code":"FR","type":"education","lineage":["https://openalex.org/I205747304","https://openalex.org/I4387153255"]},{"id":"https://openalex.org/I4405258862","display_name":"Universit\u00e9 de Toulouse","ror":"https://ror.org/01ahyrz84","country_code":"FR","type":"education","lineage":["https://openalex.org/I4405258862"]}],"countries":["FR"],"is_corresponding":false,"raw_author_name":"Riccietti, Elisa","raw_affiliation_strings":["INPT-IRIT , University of Toulouse and ENSEEIHT , 2 Rue Camichel , BP 7122 , F-31071 Toulouse Cedex 7 , France","ENS de Lyon - \u00c9cole normale sup\u00e9rieure de Lyon (15 parvis Ren\u00e9 Descartes - BP 7000 - 69342 Lyon Cedex 07 - France)"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"INPT-IRIT , University of Toulouse and ENSEEIHT , 2 Rue Camichel , BP 7122 , F-31071 Toulouse Cedex 7 , France","institution_ids":["https://openalex.org/I205747304","https://openalex.org/I4210119061","https://openalex.org/I4387153255","https://openalex.org/I4405258862"]},{"raw_affiliation_string":"ENS de Lyon - \u00c9cole normale sup\u00e9rieure de Lyon (15 parvis Ren\u00e9 Descartes - BP 7000 - 69342 Lyon Cedex 07 - France)","institution_ids":["https://openalex.org/I113428412"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5102934326","display_name":"Xavier Vasseur","orcid":"https://orcid.org/0000-0002-2478-2524","observed_orcids":["https://orcid.org/0000-0002-2478-2524"]},"institutions":[{"id":"https://openalex.org/I193033237","display_name":"Institut Sup\u00e9rieur de l'A\u00e9ronautique et de l'Espace (ISAE-SUPAERO)","ror":"https://ror.org/04gyj6s21","country_code":"FR","type":"education","lineage":["https://openalex.org/I193033237"]},{"id":"https://openalex.org/I4405258862","display_name":"Universit\u00e9 de Toulouse","ror":"https://ror.org/01ahyrz84","country_code":"FR","type":"education","lineage":["https://openalex.org/I4405258862"]}],"countries":["FR"],"is_corresponding":false,"raw_author_name":"Vasseur, Xavier","raw_affiliation_strings":["ISAE-SUPAERO , University of Toulouse , 10, avenue Edouard Belin , BP 54032 , F-31055 Toulouse Cedex 4 , France","ISAE-SUPAERO - Institut Sup\u00e9rieur de l'A\u00e9ronautique et de l'Espace (ISAE - 10 av. Edouard Belin - BP 54032 - 31055 TOULOUSE Cedex 4 - France)"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"ISAE-SUPAERO , University of Toulouse , 10, avenue Edouard Belin , BP 54032 , F-31055 Toulouse Cedex 4 , France","institution_ids":["https://openalex.org/I193033237","https://openalex.org/I4405258862"]},{"raw_affiliation_string":"ISAE-SUPAERO - Institut Sup\u00e9rieur de l'A\u00e9ronautique et de l'Espace (ISAE - 10 av. Edouard Belin - BP 54032 - 31055 TOULOUSE Cedex 4 - France)","institution_ids":["https://openalex.org/I193033237"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":7,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":8,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":"16 pages","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":0.8715999722480774,"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":0.8715999722480774,"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/T10320","display_name":"Neural Networks and Applications","score":0.1103999987244606,"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"}},{"id":"https://openalex.org/T10339","display_name":"Advanced Numerical Methods in Computational Mathematics","score":0.0052999998442828655,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/partial-differential-equations","display_name":"partial differential equations","score":0.8299999833106995},{"id":"https://openalex.org/keywords/neural-network","display_name":"artificial neural networks","score":0.8050000071525574},{"id":"https://openalex.org/keywords/feedforward-neural-networks","display_name":"feedforward neural networks","score":0.8700000047683716},{"id":"https://openalex.org/keywords/least-squares-training","display_name":"least squares training","score":0.7289999723434448},{"id":"https://openalex.org/keywords/hierarchical-optimization","display_name":"hierarchical optimization","score":0.7960000038146973},{"id":"https://openalex.org/keywords/algebraic-multigrid","display_name":"algebraic multigrid","score":0.9240000247955322},{"id":"https://openalex.org/keywords/neural-network-training","display_name":"neural network training","score":0.4819999933242798}],"concepts":[{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.7442439794540405},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.5828254222869873},{"id":"https://openalex.org/C47702885","wikidata":"https://www.wikidata.org/wiki/Q5441227","display_name":"Feedforward neural network","level":3,"score":0.5483877062797546},{"id":"https://openalex.org/C137119250","wikidata":"https://www.wikidata.org/wiki/Q1413101","display_name":"Multigrid method","level":3,"score":0.49277904629707336},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.44683319330215454},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.4452188313007355},{"id":"https://openalex.org/C53846429","wikidata":"https://www.wikidata.org/wiki/Q186475","display_name":"Partial derivative","level":2,"score":0.4386734068393707},{"id":"https://openalex.org/C137800194","wikidata":"https://www.wikidata.org/wiki/Q11713455","display_name":"Interpolation (computer graphics)","level":3,"score":0.43783408403396606},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.431112140417099},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4296291768550873},{"id":"https://openalex.org/C173801870","wikidata":"https://www.wikidata.org/wiki/Q201413","display_name":"Heuristic","level":2,"score":0.42840152978897095},{"id":"https://openalex.org/C155512373","wikidata":"https://www.wikidata.org/wiki/Q287450","display_name":"Residual","level":2,"score":0.41692206263542175},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.38624516129493713},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.32956409454345703},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.11453315615653992},{"id":"https://openalex.org/C104114177","wikidata":"https://www.wikidata.org/wiki/Q79782","display_name":"Motion (physics)","level":2,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"pmh:oai:arXiv.org:1904.04685","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/1904.04685","pdf_url":"https://arxiv.org/pdf/1904.04685","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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-02943228v1","doi":null,"is_oa":true,"landing_page_url":"https://hal.science/hal-02943228","pdf_url":null,"source":{"id":"https://openalex.org/S4306402512","display_name":"HAL","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I4405252976","host_organization_name":"Centre pour la Communication Scientifique Directe","host_organization_lineage":["https://openalex.org/I4405252976"],"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":"Optimization Methods and Software, 2020, 16 p","raw_type":"info:eu-repo/semantics/article"},{"id":"doi:10.48550/arxiv.1904.04685","doi":"https://doi.org/10.48550/arxiv.1904.04685","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.1904.04685","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:1904.04685","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/1904.04685","pdf_url":"https://arxiv.org/pdf/1904.04685","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2937868358.pdf","grobid_xml":"https://content.openalex.org/works/W2937868358.grobid-xml"},"referenced_works_count":50,"referenced_works":["https://openalex.org/W7731662","https://openalex.org/W26823901","https://openalex.org/W652914187","https://openalex.org/W1663973292","https://openalex.org/W1976792725","https://openalex.org/W1978075626","https://openalex.org/W1980427518","https://openalex.org/W1997542937","https://openalex.org/W1998720717","https://openalex.org/W2007008305","https://openalex.org/W2010315317","https://openalex.org/W2028166238","https://openalex.org/W2039708501","https://openalex.org/W2048763981","https://openalex.org/W2049974290","https://openalex.org/W2077008103","https://openalex.org/W2083613895","https://openalex.org/W2085496778","https://openalex.org/W2102871258","https://openalex.org/W2120575449","https://openalex.org/W2124776405","https://openalex.org/W2127981460","https://openalex.org/W2131677235","https://openalex.org/W2135479785","https://openalex.org/W2149167810","https://openalex.org/W2156005216","https://openalex.org/W2158985775","https://openalex.org/W2163605009","https://openalex.org/W2269075851","https://openalex.org/W2436219157","https://openalex.org/W2498157954","https://openalex.org/W2510806995","https://openalex.org/W2525748878","https://openalex.org/W2573798107","https://openalex.org/W2605147767","https://openalex.org/W2616324011","https://openalex.org/W2745110207","https://openalex.org/W2760972773","https://openalex.org/W2765872171","https://openalex.org/W2772097715","https://openalex.org/W2788421913","https://openalex.org/W2803629276","https://openalex.org/W2810544715","https://openalex.org/W2884714875","https://openalex.org/W2891311596","https://openalex.org/W2937835837","https://openalex.org/W2963533707","https://openalex.org/W2963653623","https://openalex.org/W4212863985","https://openalex.org/W4394649613"],"related_works":["https://openalex.org/W3036590103","https://openalex.org/W4324375243","https://openalex.org/W2953464083","https://openalex.org/W2770250658","https://openalex.org/W2994074087","https://openalex.org/W4401698771","https://openalex.org/W2982346055","https://openalex.org/W2158985775","https://openalex.org/W4367728595","https://openalex.org/W7155176274"],"abstract_inverted_index":{"This":[0,69],"paper":[1],"is":[2,25,39,63,128],"concerned":[3],"with":[4],"the":[5,8,29,32,47,50,78,85,91,97,110,133,144,147,153,161],"approximation":[6],"of":[7,10,16,31,49,80,93,146,155],"solution":[9,30],"partial":[11,33,51],"differential":[12,34,52],"equations":[13],"by":[14,105,124],"means":[15],"artificial":[17,94,156],"neural":[18,23,95,157],"networks.":[19],"Here":[20],"a":[21,42,55,59,66],"feedforward":[22],"network":[24],"used":[26],"to":[27,73,100,131,143,160],"approximate":[28],"equation.":[35],"The":[36],"learning":[37],"problem":[38,88],"formulated":[40],"as":[41,54,65],"least":[43,86],"squares":[44,87],"problem,":[45],"choosing":[46],"residual":[48],"equation":[53],"loss":[56],"function,":[57],"whereas":[58],"multilevel":[60,81,134,149],"Levenberg-Marquardt":[61],"method":[62,151],"employed":[64,118],"training":[67,92,154],"method.":[68],"setting":[70],"allows":[71],"us":[72],"get":[74],"further":[75],"insight":[76],"into":[77],"potential":[79],"methods.":[82],"Indeed,":[83],"when":[84],"arises":[89],"from":[90],"networks,":[96,158],"variables":[98],"subject":[99],"optimization":[101,150],"are":[102],"not":[103],"related":[104,142],"any":[106,119],"geometrical":[107],"constraints":[108],"and":[109,113],"standard":[111,162],"interpolation":[112],"restriction":[114],"operators":[115],"cannot":[116],"be":[117],"longer.":[120],"A":[121],"heuristic,":[122],"inspired":[123],"algebraic":[125],"multigrid":[126],"methods,":[127],"then":[129],"proposed":[130],"construct":[132],"transfer":[135],"operators.":[136],"Numerical":[137],"experiments":[138],"show":[139],"encouraging":[140],"results":[141],"efficiency":[145],"new":[148],"for":[152],"compared":[159],"corresponding":[163],"one-level":[164],"procedure.":[165]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2023,"cited_by_count":2},{"year":2022,"cited_by_count":2},{"year":2021,"cited_by_count":2},{"year":2019,"cited_by_count":1}],"updated_date":"2026-11-08T12:12:28.789561","created_date":"2019-04-25T00:00:00"}