{"id":"https://openalex.org/W4401567834","doi":"https://doi.org/10.1109/tii.2024.3438277","title":"Heat Equation Stein Variational Ensemble: Rethinking and Advancing Uncertainty-Aware Soft Sensor Modeling","display_name":"Heat Equation Stein Variational Ensemble: Rethinking and Advancing Uncertainty-Aware Soft Sensor Modeling","publication_year":2024,"publication_date":"2024-08-14","ids":{"openalex":"https://openalex.org/W4401567834","doi":"https://doi.org/10.1109/tii.2024.3438277"},"language":"en","primary_location":{"id":"doi:10.1109/tii.2024.3438277","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tii.2024.3438277","pdf_url":null,"source":{"id":"https://openalex.org/S184777250","display_name":"IEEE Transactions on Industrial Informatics","issn_l":"1551-3203","issn":["1551-3203","1941-0050"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Industrial Informatics","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/A5104335861","display_name":"Yiran Ma","orcid":"https://orcid.org/0009-0004-6749-5121"},"institutions":[{"id":"https://openalex.org/I4391767838","display_name":"State Key Laboratory of Industrial Control Technology","ror":"https://ror.org/03a33a786","country_code":"CN","type":"facility","lineage":["https://openalex.org/I4391767838","https://openalex.org/I76130692"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Yiran Ma","raw_affiliation_strings":["State Key Laboratory of Industrial Control Technology, College of Control Science and Engineering, Zhejiang University, Hangzhou, China"],"raw_orcid":"https://orcid.org/0009-0004-6749-5121","affiliations":[{"raw_affiliation_string":"State Key Laboratory of Industrial Control Technology, College of Control Science and Engineering, Zhejiang University, Hangzhou, China","institution_ids":["https://openalex.org/I4391767838"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5100619470","display_name":"Zhichao Chen","orcid":"https://orcid.org/0000-0001-5785-0741"},"institutions":[{"id":"https://openalex.org/I4391767838","display_name":"State Key Laboratory of Industrial Control Technology","ror":"https://ror.org/03a33a786","country_code":"CN","type":"facility","lineage":["https://openalex.org/I4391767838","https://openalex.org/I76130692"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Zhichao Chen","raw_affiliation_strings":["State Key Laboratory of Industrial Control Technology, College of Control Science and Engineering, Zhejiang University, Hangzhou, China"],"raw_orcid":"https://orcid.org/0000-0001-5785-0741","affiliations":[{"raw_affiliation_string":"State Key Laboratory of Industrial Control Technology, College of Control Science and Engineering, Zhejiang University, Hangzhou, China","institution_ids":["https://openalex.org/I4391767838"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5017990705","display_name":"Zeyu Yang","orcid":"https://orcid.org/0000-0003-0253-9053"},"institutions":[{"id":"https://openalex.org/I3018263800","display_name":"Huzhou University","ror":"https://ror.org/04mvpxy20","country_code":"CN","type":"education","lineage":["https://openalex.org/I3018263800"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Zeyu Yang","raw_affiliation_strings":["Huzhou Key Laboratory of Intelligent Sensing and Optimal Control for Industrial Systems, School of Engineering, Huzhou University, Huzhou, China"],"raw_orcid":"https://orcid.org/0000-0003-0253-9053","affiliations":[{"raw_affiliation_string":"Huzhou Key Laboratory of Intelligent Sensing and Optimal Control for Industrial Systems, School of Engineering, Huzhou University, Huzhou, China","institution_ids":["https://openalex.org/I3018263800"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5100654679","display_name":"Xinmin Zhang","orcid":"https://orcid.org/0000-0002-4761-3969"},"institutions":[{"id":"https://openalex.org/I4391767838","display_name":"State Key Laboratory of Industrial Control Technology","ror":"https://ror.org/03a33a786","country_code":"CN","type":"facility","lineage":["https://openalex.org/I4391767838","https://openalex.org/I76130692"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Xinmin Zhang","raw_affiliation_strings":["State Key Laboratory of Industrial Control Technology, College of Control Science and Engineering, Zhejiang University, Hangzhou, China"],"raw_orcid":"https://orcid.org/0000-0002-4761-3969","affiliations":[{"raw_affiliation_string":"State Key Laboratory of Industrial Control Technology, College of Control Science and Engineering, Zhejiang University, Hangzhou, China","institution_ids":["https://openalex.org/I4391767838"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5048733958","display_name":"Zhihuan Song","orcid":"https://orcid.org/0000-0003-4098-6479"},"institutions":[{"id":"https://openalex.org/I4391767838","display_name":"State Key Laboratory of Industrial Control Technology","ror":"https://ror.org/03a33a786","country_code":"CN","type":"facility","lineage":["https://openalex.org/I4391767838","https://openalex.org/I76130692"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Zhihuan Song","raw_affiliation_strings":["State Key Laboratory of Industrial Control Technology, College of Control Science and Engineering, Zhejiang University, Hangzhou, China"],"raw_orcid":"https://orcid.org/0000-0003-4098-6479","affiliations":[{"raw_affiliation_string":"State Key Laboratory of Industrial Control Technology, College of Control Science and Engineering, Zhejiang University, Hangzhou, China","institution_ids":["https://openalex.org/I4391767838"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":1.1335,"has_fulltext":false,"cited_by_count":6,"citation_normalized_percentile":{"value":0.75395347,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":97,"max":98},"biblio":{"volume":"20","issue":"12","first_page":"13996","last_page":"14005"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10876","display_name":"Fault Detection and Control Systems","score":0.9969000220298767,"subfield":{"id":"https://openalex.org/subfields/2207","display_name":"Control and Systems Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10876","display_name":"Fault Detection and Control Systems","score":0.9969000220298767,"subfield":{"id":"https://openalex.org/subfields/2207","display_name":"Control and Systems Engineering"},"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":0.9965999722480774,"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/T11206","display_name":"Model Reduction and Neural Networks","score":0.9904999732971191,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/hyperparameter","display_name":"Hyperparameter","score":0.576282799243927},{"id":"https://openalex.org/keywords/uncertainty-quantification","display_name":"Uncertainty quantification","score":0.5745164752006531},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.4966304898262024},{"id":"https://openalex.org/keywords/estimator","display_name":"Estimator","score":0.4870939254760742},{"id":"https://openalex.org/keywords/gradient-descent","display_name":"Gradient descent","score":0.4319360852241516},{"id":"https://openalex.org/keywords/heat-equation","display_name":"Heat equation","score":0.41544926166534424},{"id":"https://openalex.org/keywords/noise","display_name":"Noise (video)","score":0.4101029932498932},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.38931751251220703},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.38434672355651855},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.33683234453201294},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.3087570071220398},{"id":"https://openalex.org/keywords/machine-learning","display_name":"Machine learning","score":0.24866369366645813},{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.20943179726600647},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.15096762776374817},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.0883815586566925}],"concepts":[{"id":"https://openalex.org/C8642999","wikidata":"https://www.wikidata.org/wiki/Q4171168","display_name":"Hyperparameter","level":2,"score":0.576282799243927},{"id":"https://openalex.org/C32230216","wikidata":"https://www.wikidata.org/wiki/Q7882499","display_name":"Uncertainty quantification","level":2,"score":0.5745164752006531},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.4966304898262024},{"id":"https://openalex.org/C185429906","wikidata":"https://www.wikidata.org/wiki/Q1130160","display_name":"Estimator","level":2,"score":0.4870939254760742},{"id":"https://openalex.org/C153258448","wikidata":"https://www.wikidata.org/wiki/Q1199743","display_name":"Gradient descent","level":3,"score":0.4319360852241516},{"id":"https://openalex.org/C202787564","wikidata":"https://www.wikidata.org/wiki/Q6510488","display_name":"Heat equation","level":2,"score":0.41544926166534424},{"id":"https://openalex.org/C99498987","wikidata":"https://www.wikidata.org/wiki/Q2210247","display_name":"Noise (video)","level":3,"score":0.4101029932498932},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.38931751251220703},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.38434672355651855},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.33683234453201294},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.3087570071220398},{"id":"https://openalex.org/C119857082","wikidata":"https://www.wikidata.org/wiki/Q2539","display_name":"Machine learning","level":1,"score":0.24866369366645813},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.20943179726600647},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.15096762776374817},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.0883815586566925},{"id":"https://openalex.org/C115961682","wikidata":"https://www.wikidata.org/wiki/Q860623","display_name":"Image (mathematics)","level":2,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1109/tii.2024.3438277","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tii.2024.3438277","pdf_url":null,"source":{"id":"https://openalex.org/S184777250","display_name":"IEEE Transactions on Industrial Informatics","issn_l":"1551-3203","issn":["1551-3203","1941-0050"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Industrial Informatics","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G5659945949","display_name":null,"funder_award_id":"61933013","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G7398783184","display_name":null,"funder_award_id":"62203169","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"}],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":47,"referenced_works":["https://openalex.org/W1972195634","https://openalex.org/W2000651380","https://openalex.org/W2004220372","https://openalex.org/W2043251326","https://openalex.org/W2047619877","https://openalex.org/W2083259944","https://openalex.org/W2094298886","https://openalex.org/W2254249950","https://openalex.org/W2768613868","https://openalex.org/W2888872523","https://openalex.org/W2947216528","https://openalex.org/W2985022470","https://openalex.org/W2988720209","https://openalex.org/W2992525328","https://openalex.org/W3014596384","https://openalex.org/W3030356194","https://openalex.org/W3048889688","https://openalex.org/W3093656051","https://openalex.org/W3102100346","https://openalex.org/W3104574805","https://openalex.org/W3122034064","https://openalex.org/W3123899295","https://openalex.org/W3134774296","https://openalex.org/W3148740559","https://openalex.org/W3164731060","https://openalex.org/W3173017111","https://openalex.org/W3183303935","https://openalex.org/W3207007544","https://openalex.org/W4283169343","https://openalex.org/W4285058963","https://openalex.org/W4313591689","https://openalex.org/W4322724151","https://openalex.org/W4360994956","https://openalex.org/W4387762676","https://openalex.org/W4389104991","https://openalex.org/W6684809622","https://openalex.org/W6725708968","https://openalex.org/W6736966227","https://openalex.org/W6748758124","https://openalex.org/W6750300694","https://openalex.org/W6753254651","https://openalex.org/W6754200721","https://openalex.org/W6762470948","https://openalex.org/W6771378952","https://openalex.org/W6774120203","https://openalex.org/W6803759503","https://openalex.org/W6973772974"],"related_works":["https://openalex.org/W4390421286","https://openalex.org/W4280563792","https://openalex.org/W2140186469","https://openalex.org/W4389724018","https://openalex.org/W4318719684","https://openalex.org/W2775233965","https://openalex.org/W4318559728","https://openalex.org/W4360995913","https://openalex.org/W4312206232","https://openalex.org/W2975301014"],"abstract_inverted_index":{"Data-driven":[0],"soft":[1],"sensors":[2],"have":[3],"been":[4,20],"prevalent":[5],"in":[6,27,143],"industrial":[7,28],"key":[8],"performance":[9],"indicator":[10],"prediction.":[11],"However,":[12],"rigorously":[13],"quantifying":[14],"the":[15,33,36,62,123,139],"uncertainty":[16,50,60,135],"of":[17,32,35,116,141],"predictions":[18],"has":[19],"consistently":[21],"neglected.":[22],"This":[23],"neglect":[24],"can":[25],"result":[26],"practitioners":[29],"being":[30],"ignorant":[31],"reliability":[34],"predictions,":[37],"potentially":[38],"leading":[39],"to":[40,99,127,134,147],"dangerous":[41],"misjudgments.":[42],"To":[43],"bridge":[44],"this":[45,47,73],"gap,":[46],"study":[48],"interprets":[49],"quantification":[51],"(UQ)":[52],"from":[53,68],"a":[54,75],"Bayesian":[55],"perspective,":[56],"which":[57,132],"conceptualizes":[58],"model":[59],"as":[61],"parameter":[63,101],"distribution,":[64],"thus":[65],"distinguishing":[66],"it":[67],"data":[69,130],"uncertainty.":[70],"Based":[71],"on":[72],"interpretation,":[74],"novel":[76],"UQ":[77],"method,":[78],"heat":[79,106],"equation":[80],"stein":[81],"variational":[82,95],"ensemble":[83],"(HESVE),":[84],"is":[85,109],"proposed.":[86],"HESVE":[87,142],"introduces":[88],"an":[89],"efficient":[90],"nonparametric":[91],"approach":[92],"called":[93],"Stein":[94],"gradient":[96],"descent":[97],"(SVGD)":[98],"approximate":[100],"distributions":[102],"more":[103],"precisely.":[104],"A":[105],"equation-based":[107],"method":[108,121],"also":[110],"adopted":[111],"for":[112],"adaptive":[113],"hyperparameter":[114],"tuning":[115],"SVGD.":[117],"In":[118],"addition,":[119],"our":[120],"incorporates":[122],"variable":[124],"noise":[125],"estimator":[126],"capture":[128],"heteroscedastic":[129],"noise,":[131],"contributes":[133],"decomposition.":[136],"Experiments":[137],"demonstrate":[138],"superiority":[140],"multiple":[144],"aspects":[145],"compared":[146],"other":[148],"state-of-the-art":[149],"methods.":[150]},"counts_by_year":[{"year":2026,"cited_by_count":2},{"year":2025,"cited_by_count":4}],"updated_date":"2026-07-22T07:51:19.307946","created_date":"2025-10-10T00:00:00"}