{"id":"https://openalex.org/W4392817028","doi":"https://doi.org/10.1137/22m1522504","title":"wPINNs: Weak Physics Informed Neural Networks for Approximating Entropy Solutions of Hyperbolic Conservation Laws","display_name":"wPINNs: Weak Physics Informed Neural Networks for Approximating Entropy Solutions of Hyperbolic Conservation Laws","publication_year":2024,"publication_date":"2024-03-14","ids":{"openalex":"https://openalex.org/W4392817028","doi":"https://doi.org/10.1137/22m1522504"},"language":"en","primary_location":{"id":"doi:10.1137/22m1522504","is_oa":false,"landing_page_url":"https://doi.org/10.1137/22m1522504","pdf_url":null,"source":{"id":"https://openalex.org/S203348814","display_name":"SIAM Journal on Numerical Analysis","issn_l":"0036-1429","issn":["0036-1429","1095-7170"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Numerical Analysis","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5053556239","display_name":"Tim De Ryck","orcid":"https://orcid.org/0000-0001-6860-1345"},"institutions":[{"id":"https://openalex.org/I35440088","display_name":"ETH Zurich","ror":"https://ror.org/05a28rw58","country_code":"CH","type":"education","lineage":["https://openalex.org/I2799323385","https://openalex.org/I35440088"]}],"countries":["CH"],"is_corresponding":true,"raw_author_name":"Tim De Ryck","raw_affiliation_strings":["Seminar for Applied Mathematics, ETH Z\u00fcrich, 8092 Z\u00fcrich, Switzerland"],"affiliations":[{"raw_affiliation_string":"Seminar for Applied Mathematics, ETH Z\u00fcrich, 8092 Z\u00fcrich, Switzerland","institution_ids":["https://openalex.org/I35440088"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5101961582","display_name":"Siddhartha Mishra","orcid":"https://orcid.org/0000-0002-2665-5385"},"institutions":[{"id":"https://openalex.org/I35440088","display_name":"ETH Zurich","ror":"https://ror.org/05a28rw58","country_code":"CH","type":"education","lineage":["https://openalex.org/I2799323385","https://openalex.org/I35440088"]}],"countries":["CH"],"is_corresponding":false,"raw_author_name":"Siddhartha Mishra","raw_affiliation_strings":["Seminar for Applied Mathematics, ETH Z\u00fcrich, 8092 Z\u00fcrich, Switzerland"],"affiliations":[{"raw_affiliation_string":"Seminar for Applied Mathematics, ETH Z\u00fcrich, 8092 Z\u00fcrich, Switzerland","institution_ids":["https://openalex.org/I35440088"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5103051899","display_name":"Roberto Molinaro","orcid":"https://orcid.org/0000-0002-8549-6791"},"institutions":[{"id":"https://openalex.org/I35440088","display_name":"ETH Zurich","ror":"https://ror.org/05a28rw58","country_code":"CH","type":"education","lineage":["https://openalex.org/I2799323385","https://openalex.org/I35440088"]}],"countries":["CH"],"is_corresponding":false,"raw_author_name":"Roberto Molinaro","raw_affiliation_strings":["Seminar for Applied Mathematics, ETH Z\u00fcrich, 8092 Z\u00fcrich, Switzerland"],"affiliations":[{"raw_affiliation_string":"Seminar for Applied Mathematics, ETH Z\u00fcrich, 8092 Z\u00fcrich, Switzerland","institution_ids":["https://openalex.org/I35440088"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5053556239"],"corresponding_institution_ids":["https://openalex.org/I35440088"],"apc_list":null,"apc_paid":null,"fwci":10.2685,"has_fulltext":false,"cited_by_count":37,"citation_normalized_percentile":{"value":0.99056475,"is_in_top_1_percent":true,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":97,"max":100},"biblio":{"volume":"62","issue":"2","first_page":"811","last_page":"841"},"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/T10360","display_name":"Fluid Dynamics and Turbulent Flows","score":0.9907000064849854,"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":0.9696000218391418,"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/conservation-law","display_name":"Conservation law","score":0.9232928156852722},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7508565187454224},{"id":"https://openalex.org/keywords/entropy","display_name":"Entropy (arrow of time)","score":0.6557042598724365},{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.5922060608863831},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.5543705821037292},{"id":"https://openalex.org/keywords/residual","display_name":"Residual","score":0.4807176887989044},{"id":"https://openalex.org/keywords/hyperbolic-partial-differential-equation","display_name":"Hyperbolic partial differential equation","score":0.48053035140037537},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.4466785490512848},{"id":"https://openalex.org/keywords/scalar","display_name":"Scalar (mathematics)","score":0.4421469271183014},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.3530246913433075},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.3007217347621918},{"id":"https://openalex.org/keywords/partial-differential-equation","display_name":"Partial differential equation","score":0.21140459179878235},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.18136584758758545},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.12596622109413147},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.11915707588195801},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.0868181586265564}],"concepts":[{"id":"https://openalex.org/C3445786","wikidata":"https://www.wikidata.org/wiki/Q205805","display_name":"Conservation law","level":2,"score":0.9232928156852722},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7508565187454224},{"id":"https://openalex.org/C106301342","wikidata":"https://www.wikidata.org/wiki/Q4117933","display_name":"Entropy (arrow of time)","level":2,"score":0.6557042598724365},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.5922060608863831},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5543705821037292},{"id":"https://openalex.org/C155512373","wikidata":"https://www.wikidata.org/wiki/Q287450","display_name":"Residual","level":2,"score":0.4807176887989044},{"id":"https://openalex.org/C111615704","wikidata":"https://www.wikidata.org/wiki/Q2627459","display_name":"Hyperbolic partial differential equation","level":3,"score":0.48053035140037537},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.4466785490512848},{"id":"https://openalex.org/C57691317","wikidata":"https://www.wikidata.org/wiki/Q1289248","display_name":"Scalar (mathematics)","level":2,"score":0.4421469271183014},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.3530246913433075},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3007217347621918},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.21140459179878235},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.18136584758758545},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.12596622109413147},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.11915707588195801},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0868181586265564},{"id":"https://openalex.org/C119857082","wikidata":"https://www.wikidata.org/wiki/Q2539","display_name":"Machine learning","level":1,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1137/22m1522504","is_oa":false,"landing_page_url":"https://doi.org/10.1137/22m1522504","pdf_url":null,"source":{"id":"https://openalex.org/S203348814","display_name":"SIAM Journal on Numerical Analysis","issn_l":"0036-1429","issn":["0036-1429","1095-7170"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Numerical Analysis","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[{"score":0.7099999785423279,"id":"https://metadata.un.org/sdg/15","display_name":"Life in Land"}],"awards":[{"id":"https://openalex.org/G5256065697","display_name":null,"funder_award_id":"770880","funder_id":"https://openalex.org/F4320338335","funder_display_name":"H2020 European Research Council"}],"funders":[{"id":"https://openalex.org/F4320338335","display_name":"H2020 European Research Council","ror":"https://ror.org/0472cxd90"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":32,"referenced_works":["https://openalex.org/W1555627060","https://openalex.org/W1970454553","https://openalex.org/W2034511734","https://openalex.org/W2148666448","https://openalex.org/W2158985775","https://openalex.org/W2625995436","https://openalex.org/W2745110207","https://openalex.org/W2883486956","https://openalex.org/W2899283552","https://openalex.org/W2919115771","https://openalex.org/W2921773029","https://openalex.org/W2963146412","https://openalex.org/W2969381807","https://openalex.org/W2998366519","https://openalex.org/W3010839048","https://openalex.org/W3010849941","https://openalex.org/W3015865829","https://openalex.org/W3048421397","https://openalex.org/W3088991477","https://openalex.org/W3098546160","https://openalex.org/W3099661895","https://openalex.org/W3167031531","https://openalex.org/W3176116059","https://openalex.org/W3177136884","https://openalex.org/W3194468165","https://openalex.org/W3197473870","https://openalex.org/W3199775905","https://openalex.org/W3207890637","https://openalex.org/W4200119169","https://openalex.org/W4212988227","https://openalex.org/W4213444041","https://openalex.org/W4301116206"],"related_works":["https://openalex.org/W2094862772","https://openalex.org/W4382766439","https://openalex.org/W4388397561","https://openalex.org/W4287661942","https://openalex.org/W2595463258","https://openalex.org/W2380080472","https://openalex.org/W2765737651","https://openalex.org/W2476202684","https://openalex.org/W1989462402","https://openalex.org/W2792985587"],"abstract_inverted_index":{".Physics":[0],"informed":[1],"neural":[2,83],"networks":[3,84],"(PINNs)":[4],"require":[5],"regularity":[6],"of":[7,9,25,40,50,53,64,75],"solutions":[8,24,52,118],"the":[10,62,82,86,99],"underlying":[11],"PDE":[12],"to":[13,78,111],"guarantee":[14],"accurate":[15,48],"approximation.":[16],"Consequently,":[17],"they":[18],"may":[19],"fail":[20],"at":[21],"approximating":[22,61,85],"discontinuous":[23],"PDEs":[26],"such":[27],"as":[28,43,89,91],"nonlinear":[29],"hyperbolic":[30],"equations.":[31],"To":[32],"ameliorate":[33],"this,":[34],"we":[35],"propose":[36],"a":[37,65,70],"novel":[38],"variant":[39],"PINNs,":[41],"termed":[42],"weak":[44],"PINNs":[45],"(wPINNs)":[46],"for":[47,69,81],"approximation":[49],"entropy":[51,87,117],"scalar":[54],"conservation":[55,120],"laws.":[56],"wPINNs":[57,103,114],"are":[58],"based":[59],"on":[60,98],"solution":[63,88],"min-max":[66],"optimization":[67],"problem":[68],"residual,":[71],"defined":[72],"in":[73],"terms":[74],"Kruzkhov":[76],"entropies,":[77],"determine":[79],"parameters":[80],"well":[90],"test":[92],"functions.":[93],"We":[94],"prove":[95],"rigorous":[96],"bounds":[97],"error":[100],"incurred":[101],"by":[102],"and":[104],"illustrate":[105],"their":[106],"performance":[107],"through":[108],"numerical":[109],"experiments":[110],"demonstrate":[112],"that":[113],"can":[115],"approximate":[116],"accurately.KeywordsPINNshyperbolic":[119],"lawsdeep":[121],"learningMSC":[122],"codes65M99":[123]},"counts_by_year":[{"year":2026,"cited_by_count":13},{"year":2025,"cited_by_count":20},{"year":2024,"cited_by_count":4}],"updated_date":"2026-04-07T14:57:38.498316","created_date":"2025-10-10T00:00:00"}