{"id":"https://openalex.org/W2985558461","doi":"https://doi.org/10.1137/18m1192184","title":"Mean Field Analysis of Neural Networks: A Law of Large Numbers","display_name":"Mean Field Analysis of Neural Networks: A Law of Large Numbers","publication_year":2020,"publication_date":"2020-01-01","ids":{"openalex":"https://openalex.org/W2985558461","doi":"https://doi.org/10.1137/18m1192184","mag":"2985558461"},"language":"en","primary_location":{"id":"doi:10.1137/18m1192184","is_oa":false,"landing_page_url":"https://doi.org/10.1137/18m1192184","pdf_url":null,"source":{"id":"https://openalex.org/S77200987","display_name":"SIAM Journal on Applied Mathematics","issn_l":"0036-1399","issn":["0036-1399","1095-712X"],"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 Applied Mathematics","raw_type":"journal-article"},"type":"preprint","indexed_in":["arxiv","crossref","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/1805.01053","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5103033774","display_name":"Justin Sirignano","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Justin Sirignano","raw_affiliation_strings":["Industrial and Enterprise Systems Engineering"],"raw_orcid":"https://orcid.org/0000-0002-0971-1349","affiliations":[{"raw_affiliation_string":"Industrial and Enterprise Systems Engineering","institution_ids":[]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5103169334","display_name":"Konstantinos Spiliopoulos","orcid":"https://orcid.org/0000-0003-2021-6426"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Konstantinos Spiliopoulos","raw_affiliation_strings":["Boston Univ"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Boston Univ","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":2.1369,"has_fulltext":true,"cited_by_count":40,"citation_normalized_percentile":{"value":0.86931259,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":100},"biblio":{"volume":"80","issue":"2","first_page":"725","last_page":"752"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11206","display_name":"Model Reduction and Neural Networks","score":0.9988999962806702,"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":0.9988999962806702,"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/T10067","display_name":"Stochastic processes and financial applications","score":0.9847999811172485,"subfield":{"id":"https://openalex.org/subfields/2003","display_name":"Finance"},"field":{"id":"https://openalex.org/fields/20","display_name":"Economics, Econometrics and Finance"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},{"id":"https://openalex.org/T10320","display_name":"Neural Networks and Applications","score":0.9771000146865845,"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/artificial-neural-network","display_name":"Artificial neural network","score":0.8001249432563782},{"id":"https://openalex.org/keywords/stochastic-neural-network","display_name":"Stochastic neural network","score":0.6048316359519958},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.5956425666809082},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.554207980632782},{"id":"https://openalex.org/keywords/stochastic-gradient-descent","display_name":"Stochastic gradient descent","score":0.5240012407302856},{"id":"https://openalex.org/keywords/field","display_name":"Field (mathematics)","score":0.5116551518440247},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.49395519495010376},{"id":"https://openalex.org/keywords/cellular-neural-network","display_name":"Cellular neural network","score":0.43397068977355957},{"id":"https://openalex.org/keywords/time-delay-neural-network","display_name":"Time delay neural network","score":0.25877538323402405},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.2570006251335144},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.07847589254379272}],"concepts":[{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.8001249432563782},{"id":"https://openalex.org/C86582703","wikidata":"https://www.wikidata.org/wiki/Q7617824","display_name":"Stochastic neural network","level":4,"score":0.6048316359519958},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.5956425666809082},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.554207980632782},{"id":"https://openalex.org/C206688291","wikidata":"https://www.wikidata.org/wiki/Q7617819","display_name":"Stochastic gradient descent","level":3,"score":0.5240012407302856},{"id":"https://openalex.org/C9652623","wikidata":"https://www.wikidata.org/wiki/Q190109","display_name":"Field (mathematics)","level":2,"score":0.5116551518440247},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.49395519495010376},{"id":"https://openalex.org/C812465","wikidata":"https://www.wikidata.org/wiki/Q5058375","display_name":"Cellular neural network","level":3,"score":0.43397068977355957},{"id":"https://openalex.org/C175202392","wikidata":"https://www.wikidata.org/wiki/Q2434543","display_name":"Time delay neural network","level":3,"score":0.25877538323402405},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.2570006251335144},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.07847589254379272},{"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/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.0}],"mesh":[],"locations_count":5,"locations":[{"id":"doi:10.1137/18m1192184","is_oa":false,"landing_page_url":"https://doi.org/10.1137/18m1192184","pdf_url":null,"source":{"id":"https://openalex.org/S77200987","display_name":"SIAM Journal on Applied Mathematics","issn_l":"0036-1399","issn":["0036-1399","1095-712X"],"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 Applied Mathematics","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:1805.01053","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1805.01053","pdf_url":"https://arxiv.org/pdf/1805.01053","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":"","raw_type":"text"},{"id":"mag:2985558461","is_oa":true,"landing_page_url":"https://arxiv.org/pdf/1805.01053v4","pdf_url":null,"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":"arXiv (Cornell University)","raw_type":null},{"id":"pmh:oai:ora.ox.ac.uk:uuid:8840b583-5f3d-484f-8f63-134cd35be896","is_oa":false,"landing_page_url":"https://ora.ox.ac.uk/objects/uuid:8840b583-5f3d-484f-8f63-134cd35be896","pdf_url":null,"source":{"id":"https://openalex.org/S4306402636","display_name":"Oxford University Research Archive (ORA) (University of Oxford)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I40120149","host_organization_name":"University of Oxford","host_organization_lineage":["https://openalex.org/I40120149"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symplectic Elements","raw_type":"Journal article"},{"id":"doi:10.48550/arxiv.1805.01053","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.1805.01053","pdf_url":null,"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":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:1805.01053","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1805.01053","pdf_url":"https://arxiv.org/pdf/1805.01053","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":"","raw_type":"text"},"sustainable_development_goals":[{"display_name":"Peace, Justice and strong institutions","id":"https://metadata.un.org/sdg/16","score":0.5}],"awards":[{"id":"https://openalex.org/G5599753359","display_name":null,"funder_award_id":"DMS 1550918","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G945892359","display_name":"CAREER: Multiscale stochastic processes, Monte Carlo Methods and Irreversibility","funder_award_id":"1550918","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":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2985558461.pdf","grobid_xml":"https://content.openalex.org/works/W2985558461.grobid-xml"},"referenced_works_count":59,"referenced_works":["https://openalex.org/W125715901","https://openalex.org/W182704713","https://openalex.org/W576681630","https://openalex.org/W1019830208","https://openalex.org/W1522579744","https://openalex.org/W1677890546","https://openalex.org/W1973631422","https://openalex.org/W1988115241","https://openalex.org/W2012405657","https://openalex.org/W2016833839","https://openalex.org/W2027802883","https://openalex.org/W2063450328","https://openalex.org/W2071048859","https://openalex.org/W2080834695","https://openalex.org/W2099365847","https://openalex.org/W2112796928","https://openalex.org/W2126794261","https://openalex.org/W2130942839","https://openalex.org/W2137983211","https://openalex.org/W2145287260","https://openalex.org/W2145708596","https://openalex.org/W2156937773","https://openalex.org/W2257979135","https://openalex.org/W2261689926","https://openalex.org/W2281746805","https://openalex.org/W2342840547","https://openalex.org/W2345737627","https://openalex.org/W2471243427","https://openalex.org/W2525778437","https://openalex.org/W2530876040","https://openalex.org/W2534240011","https://openalex.org/W2557283755","https://openalex.org/W2575705757","https://openalex.org/W2581082771","https://openalex.org/W2591927543","https://openalex.org/W2736506089","https://openalex.org/W2749028154","https://openalex.org/W2772785876","https://openalex.org/W2790822776","https://openalex.org/W2798986185","https://openalex.org/W2804589149","https://openalex.org/W2919115771","https://openalex.org/W2921395510","https://openalex.org/W2963095610","https://openalex.org/W2963664410","https://openalex.org/W2963777360","https://openalex.org/W2963791871","https://openalex.org/W2963929190","https://openalex.org/W2964248288","https://openalex.org/W2966530573","https://openalex.org/W2977796275","https://openalex.org/W3102149128","https://openalex.org/W3123893832","https://openalex.org/W3124423280","https://openalex.org/W3124954191","https://openalex.org/W3146803896","https://openalex.org/W4236966694","https://openalex.org/W4248961346","https://openalex.org/W4285719527"],"related_works":["https://openalex.org/W2966530573","https://openalex.org/W2798986185","https://openalex.org/W2809090039","https://openalex.org/W2798826368","https://openalex.org/W2963095610","https://openalex.org/W2899748887","https://openalex.org/W2962698540","https://openalex.org/W2900959181","https://openalex.org/W2894604724","https://openalex.org/W2804589149","https://openalex.org/W2970217468","https://openalex.org/W2963417959","https://openalex.org/W2913010492","https://openalex.org/W2796744096","https://openalex.org/W2971043187","https://openalex.org/W2964052793","https://openalex.org/W2946963372","https://openalex.org/W2918794972","https://openalex.org/W2886067286","https://openalex.org/W2194775991"],"abstract_inverted_index":{"Machine":[0],"learning,":[1],"and":[2,15,40,68,95],"in":[3,23,36,46,85],"particular":[4],"neural":[5,29,54,60,83,114,138,154],"network":[6,93,115,155],"models,":[7],"have":[8],"revolutionized":[9],"fields":[10],"such":[11],"as":[12],"image,":[13],"text,":[14],"speech":[16],"recognition.":[17],"Today,":[18],"many":[19],"important":[20],"real-world":[21],"applications":[22,35],"these":[24],"areas":[25],"are":[26,32],"driven":[27],"by":[28],"networks.":[30,55,139],"There":[31],"also":[33],"growing":[34],"engineering,":[37],"robotics,":[38],"medicine,":[39],"finance.":[41],"Despite":[42],"their":[43],"immense":[44],"success":[45],"practice,":[47],"there":[48],"is":[49,147,162],"limited":[50],"mathematical":[51],"understanding":[52],"of":[53,74,89,99,112,121,134,144,152,166],"This":[56,127],"paper":[57],"illustrates":[58],"how":[59],"networks":[61,84],"can":[62,129],"be":[63,130],"studied":[64],"via":[65],"stochastic":[66,100],"analysis":[67,146],"develops":[69],"approaches":[70],"for":[71,137],"addressing":[72],"some":[73],"the":[75,86,109,113,119,149,153],"technical":[76],"challenges":[77],"which":[78,161],"arise.":[79],"We":[80,105],"analyze":[81],"one-layer":[82],"asymptotic":[87],"regime":[88],"simultaneously":[90],"(a)":[91],"large":[92,97,135],"sizes":[94],"(b)":[96],"numbers":[98,136],"gradient":[101],"descent":[102],"training":[103],"iterations.":[104],"rigorously":[106],"prove":[107],"that":[108,148],"empirical":[110],"distribution":[111],"parameters":[116,151],"converges":[117],"to":[118],"solution":[120],"a":[122,132,142,159],"nonlinear":[123],"partial":[124],"differential":[125],"equation.":[126],"result":[128],"considered":[131],"law":[133],"In":[140],"addition,":[141],"consequence":[143],"our":[145],"trained":[150],"asymptotically":[156],"become":[157],"independent,":[158],"property":[160],"commonly":[163],"called":[164],"\u201cpropagation":[165],"chaos.\u201d":[167]},"counts_by_year":[{"year":2025,"cited_by_count":4},{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":3},{"year":2020,"cited_by_count":11},{"year":2019,"cited_by_count":15},{"year":2018,"cited_by_count":4}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}