{"id":"https://openalex.org/W4282837570","doi":"https://doi.org/10.1137/20m1344986","title":"Neural Parametric Fokker--Planck Equation","display_name":"Neural Parametric Fokker--Planck Equation","publication_year":2022,"publication_date":"2022-06-01","ids":{"openalex":"https://openalex.org/W4282837570","doi":"https://doi.org/10.1137/20m1344986"},"language":"en","primary_location":{"id":"doi:10.1137/20m1344986","is_oa":false,"landing_page_url":"https://doi.org/10.1137/20m1344986","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/A5100453848","display_name":"Shu Liu","orcid":"https://orcid.org/0000-0001-6907-8787"},"institutions":[{"id":"https://openalex.org/I130701444","display_name":"Georgia Institute of Technology","ror":"https://ror.org/01zkghx44","country_code":"US","type":"education","lineage":["https://openalex.org/I130701444"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Shu Liu","raw_affiliation_strings":["School of Mathematics, Georgia Institute of Technology, Atlanta, GA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematics, Georgia Institute of Technology, Atlanta, GA","institution_ids":["https://openalex.org/I130701444"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5043410025","display_name":"Wuchen Li","orcid":"https://orcid.org/0000-0002-2218-5734"},"institutions":[{"id":"https://openalex.org/I155781252","display_name":"University of South Carolina","ror":"https://ror.org/02b6qw903","country_code":"US","type":"education","lineage":["https://openalex.org/I155781252"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Wuchen Li","raw_affiliation_strings":["Department of Mathematics, University of South Carolina, Columbia, SC 29208 USA"],"raw_orcid":"https://orcid.org/0000-0002-2218-5734","affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of South Carolina, Columbia, SC 29208 USA","institution_ids":["https://openalex.org/I155781252"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5046703129","display_name":"Hongyuan Zha","orcid":"https://orcid.org/0000-0001-7493-0911"},"institutions":[{"id":"https://openalex.org/I4210099586","display_name":"Shenzhen Research Institute of Big Data","ror":"https://ror.org/00z1gwf89","country_code":"CN","type":"education","lineage":["https://openalex.org/I4210099586"]},{"id":"https://openalex.org/I4210116924","display_name":"Chinese University of Hong Kong, Shenzhen","ror":"https://ror.org/02d5ks197","country_code":"CN","type":"education","lineage":["https://openalex.org/I177725633","https://openalex.org/I180726961","https://openalex.org/I4210116924"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Hongyuan Zha","raw_affiliation_strings":["School of Data Science, Shenzhen Research Institute of Big Data, The Chinese University of Hong Kong, Shenzhen, China, 518172"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Data Science, Shenzhen Research Institute of Big Data, The Chinese University of Hong Kong, Shenzhen, China, 518172","institution_ids":["https://openalex.org/I4210116924","https://openalex.org/I4210099586"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5102878627","display_name":"Haomin Zhou","orcid":"https://orcid.org/0000-0001-7647-2600"},"institutions":[{"id":"https://openalex.org/I130701444","display_name":"Georgia Institute of Technology","ror":"https://ror.org/01zkghx44","country_code":"US","type":"education","lineage":["https://openalex.org/I130701444"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Haomin Zhou","raw_affiliation_strings":["School of Mathematics, Georgia Institute of Technology, Atlanta, GA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematics, Georgia Institute of Technology, Atlanta, GA","institution_ids":["https://openalex.org/I130701444"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5100453848"],"corresponding_institution_ids":["https://openalex.org/I130701444"],"apc_list":null,"apc_paid":null,"fwci":2.0029,"has_fulltext":false,"cited_by_count":20,"citation_normalized_percentile":{"value":0.86605164,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":96,"max":100},"biblio":{"volume":"60","issue":"3","first_page":"1385","last_page":"1449"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11206","display_name":"Model Reduction and Neural Networks","score":0.9995999932289124,"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":0.9995999932289124,"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/T10928","display_name":"Probabilistic and Robust Engineering Design","score":0.9969000220298767,"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"}},{"id":"https://openalex.org/T12814","display_name":"Gaussian Processes and Bayesian Inference","score":0.9804999828338623,"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/fokker\u2013planck-equation","display_name":"Fokker\u2013Planck equation","score":0.8478221893310547},{"id":"https://openalex.org/keywords/ode","display_name":"Ode","score":0.7249253988265991},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6576313972473145},{"id":"https://openalex.org/keywords/discretization","display_name":"Discretization","score":0.6167158484458923},{"id":"https://openalex.org/keywords/divergence","display_name":"Divergence (linguistics)","score":0.5850094556808472},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.5412907004356384},{"id":"https://openalex.org/keywords/ordinary-differential-equation","display_name":"Ordinary differential equation","score":0.5096356868743896},{"id":"https://openalex.org/keywords/parametric-statistics","display_name":"Parametric statistics","score":0.4890216290950775},{"id":"https://openalex.org/keywords/balanced-flow","display_name":"Balanced flow","score":0.44920575618743896},{"id":"https://openalex.org/keywords/flow","display_name":"Flow (mathematics)","score":0.44776707887649536},{"id":"https://openalex.org/keywords/partial-differential-equation","display_name":"Partial differential equation","score":0.4388121962547302},{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.4338149130344391},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.4329095482826233},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.36239057779312134},{"id":"https://openalex.org/keywords/differential-equation","display_name":"Differential equation","score":0.35472315549850464},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.18524014949798584},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.12847214937210083},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.12220531702041626}],"concepts":[{"id":"https://openalex.org/C69123182","wikidata":"https://www.wikidata.org/wiki/Q891766","display_name":"Fokker\u2013Planck equation","level":3,"score":0.8478221893310547},{"id":"https://openalex.org/C34862557","wikidata":"https://www.wikidata.org/wiki/Q178985","display_name":"Ode","level":2,"score":0.7249253988265991},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6576313972473145},{"id":"https://openalex.org/C73000952","wikidata":"https://www.wikidata.org/wiki/Q17007827","display_name":"Discretization","level":2,"score":0.6167158484458923},{"id":"https://openalex.org/C207390915","wikidata":"https://www.wikidata.org/wiki/Q1230525","display_name":"Divergence (linguistics)","level":2,"score":0.5850094556808472},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5412907004356384},{"id":"https://openalex.org/C51544822","wikidata":"https://www.wikidata.org/wiki/Q465274","display_name":"Ordinary differential equation","level":3,"score":0.5096356868743896},{"id":"https://openalex.org/C117251300","wikidata":"https://www.wikidata.org/wiki/Q1849855","display_name":"Parametric statistics","level":2,"score":0.4890216290950775},{"id":"https://openalex.org/C167879884","wikidata":"https://www.wikidata.org/wiki/Q727568","display_name":"Balanced flow","level":2,"score":0.44920575618743896},{"id":"https://openalex.org/C38349280","wikidata":"https://www.wikidata.org/wiki/Q1434290","display_name":"Flow (mathematics)","level":2,"score":0.44776707887649536},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.4388121962547302},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.4338149130344391},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.4329095482826233},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.36239057779312134},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.35472315549850464},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.18524014949798584},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.12847214937210083},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.12220531702041626},{"id":"https://openalex.org/C50522688","wikidata":"https://www.wikidata.org/wiki/Q189833","display_name":"Economic growth","level":1,"score":0.0},{"id":"https://openalex.org/C41895202","wikidata":"https://www.wikidata.org/wiki/Q8162","display_name":"Linguistics","level":1,"score":0.0},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1137/20m1344986","is_oa":false,"landing_page_url":"https://doi.org/10.1137/20m1344986","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":[],"awards":[{"id":"https://openalex.org/G1696347033","display_name":null,"funder_award_id":"DMS-1620345","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G5796604383","display_name":null,"funder_award_id":"RTG:2038080","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G6624031046","display_name":null,"funder_award_id":"N000141310408","funder_id":"https://openalex.org/F4320337345","funder_display_name":"Office of Naval Research"},{"id":"https://openalex.org/G7487623949","display_name":null,"funder_award_id":"DMS-1830225","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"},{"id":"https://openalex.org/F4320310846","display_name":"University of South Carolina","ror":"https://ror.org/02b6qw903"},{"id":"https://openalex.org/F4320331102","display_name":"Shenzhen Research Institute of Big Data","ror":"https://ror.org/00z1gwf89"},{"id":"https://openalex.org/F4320337345","display_name":"Office of Naval Research","ror":"https://ror.org/00rk2pe57"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":82,"referenced_works":["https://openalex.org/W385466589","https://openalex.org/W1174634663","https://openalex.org/W1522301498","https://openalex.org/W1522579744","https://openalex.org/W1555683961","https://openalex.org/W1566694705","https://openalex.org/W1585160083","https://openalex.org/W1591074427","https://openalex.org/W1970789124","https://openalex.org/W1974932891","https://openalex.org/W1976284739","https://openalex.org/W1980300394","https://openalex.org/W1983452151","https://openalex.org/W2003297161","https://openalex.org/W2005047575","https://openalex.org/W2013436022","https://openalex.org/W2026258334","https://openalex.org/W2027258435","https://openalex.org/W2027399303","https://openalex.org/W2047352485","https://openalex.org/W2054723044","https://openalex.org/W2071048859","https://openalex.org/W2080993911","https://openalex.org/W2084233335","https://openalex.org/W2119323323","https://openalex.org/W2130401121","https://openalex.org/W2155894447","https://openalex.org/W2156387975","https://openalex.org/W2167433878","https://openalex.org/W2320126040","https://openalex.org/W2323824333","https://openalex.org/W2401363764","https://openalex.org/W2483948083","https://openalex.org/W2485135680","https://openalex.org/W2511837229","https://openalex.org/W2523246573","https://openalex.org/W2528305538","https://openalex.org/W2555799101","https://openalex.org/W2559655401","https://openalex.org/W2625995436","https://openalex.org/W2739748921","https://openalex.org/W2792596237","https://openalex.org/W2798826368","https://openalex.org/W2806862445","https://openalex.org/W2809090039","https://openalex.org/W2885496552","https://openalex.org/W2899283552","https://openalex.org/W2913214425","https://openalex.org/W2944394121","https://openalex.org/W2952716587","https://openalex.org/W2963090522","https://openalex.org/W2963139910","https://openalex.org/W2963146412","https://openalex.org/W2963172141","https://openalex.org/W2963645788","https://openalex.org/W2964110066","https://openalex.org/W2967196698","https://openalex.org/W2969211630","https://openalex.org/W2992915149","https://openalex.org/W3032032567","https://openalex.org/W3098175809","https://openalex.org/W3098813942","https://openalex.org/W3099661895","https://openalex.org/W3103271357","https://openalex.org/W3104332327","https://openalex.org/W3158386662","https://openalex.org/W3180106877","https://openalex.org/W3186981831","https://openalex.org/W4212774754","https://openalex.org/W4229827205","https://openalex.org/W4230394162","https://openalex.org/W4231021340","https://openalex.org/W4236312485","https://openalex.org/W4236994424","https://openalex.org/W4238063566","https://openalex.org/W4238209390","https://openalex.org/W4240788980","https://openalex.org/W4248701743","https://openalex.org/W4251193772","https://openalex.org/W4297736277","https://openalex.org/W4299828299","https://openalex.org/W4320013936"],"related_works":["https://openalex.org/W3082668976","https://openalex.org/W2517973022","https://openalex.org/W1543121148","https://openalex.org/W1580703421","https://openalex.org/W2996841952","https://openalex.org/W2618518959","https://openalex.org/W2231364979","https://openalex.org/W4295745414","https://openalex.org/W4288093367","https://openalex.org/W4382202915"],"abstract_inverted_index":{"In":[0],"this":[1],"paper,":[2],"we":[3,107,138],"develop":[4],"and":[5,163,179],"analyze":[6],"numerical":[7,105,167],"methods":[8],"for":[9,113,142,159],"high-dimensional":[10],"Fokker--Planck":[11,28,59,65,131,151],"equations":[12,36,132],"by":[13,101],"leveraging":[14],"generative":[15,47],"models":[16,48],"from":[17,46],"deep":[18],"learning.":[19],"Our":[20],"starting":[21],"point":[22],"is":[23,124],"a":[24,31,109],"formulation":[25],"of":[26,33,75,91,97,117,147,175],"the":[27,43,64,71,83,86,95,114,118,130,143,148,156,161,173,176],"equation":[29,66,152],"as":[30,50,70,85,153,155],"system":[32],"ordinary":[34],"differential":[35],"(ODEs)":[37],"on":[38,94],"finite-dimensional":[39],"parameter":[40],"space":[41],"with":[42],"parameters":[44],"inherited":[45],"such":[49,55],"normalizing":[51],"flows.":[52],"We":[53],"call":[54],"ODEs":[56,84],"neural":[57,102,149],"parametric":[58,150],"equations.":[60],"The":[61],"fact":[62],"that":[63],"can":[67,127],"be":[68],"viewed":[69],"$L^2$-Wasserstein":[72,88],"gradient":[73,89],"flow":[74,90],"Kullback--Leibler":[76],"(KL)":[77],"divergence":[78,93],"allows":[79],"us":[80],"to":[81,171],"derive":[82],"constrained":[87],"KL":[92],"set":[96],"probability":[98],"densities":[99],"generated":[100],"networks.":[103],"For":[104],"computation,":[106],"design":[108],"variational":[110],"semi-implicit":[111],"scheme":[112],"time":[115],"discretization":[116],"proposed":[119,177],"ODE.":[120],"Such":[121],"an":[122],"algorithm":[123],"sampling-based,":[125],"which":[126],"readily":[128],"handle":[129],"in":[133],"higher":[134],"dimensional":[135],"spaces.":[136],"Moreover,":[137],"also":[139],"establish":[140],"bounds":[141],"asymptotic":[144],"convergence":[145],"analysis":[146,158],"well":[154],"error":[157],"both":[160],"continuous":[162],"discrete":[164],"versions.":[165],"Several":[166],"examples":[168],"are":[169],"provided":[170],"illustrate":[172],"performance":[174],"algorithms":[178],"analysis.":[180]},"counts_by_year":[{"year":2026,"cited_by_count":4},{"year":2025,"cited_by_count":8},{"year":2024,"cited_by_count":5},{"year":2023,"cited_by_count":3}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2022-06-15T00:00:00"}