{"id":"https://openalex.org/W4390188840","doi":"https://doi.org/10.1109/hpec58863.2023.10363574","title":"Continuous Deep Equilibrium Models: Training Neural ODEs Faster by Integrating Them to Infinity","display_name":"Continuous Deep Equilibrium Models: Training Neural ODEs Faster by Integrating Them to Infinity","publication_year":2023,"publication_date":"2023-09-25","ids":{"openalex":"https://openalex.org/W4390188840","doi":"https://doi.org/10.1109/hpec58863.2023.10363574"},"language":"en","primary_location":{"id":"doi:10.1109/hpec58863.2023.10363574","is_oa":false,"landing_page_url":"https://doi.org/10.1109/hpec58863.2023.10363574","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"2023 IEEE High Performance Extreme Computing Conference (HPEC)","raw_type":"proceedings-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/A5101976851","display_name":"Avik Pal","orcid":"https://orcid.org/0000-0003-1171-743X"},"institutions":[{"id":"https://openalex.org/I4210110987","display_name":"IIT@MIT","ror":"https://ror.org/01wp8zh54","country_code":"US","type":"facility","lineage":["https://openalex.org/I30771326","https://openalex.org/I4210110987"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Avik Pal","raw_affiliation_strings":["MIT,Electrical Engg. &#x0026; Computer Science,Cambridge,U.S.A"],"affiliations":[{"raw_affiliation_string":"MIT,Electrical Engg. &#x0026; Computer Science,Cambridge,U.S.A","institution_ids":["https://openalex.org/I4210110987"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5029673947","display_name":"Alan Edelman","orcid":"https://orcid.org/0000-0001-7676-3133"},"institutions":[{"id":"https://openalex.org/I4210110987","display_name":"IIT@MIT","ror":"https://ror.org/01wp8zh54","country_code":"US","type":"facility","lineage":["https://openalex.org/I30771326","https://openalex.org/I4210110987"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Alan Edelman","raw_affiliation_strings":["Mathematics MIT,Cambridge,U.S.A","Mathematics MIT, Cambridge, U.S.A"],"affiliations":[{"raw_affiliation_string":"Mathematics MIT,Cambridge,U.S.A","institution_ids":["https://openalex.org/I4210110987"]},{"raw_affiliation_string":"Mathematics MIT, Cambridge, U.S.A","institution_ids":["https://openalex.org/I4210110987"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5043195923","display_name":"Christopher Rackauckas","orcid":"https://orcid.org/0000-0001-5850-0663"},"institutions":[{"id":"https://openalex.org/I4210110987","display_name":"IIT@MIT","ror":"https://ror.org/01wp8zh54","country_code":"US","type":"facility","lineage":["https://openalex.org/I30771326","https://openalex.org/I4210110987"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Chris Rackauckas","raw_affiliation_strings":["Mathematics MIT,Cambridge,U.S.A","Mathematics MIT, Cambridge, U.S.A"],"affiliations":[{"raw_affiliation_string":"Mathematics MIT,Cambridge,U.S.A","institution_ids":["https://openalex.org/I4210110987"]},{"raw_affiliation_string":"Mathematics MIT, Cambridge, U.S.A","institution_ids":["https://openalex.org/I4210110987"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5101976851"],"corresponding_institution_ids":["https://openalex.org/I4210110987"],"apc_list":null,"apc_paid":null,"fwci":0.6503,"has_fulltext":false,"cited_by_count":4,"citation_normalized_percentile":{"value":0.67653236,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":95,"max":99},"biblio":{"volume":null,"issue":null,"first_page":"1","last_page":"9"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11206","display_name":"Model Reduction and Neural Networks","score":0.9987000226974487,"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":0.9987000226974487,"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":0.9894000291824341,"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/T10462","display_name":"Reinforcement Learning in Robotics","score":0.9825000166893005,"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/ode","display_name":"Ode","score":0.7068368196487427},{"id":"https://openalex.org/keywords/infinity","display_name":"Infinity","score":0.6596700549125671},{"id":"https://openalex.org/keywords/training","display_name":"Training (meteorology)","score":0.5404887795448303},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.512969434261322},{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.49921655654907227},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.40803465247154236},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.357828289270401},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.2908160388469696},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.15060311555862427},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.1299038529396057},{"id":"https://openalex.org/keywords/meteorology","display_name":"Meteorology","score":0.059336572885513306}],"concepts":[{"id":"https://openalex.org/C34862557","wikidata":"https://www.wikidata.org/wiki/Q178985","display_name":"Ode","level":2,"score":0.7068368196487427},{"id":"https://openalex.org/C7321624","wikidata":"https://www.wikidata.org/wiki/Q205","display_name":"Infinity","level":2,"score":0.6596700549125671},{"id":"https://openalex.org/C2777211547","wikidata":"https://www.wikidata.org/wiki/Q17141490","display_name":"Training (meteorology)","level":2,"score":0.5404887795448303},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.512969434261322},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.49921655654907227},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.40803465247154236},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.357828289270401},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.2908160388469696},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.15060311555862427},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.1299038529396057},{"id":"https://openalex.org/C153294291","wikidata":"https://www.wikidata.org/wiki/Q25261","display_name":"Meteorology","level":1,"score":0.059336572885513306}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1109/hpec58863.2023.10363574","is_oa":false,"landing_page_url":"https://doi.org/10.1109/hpec58863.2023.10363574","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"2023 IEEE High Performance Extreme Computing Conference (HPEC)","raw_type":"proceedings-article"}],"best_oa_location":null,"sustainable_development_goals":[{"score":0.6299999952316284,"id":"https://metadata.un.org/sdg/13","display_name":"Climate action"}],"awards":[{"id":"https://openalex.org/G3857680246","display_name":null,"funder_award_id":"OAC-1835443,SII-2029670,ECCS-2029670,OAC-2103804,PHY-2021825","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":52,"referenced_works":["https://openalex.org/W1522301498","https://openalex.org/W1583837637","https://openalex.org/W1976068300","https://openalex.org/W1977822596","https://openalex.org/W1995504856","https://openalex.org/W2022508996","https://openalex.org/W2108598243","https://openalex.org/W2112796928","https://openalex.org/W2116055501","https://openalex.org/W2120575449","https://openalex.org/W2140153041","https://openalex.org/W2335728318","https://openalex.org/W2412782625","https://openalex.org/W2619381903","https://openalex.org/W2902809050","https://openalex.org/W2908510526","https://openalex.org/W2948977392","https://openalex.org/W2958796454","https://openalex.org/W2962891704","https://openalex.org/W2963641970","https://openalex.org/W2963755523","https://openalex.org/W2963840672","https://openalex.org/W2981603589","https://openalex.org/W2999026783","https://openalex.org/W3005235017","https://openalex.org/W3034634365","https://openalex.org/W3035935089","https://openalex.org/W3040996572","https://openalex.org/W3118608800","https://openalex.org/W3125148525","https://openalex.org/W3176131920","https://openalex.org/W3201167453","https://openalex.org/W3204483762","https://openalex.org/W3207314135","https://openalex.org/W4205799487","https://openalex.org/W4231697575","https://openalex.org/W4234650083","https://openalex.org/W4244155866","https://openalex.org/W4285607191","https://openalex.org/W4289143709","https://openalex.org/W6621110542","https://openalex.org/W6631190155","https://openalex.org/W6686368498","https://openalex.org/W6696085341","https://openalex.org/W6734122017","https://openalex.org/W6752307458","https://openalex.org/W6755056417","https://openalex.org/W6757817989","https://openalex.org/W6789981588","https://openalex.org/W6795275719","https://openalex.org/W6796055584","https://openalex.org/W6802475320"],"related_works":["https://openalex.org/W2371448224","https://openalex.org/W3134495997","https://openalex.org/W2363680170","https://openalex.org/W2376428685","https://openalex.org/W4256082577","https://openalex.org/W625783435","https://openalex.org/W2048742619","https://openalex.org/W2384641672","https://openalex.org/W2742179975","https://openalex.org/W2362647823"],"abstract_inverted_index":{"Implicit":[0],"models":[1],"separate":[2],"the":[3,9,45,48,52,65,81,90,97,142,156,166],"definition":[4],"of":[5,11,47,56,92,101,158,168],"a":[6,69,85],"layer":[7,118],"from":[8],"description":[10],"its":[12,34],"solution":[13],"process.":[14],"While":[15],"implicit":[16,93,128,159],"layers":[17,94],"allow":[18],"features":[19],"such":[20],"as":[21],"depth":[22],"to":[23,26,38,87],"adapt":[24],"automatically":[25],"new":[27],"scenarios":[28],"and":[29,111,139,160],"inputs,":[30],"this":[31,41,134,151],"adaptivity":[32],"makes":[33],"computational":[35,99],"expense":[36,100],"challenging":[37],"predict.":[39],"In":[40],"manuscript,":[42],"we":[43,79,108],"increase":[44],"\u201cimplicitness\u201d":[46],"DEQ":[49],"by":[50,73,126,145],"redefining":[51],"method":[53],"in":[54],"terms":[55],"an":[57,102,115,122,127],"infinite":[58],"time":[59,144],"neural":[60,71],"ODE,":[61],"which":[62],"paradoxically":[63],"decreases":[64,141],"training":[66,133,143],"cost":[67],"over":[68],"standard":[70],"ODE":[72],"<tex":[74,146],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[75,147],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">$2-4\\times$</tex>":[76],".":[77,149],"Additionally,":[78],"address":[80],"question:":[82],"is":[83,137],"there":[84],"way":[86],"simultaneously":[88,120],"achieve":[89],"robustness":[91],"while":[95],"allowing":[96],"reduced":[98],"explicit":[103,123,135,161],"layer?":[104],"To":[105],"solve":[106],"this,":[107],"develop":[109],"Skip":[110,112],"Reg.":[113],"DEQ,":[114],"implicit-explicit":[116],"(IMEX)":[117],"that":[119,132],"trains":[121],"prediction":[124],"followed":[125],"correction.":[129],"We":[130],"show":[131],"predictor":[136],"free":[138],"even":[140],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">$1.11-3.19\\times$</tex>":[148],"Together,":[150],"manuscript":[152],"shows":[153],"how":[154],"bridging":[155],"dichotomy":[157],"deep":[162],"learning":[163],"can":[164],"combine":[165],"advantages":[167],"both":[169],"techniques.":[170]},"counts_by_year":[{"year":2026,"cited_by_count":2},{"year":2025,"cited_by_count":2}],"updated_date":"2026-04-15T08:11:43.952461","created_date":"2025-10-10T00:00:00"}