{"id":"https://openalex.org/W7165920202","doi":"https://doi.org/10.1007/s10915-026-03379-5","title":"An Optimal-order Backpropagation Algorithm for Stochastic Neural Networks","display_name":"An Optimal-order Backpropagation Algorithm for Stochastic Neural Networks","publication_year":2026,"publication_date":"2026-06-25","ids":{"openalex":"https://openalex.org/W7165920202","doi":"https://doi.org/10.1007/s10915-026-03379-5"},"language":"en","primary_location":{"id":"doi:10.1007/s10915-026-03379-5","is_oa":false,"landing_page_url":"https://doi.org/10.1007/s10915-026-03379-5","pdf_url":null,"source":{"id":"https://openalex.org/S151550017","display_name":"Journal of Scientific Computing","issn_l":"0885-7474","issn":["0885-7474","1573-7691"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Scientific Computing","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/A5104236605","display_name":"Daili Sheng","orcid":null},"institutions":[{"id":"https://openalex.org/I204983213","display_name":"Harbin Institute of Technology","ror":"https://ror.org/01yqg2h08","country_code":"CN","type":"education","lineage":["https://openalex.org/I204983213"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Daili Sheng","raw_affiliation_strings":["School of Mathematics, Harbin Institute of Technology, Harbin, 150001, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematics, Harbin Institute of Technology, Harbin, 150001, China","institution_ids":["https://openalex.org/I204983213"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5040187407","display_name":"Minghui Song","orcid":"https://orcid.org/0000-0002-0570-0124"},"institutions":[{"id":"https://openalex.org/I204983213","display_name":"Harbin Institute of Technology","ror":"https://ror.org/01yqg2h08","country_code":"CN","type":"education","lineage":["https://openalex.org/I204983213"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Minghui Song","raw_affiliation_strings":["School of Mathematics, Harbin Institute of Technology, Harbin, 150001, China"],"raw_orcid":"https://orcid.org/0000-0002-0570-0124","affiliations":[{"raw_affiliation_string":"School of Mathematics, Harbin Institute of Technology, Harbin, 150001, China","institution_ids":["https://openalex.org/I204983213"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5067202795","display_name":"Xiang Peng","orcid":"https://orcid.org/0000-0003-0815-5122"},"institutions":[{"id":"https://openalex.org/I204983213","display_name":"Harbin Institute of Technology","ror":"https://ror.org/01yqg2h08","country_code":"CN","type":"education","lineage":["https://openalex.org/I204983213"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Xiang Peng","raw_affiliation_strings":["School of Mathematics, Harbin Institute of Technology, Harbin, 150001, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematics, Harbin Institute of Technology, Harbin, 150001, China","institution_ids":["https://openalex.org/I204983213"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5101278284","display_name":"Xuanqi Dong","orcid":null},"institutions":[{"id":"https://openalex.org/I204983213","display_name":"Harbin Institute of Technology","ror":"https://ror.org/01yqg2h08","country_code":"CN","type":"education","lineage":["https://openalex.org/I204983213"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Xuanqi Dong","raw_affiliation_strings":["School of Mathematics, Harbin Institute of Technology, Harbin, 150001, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematics, Harbin Institute of Technology, Harbin, 150001, China","institution_ids":["https://openalex.org/I204983213"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5040187407"],"corresponding_institution_ids":["https://openalex.org/I204983213"],"apc_list":{"value":2990,"currency":"USD","value_usd":2990},"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.80738822,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"108","issue":"2","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11612","display_name":"Stochastic Gradient Optimization Techniques","score":0.7778000235557556,"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"}},"topics":[{"id":"https://openalex.org/T11612","display_name":"Stochastic Gradient Optimization Techniques","score":0.7778000235557556,"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/T12676","display_name":"Machine Learning and ELM","score":0.07689999788999557,"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/T11206","display_name":"Model Reduction and Neural Networks","score":0.03669999912381172,"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/backpropagation","display_name":"Backpropagation","score":0.8928999900817871},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.6593999862670898},{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.6284000277519226},{"id":"https://openalex.org/keywords/stochastic-gradient-descent","display_name":"Stochastic gradient descent","score":0.5088000297546387},{"id":"https://openalex.org/keywords/convexity","display_name":"Convexity","score":0.5077000260353088},{"id":"https://openalex.org/keywords/process","display_name":"Process (computing)","score":0.43149998784065247},{"id":"https://openalex.org/keywords/stochastic-neural-network","display_name":"Stochastic neural network","score":0.4108999967575073},{"id":"https://openalex.org/keywords/stochastic-process","display_name":"Stochastic process","score":0.3978999853134155}],"concepts":[{"id":"https://openalex.org/C155032097","wikidata":"https://www.wikidata.org/wiki/Q798503","display_name":"Backpropagation","level":3,"score":0.8928999900817871},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.6593999862670898},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.6284000277519226},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.5812000036239624},{"id":"https://openalex.org/C206688291","wikidata":"https://www.wikidata.org/wiki/Q7617819","display_name":"Stochastic gradient descent","level":3,"score":0.5088000297546387},{"id":"https://openalex.org/C72134830","wikidata":"https://www.wikidata.org/wiki/Q5166524","display_name":"Convexity","level":2,"score":0.5077000260353088},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.492900013923645},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.43309998512268066},{"id":"https://openalex.org/C98045186","wikidata":"https://www.wikidata.org/wiki/Q205663","display_name":"Process (computing)","level":2,"score":0.43149998784065247},{"id":"https://openalex.org/C86582703","wikidata":"https://www.wikidata.org/wiki/Q7617824","display_name":"Stochastic neural network","level":4,"score":0.4108999967575073},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.4065000116825104},{"id":"https://openalex.org/C8272713","wikidata":"https://www.wikidata.org/wiki/Q176737","display_name":"Stochastic process","level":2,"score":0.3978999853134155},{"id":"https://openalex.org/C55479107","wikidata":"https://www.wikidata.org/wiki/Q97663916","display_name":"Stochastic approximation","level":3,"score":0.39489999413490295},{"id":"https://openalex.org/C57869625","wikidata":"https://www.wikidata.org/wiki/Q1783502","display_name":"Rate of convergence","level":3,"score":0.3513999879360199},{"id":"https://openalex.org/C153258448","wikidata":"https://www.wikidata.org/wiki/Q1199743","display_name":"Gradient descent","level":3,"score":0.34940001368522644},{"id":"https://openalex.org/C77618280","wikidata":"https://www.wikidata.org/wiki/Q1155772","display_name":"Scheme (mathematics)","level":2,"score":0.3310999870300293},{"id":"https://openalex.org/C51955184","wikidata":"https://www.wikidata.org/wiki/Q1545585","display_name":"Stochastic differential equation","level":2,"score":0.321399986743927},{"id":"https://openalex.org/C148764684","wikidata":"https://www.wikidata.org/wiki/Q621751","display_name":"Approximation algorithm","level":2,"score":0.2849000096321106},{"id":"https://openalex.org/C112972136","wikidata":"https://www.wikidata.org/wiki/Q7595718","display_name":"Stability (learning theory)","level":2,"score":0.2644999921321869},{"id":"https://openalex.org/C91575142","wikidata":"https://www.wikidata.org/wiki/Q1971426","display_name":"Optimal control","level":2,"score":0.2621000111103058}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1007/s10915-026-03379-5","is_oa":false,"landing_page_url":"https://doi.org/10.1007/s10915-026-03379-5","pdf_url":null,"source":{"id":"https://openalex.org/S151550017","display_name":"Journal of Scientific Computing","issn_l":"0885-7474","issn":["0885-7474","1573-7691"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Scientific Computing","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G6300377352","display_name":null,"funder_award_id":"12471372","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":30,"referenced_works":["https://openalex.org/W94839203","https://openalex.org/W592962635","https://openalex.org/W1434088925","https://openalex.org/W1980091117","https://openalex.org/W2033093836","https://openalex.org/W2033603136","https://openalex.org/W2052883417","https://openalex.org/W2093487107","https://openalex.org/W2164907988","https://openalex.org/W2328323135","https://openalex.org/W2600297185","https://openalex.org/W2612252966","https://openalex.org/W2770316314","https://openalex.org/W2903857590","https://openalex.org/W3035618986","https://openalex.org/W3035711539","https://openalex.org/W3049221496","https://openalex.org/W3102100346","https://openalex.org/W3103002366","https://openalex.org/W3150057971","https://openalex.org/W3205746738","https://openalex.org/W4212780424","https://openalex.org/W4236109675","https://openalex.org/W4377101750","https://openalex.org/W4386542021","https://openalex.org/W4388750071","https://openalex.org/W4392385407","https://openalex.org/W4408768617","https://openalex.org/W4411667927","https://openalex.org/W7127279057"],"related_works":[],"abstract_inverted_index":null,"counts_by_year":[],"updated_date":"2026-07-23T08:03:31.855105","created_date":"2026-06-26T00:00:00"}