{"id":"https://openalex.org/W4402004978","doi":"https://doi.org/10.1038/s42256-024-00886-8","title":"Learning integral operators via neural integral equations","display_name":"Learning integral operators via neural integral equations","publication_year":2024,"publication_date":"2024-08-29","ids":{"openalex":"https://openalex.org/W4402004978","doi":"https://doi.org/10.1038/s42256-024-00886-8"},"language":"en","primary_location":{"id":"doi:10.1038/s42256-024-00886-8","is_oa":true,"landing_page_url":"https://doi.org/10.1038/s42256-024-00886-8","pdf_url":"https://www.nature.com/articles/s42256-024-00886-8.pdf","source":{"id":"https://openalex.org/S2912241403","display_name":"Nature Machine Intelligence","issn_l":"2522-5839","issn":["2522-5839"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319908","host_organization_name":"Nature Portfolio","host_organization_lineage":["https://openalex.org/P4310319908","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Nature Portfolio","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":"Nature Machine Intelligence","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://www.nature.com/articles/s42256-024-00886-8.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5006040096","display_name":"Emanuele Zappala","orcid":"https://orcid.org/0000-0002-9684-9441"},"institutions":[{"id":"https://openalex.org/I106969075","display_name":"Idaho State University","ror":"https://ror.org/0162z8b04","country_code":"US","type":"education","lineage":["https://openalex.org/I106969075"]},{"id":"https://openalex.org/I32971472","display_name":"Yale University","ror":"https://ror.org/03v76x132","country_code":"US","type":"education","lineage":["https://openalex.org/I32971472"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Emanuele Zappala","raw_affiliation_strings":["Department of Mathematics and Statistics (Idaho State University), and Yale School of Medicine (Yale University), New Haven, CT, USA"],"raw_orcid":"https://orcid.org/0000-0002-9684-9441","affiliations":[{"raw_affiliation_string":"Department of Mathematics and Statistics (Idaho State University), and Yale School of Medicine (Yale University), New Haven, CT, USA","institution_ids":["https://openalex.org/I106969075","https://openalex.org/I32971472"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5025440268","display_name":"Antonio Henrique de Oliveira Fonseca","orcid":null},"institutions":[{"id":"https://openalex.org/I32971472","display_name":"Yale University","ror":"https://ror.org/03v76x132","country_code":"US","type":"education","lineage":["https://openalex.org/I32971472"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Antonio Henrique de Oliveira Fonseca","raw_affiliation_strings":["Interdepartmental Neuroscience Program, Yale University, New Haven, CT, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Interdepartmental Neuroscience Program, Yale University, New Haven, CT, USA","institution_ids":["https://openalex.org/I32971472"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5083894958","display_name":"Josue Ortega","orcid":"https://orcid.org/0000-0001-7061-1112"},"institutions":[{"id":"https://openalex.org/I32971472","display_name":"Yale University","ror":"https://ror.org/03v76x132","country_code":"US","type":"education","lineage":["https://openalex.org/I32971472"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Josue Ortega Caro","raw_affiliation_strings":["Wu Tsai Institute, Yale University, New Haven, CT, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Wu Tsai Institute, Yale University, New Haven, CT, USA","institution_ids":["https://openalex.org/I32971472"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5068488561","display_name":"Andrew H. Moberly","orcid":null},"institutions":[{"id":"https://openalex.org/I32971472","display_name":"Yale University","ror":"https://ror.org/03v76x132","country_code":"US","type":"education","lineage":["https://openalex.org/I32971472"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Andrew Henry Moberly","raw_affiliation_strings":["Department of Neuroscience, Yale University, New Haven, CT, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Neuroscience, Yale University, New Haven, CT, USA","institution_ids":["https://openalex.org/I32971472"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5031151569","display_name":"Michael J. Higley","orcid":"https://orcid.org/0000-0002-5424-6730"},"institutions":[{"id":"https://openalex.org/I32971472","display_name":"Yale University","ror":"https://ror.org/03v76x132","country_code":"US","type":"education","lineage":["https://openalex.org/I32971472"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Michael James Higley","raw_affiliation_strings":["Department of Neuroscience and Department of Biomedical Engineering, Yale University, New Haven, CT, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Neuroscience and Department of Biomedical Engineering, Yale University, New Haven, CT, USA","institution_ids":["https://openalex.org/I32971472"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5057862535","display_name":"Jessica A. Cardin","orcid":"https://orcid.org/0000-0002-8209-5466"},"institutions":[{"id":"https://openalex.org/I32971472","display_name":"Yale University","ror":"https://ror.org/03v76x132","country_code":"US","type":"education","lineage":["https://openalex.org/I32971472"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Jessica Cardin","raw_affiliation_strings":["Department of Neuroscience and Department of Psychiatry, Yale university, New Haven, CT, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Neuroscience and Department of Psychiatry, Yale university, New Haven, CT, USA","institution_ids":["https://openalex.org/I32971472"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5019679682","display_name":"David van Dijk","orcid":"https://orcid.org/0000-0003-3911-9925"},"institutions":[{"id":"https://openalex.org/I32971472","display_name":"Yale University","ror":"https://ror.org/03v76x132","country_code":"US","type":"education","lineage":["https://openalex.org/I32971472"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"David van Dijk","raw_affiliation_strings":["Department of Computer Science, Yale University, New Haven, CT, USA"],"raw_orcid":"https://orcid.org/0000-0003-3911-9925","affiliations":[{"raw_affiliation_string":"Department of Computer Science, Yale University, New Haven, CT, USA","institution_ids":["https://openalex.org/I32971472"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":7,"corresponding_author_ids":["https://openalex.org/A5006040096"],"corresponding_institution_ids":["https://openalex.org/I106969075","https://openalex.org/I32971472"],"apc_list":{"value":9750,"currency":"EUR","value_usd":11690},"apc_paid":{"value":9750,"currency":"EUR","value_usd":11690},"fwci":3.9912,"has_fulltext":true,"cited_by_count":16,"citation_normalized_percentile":{"value":0.94433642,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":94,"max":99},"biblio":{"volume":"6","issue":"9","first_page":"1046","last_page":"1062"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11206","display_name":"Model Reduction and Neural Networks","score":0.9943000078201294,"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":0.9943000078201294,"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/T10288","display_name":"Fractional Differential Equations Solutions","score":0.9921000003814697,"subfield":{"id":"https://openalex.org/subfields/2611","display_name":"Modeling and Simulation"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11191","display_name":"Mathematical functions and polynomials","score":0.9690999984741211,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/integral-equation","display_name":"Integral equation","score":0.797346830368042},{"id":"https://openalex.org/keywords/fourier-integral-operator","display_name":"Fourier integral operator","score":0.6776933670043945},{"id":"https://openalex.org/keywords/volterra-integral-equation","display_name":"Volterra integral equation","score":0.6606287956237793},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.5111985802650452},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.5043236017227173},{"id":"https://openalex.org/keywords/operator","display_name":"Operator (biology)","score":0.48806846141815186},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.48260295391082764},{"id":"https://openalex.org/keywords/range","display_name":"Range (aeronautics)","score":0.47520145773887634},{"id":"https://openalex.org/keywords/solver","display_name":"Solver","score":0.45219478011131287},{"id":"https://openalex.org/keywords/operator-theory","display_name":"Operator theory","score":0.4157443642616272},{"id":"https://openalex.org/keywords/positive-definite-kernel","display_name":"Positive-definite kernel","score":0.41256946325302124},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.38509348034858704},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.2451072633266449},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.2118528187274933},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.13244298100471497}],"concepts":[{"id":"https://openalex.org/C27016315","wikidata":"https://www.wikidata.org/wiki/Q580101","display_name":"Integral equation","level":2,"score":0.797346830368042},{"id":"https://openalex.org/C191832335","wikidata":"https://www.wikidata.org/wiki/Q4262191","display_name":"Fourier integral operator","level":3,"score":0.6776933670043945},{"id":"https://openalex.org/C21965488","wikidata":"https://www.wikidata.org/wiki/Q2632613","display_name":"Volterra integral equation","level":3,"score":0.6606287956237793},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.5111985802650452},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.5043236017227173},{"id":"https://openalex.org/C17020691","wikidata":"https://www.wikidata.org/wiki/Q139677","display_name":"Operator (biology)","level":5,"score":0.48806846141815186},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.48260295391082764},{"id":"https://openalex.org/C204323151","wikidata":"https://www.wikidata.org/wiki/Q905424","display_name":"Range (aeronautics)","level":2,"score":0.47520145773887634},{"id":"https://openalex.org/C2778770139","wikidata":"https://www.wikidata.org/wiki/Q1966904","display_name":"Solver","level":2,"score":0.45219478011131287},{"id":"https://openalex.org/C43929395","wikidata":"https://www.wikidata.org/wiki/Q1198874","display_name":"Operator theory","level":2,"score":0.4157443642616272},{"id":"https://openalex.org/C117204833","wikidata":"https://www.wikidata.org/wiki/Q7233207","display_name":"Positive-definite kernel","level":4,"score":0.41256946325302124},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.38509348034858704},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.2451072633266449},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.2118528187274933},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.13244298100471497},{"id":"https://openalex.org/C158448853","wikidata":"https://www.wikidata.org/wiki/Q425218","display_name":"Repressor","level":4,"score":0.0},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C192562407","wikidata":"https://www.wikidata.org/wiki/Q228736","display_name":"Materials science","level":0,"score":0.0},{"id":"https://openalex.org/C104317684","wikidata":"https://www.wikidata.org/wiki/Q7187","display_name":"Gene","level":2,"score":0.0},{"id":"https://openalex.org/C86339819","wikidata":"https://www.wikidata.org/wiki/Q407384","display_name":"Transcription factor","level":3,"score":0.0},{"id":"https://openalex.org/C159985019","wikidata":"https://www.wikidata.org/wiki/Q181790","display_name":"Composite material","level":1,"score":0.0},{"id":"https://openalex.org/C55493867","wikidata":"https://www.wikidata.org/wiki/Q7094","display_name":"Biochemistry","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1038/s42256-024-00886-8","is_oa":true,"landing_page_url":"https://doi.org/10.1038/s42256-024-00886-8","pdf_url":"https://www.nature.com/articles/s42256-024-00886-8.pdf","source":{"id":"https://openalex.org/S2912241403","display_name":"Nature Machine Intelligence","issn_l":"2522-5839","issn":["2522-5839"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319908","host_organization_name":"Nature Portfolio","host_organization_lineage":["https://openalex.org/P4310319908","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Nature Portfolio","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":"Nature Machine Intelligence","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.1038/s42256-024-00886-8","is_oa":true,"landing_page_url":"https://doi.org/10.1038/s42256-024-00886-8","pdf_url":"https://www.nature.com/articles/s42256-024-00886-8.pdf","source":{"id":"https://openalex.org/S2912241403","display_name":"Nature Machine Intelligence","issn_l":"2522-5839","issn":["2522-5839"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319908","host_organization_name":"Nature Portfolio","host_organization_lineage":["https://openalex.org/P4310319908","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Nature Portfolio","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":"Nature Machine Intelligence","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G630046399","display_name":null,"funder_award_id":"EY031133","funder_id":"https://openalex.org/F4320332161","funder_display_name":"National Institutes of Health"},{"id":"https://openalex.org/G8168324890","display_name":null,"funder_award_id":"EY026878","funder_id":"https://openalex.org/F4320332161","funder_display_name":"National Institutes of Health"}],"funders":[{"id":"https://openalex.org/F4320308380","display_name":"Yale University","ror":"https://ror.org/03v76x132"},{"id":"https://openalex.org/F4320315019","display_name":"Simons Foundation Autism Research Initiative","ror":"https://ror.org/01cmst727"},{"id":"https://openalex.org/F4320321091","display_name":"Coordena\u00e7\u00e3o de Aperfei\u00e7oamento de Pessoal de N\u00edvel Superior","ror":"https://ror.org/00x0ma614"},{"id":"https://openalex.org/F4320332161","display_name":"National Institutes of Health","ror":"https://ror.org/01cwqze88"}],"has_content":{"grobid_xml":false,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4402004978.pdf"},"referenced_works_count":56,"referenced_works":["https://openalex.org/W175461187","https://openalex.org/W1530700356","https://openalex.org/W1661892988","https://openalex.org/W1982115588","https://openalex.org/W1996097836","https://openalex.org/W1998041037","https://openalex.org/W2018495737","https://openalex.org/W2029374903","https://openalex.org/W2064519517","https://openalex.org/W2076077791","https://openalex.org/W2078134984","https://openalex.org/W2089210221","https://openalex.org/W2137983211","https://openalex.org/W2167845354","https://openalex.org/W2223847291","https://openalex.org/W2476507202","https://openalex.org/W2478495805","https://openalex.org/W2531136704","https://openalex.org/W2566965042","https://openalex.org/W2878370351","https://openalex.org/W2898801573","https://openalex.org/W2913085834","https://openalex.org/W2934992892","https://openalex.org/W2979786244","https://openalex.org/W2983592652","https://openalex.org/W2983902802","https://openalex.org/W3010501116","https://openalex.org/W3027912155","https://openalex.org/W3041725488","https://openalex.org/W3092923133","https://openalex.org/W3096414164","https://openalex.org/W3126301767","https://openalex.org/W3128976935","https://openalex.org/W3171353004","https://openalex.org/W3189185287","https://openalex.org/W3194436063","https://openalex.org/W3197395406","https://openalex.org/W3207021380","https://openalex.org/W3216107495","https://openalex.org/W4220824047","https://openalex.org/W4294170298","https://openalex.org/W4310151068","https://openalex.org/W4310413705","https://openalex.org/W4313125078","https://openalex.org/W4313178187","https://openalex.org/W4322759863","https://openalex.org/W4379089545","https://openalex.org/W4384807914","https://openalex.org/W4398225916","https://openalex.org/W4402494745","https://openalex.org/W6739901393","https://openalex.org/W6838718877","https://openalex.org/W6939880229","https://openalex.org/W6958244976","https://openalex.org/W6958614160","https://openalex.org/W6976886995"],"related_works":["https://openalex.org/W2517687681","https://openalex.org/W4401089097","https://openalex.org/W2964031424","https://openalex.org/W325682659","https://openalex.org/W4245032361","https://openalex.org/W4317832240","https://openalex.org/W1775501858","https://openalex.org/W4252257134","https://openalex.org/W2503953152","https://openalex.org/W4236125184"],"abstract_inverted_index":{"Abstract":[0],"Nonlinear":[1],"operators":[2,19,53,114,203],"with":[3,79,214],"long-distance":[4],"spatiotemporal":[5],"dependencies":[6],"are":[7,83],"fundamental":[8],"in":[9,22,35,85,198],"modelling":[10],"complex":[11,212],"systems":[12,213],"across":[13],"sciences;":[14],"yet,":[15],"learning":[16,50,200],"these":[17],"non-local":[18,30,202],"remains":[20],"challenging":[21],"machine":[23,199],"learning.":[24],"Integral":[25],"equations,":[26,91,151,159],"which":[27,75],"model":[28,65],"such":[29],"systems,":[31],"have":[32],"wide-ranging":[33],"applications":[34],"physics,":[36],"chemistry,":[37],"biology":[38],"and":[39,64,98,126,143,149,157,165,189,204],"engineering.":[40],"We":[41,103],"introduce":[42],"the":[43,70,77,86,93,100,162],"neural":[44,72,178],"integral":[45,52,58,73,78,90,101,113,136,158,179],"equation,":[46,74],"a":[47,105,195,206],"method":[48],"for":[49,135,185,201,209],"unknown":[51,211],"from":[54],"data":[55],"using":[56],"an":[57],"equation":[59],"solver.":[60],"To":[61],"improve":[62],"scalability":[63],"capacity,":[66],"we":[67,160],"also":[68],"present":[69],"attentional":[71,177],"replaces":[76],"self-attention.":[80],"Both":[81],"models":[82],"grounded":[84],"theory":[87],"of":[88],"second-kind":[89],"where":[92],"indeterminate":[94],"appears":[95],"both":[96],"inside":[97],"outside":[99],"operator.":[102],"provide":[104],"theoretical":[106],"analysis":[107],"showing":[108],"how":[109],"self-attention":[110],"can":[111],"approximate":[112],"under":[115],"mild":[116],"regularity":[117],"assumptions,":[118],"further":[119],"deepening":[120],"previously":[121],"reported":[122],"connections":[123],"between":[124],"transformers":[125],"integration,":[127],"as":[128,130,152,154],"well":[129,153],"deriving":[131],"corresponding":[132],"approximation":[133],"results":[134],"operators.":[137],"Through":[138],"numerical":[139],"benchmarks":[140],"on":[141],"synthetic":[142],"real-world":[144],"data,":[145],"including":[146],"Lotka\u2013Volterra,":[147],"Navier\u2013Stokes":[148],"Burgers\u2019":[150],"brain":[155],"dynamics":[156,171],"showcase":[161],"models\u2019":[163],"capabilities":[164],"their":[166],"ability":[167],"to":[168],"derive":[169],"interpretable":[170],"embeddings.":[172],"Our":[173,192],"experiments":[174],"demonstrate":[175],"that":[176],"equations":[180],"outperform":[181],"existing":[182],"methods,":[183],"especially":[184],"longer":[186],"time":[187],"intervals":[188],"higher-dimensional":[190],"problems.":[191],"work":[193],"addresses":[194],"critical":[196],"gap":[197],"offers":[205],"powerful":[207],"tool":[208],"studying":[210],"long-range":[215],"dependencies.":[216]},"counts_by_year":[{"year":2026,"cited_by_count":4},{"year":2025,"cited_by_count":10},{"year":2024,"cited_by_count":2}],"updated_date":"2026-06-13T06:13:01.061226","created_date":"2024-08-30T00:00:00"}
