{"id":"https://openalex.org/W7125515233","doi":"https://doi.org/10.3390/e28020133","title":"Time Series Prediction of Open Quantum System Dynamics by Transformer Neural Networks","display_name":"Time Series Prediction of Open Quantum System Dynamics by Transformer Neural Networks","publication_year":2026,"publication_date":"2026-01-23","ids":{"openalex":"https://openalex.org/W7125515233","doi":"https://doi.org/10.3390/e28020133","pmid":"https://pubmed.ncbi.nlm.nih.gov/41751636"},"language":"en","primary_location":{"id":"doi:10.3390/e28020133","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e28020133","pdf_url":"https://www.mdpi.com/1099-4300/28/2/133/pdf?version=1769169187","source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Entropy","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj","pubmed"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/1099-4300/28/2/133/pdf?version=1769169187","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5078108772","display_name":"Z. M. Wang","orcid":null},"institutions":[{"id":"https://openalex.org/I4210098042","display_name":"Integrated Optoelectronics (Norway)","ror":"https://ror.org/00wsgg405","country_code":"NO","type":"company","lineage":["https://openalex.org/I4210098042"]},{"id":"https://openalex.org/I59028903","display_name":"Ocean University of China","ror":"https://ror.org/04rdtx186","country_code":"CN","type":"education","lineage":["https://openalex.org/I59028903"]}],"countries":["CN","NO"],"is_corresponding":false,"raw_author_name":"Zhao-Wei Wang","raw_affiliation_strings":["College of Physics and Optoelectronic Engineering, Ocean University of China, Qingdao 266100, China","Engineering Research Center of Advanced Marine Physical Instruments and Equipment of Ministry of Education, Ocean University of China, Qingdao 266100, China","Qingdao Key Laboratory of Optics and Optoelectronics, College of Physics and Optoelectronic Engineering, Ocean University of China, Qingdao 266100, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"College of Physics and Optoelectronic Engineering, Ocean University of China, Qingdao 266100, China","institution_ids":["https://openalex.org/I59028903"]},{"raw_affiliation_string":"Engineering Research Center of Advanced Marine Physical Instruments and Equipment of Ministry of Education, Ocean University of China, Qingdao 266100, China","institution_ids":["https://openalex.org/I59028903"]},{"raw_affiliation_string":"Qingdao Key Laboratory of Optics and Optoelectronics, College of Physics and Optoelectronic Engineering, Ocean University of China, Qingdao 266100, China","institution_ids":["https://openalex.org/I4210098042","https://openalex.org/I59028903"]}]},{"author_position":"middle","author":{"id":null,"display_name":"Zhao-Wei Wang","orcid":null},"institutions":[{"id":"https://openalex.org/I4210098042","display_name":"Integrated Optoelectronics (Norway)","ror":"https://ror.org/00wsgg405","country_code":"NO","type":"company","lineage":["https://openalex.org/I4210098042"]},{"id":"https://openalex.org/I59028903","display_name":"Ocean University of China","ror":"https://ror.org/04rdtx186","country_code":"CN","type":"education","lineage":["https://openalex.org/I59028903"]}],"countries":["CN","NO"],"is_corresponding":true,"raw_author_name":"Zhao-Wei Wang","raw_affiliation_strings":["College of Physics and Optoelectronic Engineering, Ocean University of China, Qingdao 266100, China","Engineering Research Center of Advanced Marine Physical Instruments and Equipment of Ministry of Education, Ocean University of China, Qingdao 266100, China","Qingdao Key Laboratory of Optics and Optoelectronics, College of Physics and Optoelectronic Engineering, Ocean University of China, Qingdao 266100, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"College of Physics and Optoelectronic Engineering, Ocean University of China, Qingdao 266100, China","institution_ids":["https://openalex.org/I59028903"]},{"raw_affiliation_string":"Engineering Research Center of Advanced Marine Physical Instruments and Equipment of Ministry of Education, Ocean University of China, Qingdao 266100, China","institution_ids":["https://openalex.org/I59028903"]},{"raw_affiliation_string":"Qingdao Key Laboratory of Optics and Optoelectronics, College of Physics and Optoelectronic Engineering, Ocean University of China, Qingdao 266100, China","institution_ids":["https://openalex.org/I4210098042","https://openalex.org/I59028903"]}]},{"author_position":"middle","author":{"id":null,"display_name":"Lian-Ao Wu","orcid":null},"institutions":[{"id":"https://openalex.org/I110594554","display_name":"Ikerbasque","ror":"https://ror.org/01cc3fy72","country_code":"ES","type":"other","lineage":["https://openalex.org/I110594554"]},{"id":"https://openalex.org/I169108374","display_name":"University of the Basque Country","ror":"https://ror.org/000xsnr85","country_code":"ES","type":"education","lineage":["https://openalex.org/I169108374"]}],"countries":["ES"],"is_corresponding":false,"raw_author_name":"Lian-Ao Wu","raw_affiliation_strings":["Department of Physics, University of the Basque Country UPV/EHU, 48080 Bilbao, Spain","EHU Quantum Center, University of the Basque Country UPV/EHU, 48940 Leioa, Spain","IKERBASQUE, Basque Foundation for Science, 48013 Bilbao, Spain"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Physics, University of the Basque Country UPV/EHU, 48080 Bilbao, Spain","institution_ids":["https://openalex.org/I169108374"]},{"raw_affiliation_string":"EHU Quantum Center, University of the Basque Country UPV/EHU, 48940 Leioa, Spain","institution_ids":["https://openalex.org/I169108374"]},{"raw_affiliation_string":"IKERBASQUE, Basque Foundation for Science, 48013 Bilbao, Spain","institution_ids":["https://openalex.org/I110594554"]}]},{"author_position":"middle","author":{"id":null,"display_name":"Zhao-Ming Wang","orcid":"https://orcid.org/0000-0002-8976-9515"},"institutions":[{"id":"https://openalex.org/I4210098042","display_name":"Integrated Optoelectronics (Norway)","ror":"https://ror.org/00wsgg405","country_code":"NO","type":"company","lineage":["https://openalex.org/I4210098042"]},{"id":"https://openalex.org/I59028903","display_name":"Ocean University of China","ror":"https://ror.org/04rdtx186","country_code":"CN","type":"education","lineage":["https://openalex.org/I59028903"]}],"countries":["CN","NO"],"is_corresponding":true,"raw_author_name":"Zhao-Ming Wang","raw_affiliation_strings":["College of Physics and Optoelectronic Engineering, Ocean University of China, Qingdao 266100, China","Engineering Research Center of Advanced Marine Physical Instruments and Equipment of Ministry of Education, Ocean University of China, Qingdao 266100, China","Qingdao Key Laboratory of Optics and Optoelectronics, College of Physics and Optoelectronic Engineering, Ocean University of China, Qingdao 266100, China"],"raw_orcid":"https://orcid.org/0000-0002-8976-9515","affiliations":[{"raw_affiliation_string":"College of Physics and Optoelectronic Engineering, Ocean University of China, Qingdao 266100, China","institution_ids":["https://openalex.org/I59028903"]},{"raw_affiliation_string":"Engineering Research Center of Advanced Marine Physical Instruments and Equipment of Ministry of Education, Ocean University of China, Qingdao 266100, China","institution_ids":["https://openalex.org/I59028903"]},{"raw_affiliation_string":"Qingdao Key Laboratory of Optics and Optoelectronics, College of Physics and Optoelectronic Engineering, Ocean University of China, Qingdao 266100, China","institution_ids":["https://openalex.org/I4210098042","https://openalex.org/I59028903"]}]},{"author_position":"last","author":{"id":null,"display_name":"Zhao-Ming Wang","orcid":"https://orcid.org/0000-0002-8976-9515"},"institutions":[{"id":"https://openalex.org/I4210098042","display_name":"Integrated Optoelectronics (Norway)","ror":"https://ror.org/00wsgg405","country_code":"NO","type":"company","lineage":["https://openalex.org/I4210098042"]},{"id":"https://openalex.org/I59028903","display_name":"Ocean University of China","ror":"https://ror.org/04rdtx186","country_code":"CN","type":"education","lineage":["https://openalex.org/I59028903"]}],"countries":["CN","NO"],"is_corresponding":false,"raw_author_name":"Zhao-Ming Wang","raw_affiliation_strings":["College of Physics and Optoelectronic Engineering, Ocean University of China, Qingdao 266100, China","Engineering Research Center of Advanced Marine Physical Instruments and Equipment of Ministry of Education, Ocean University of China, Qingdao 266100, China","Qingdao Key Laboratory of Optics and Optoelectronics, College of Physics and Optoelectronic Engineering, Ocean University of China, Qingdao 266100, China"],"raw_orcid":"https://orcid.org/0000-0002-8976-9515","affiliations":[{"raw_affiliation_string":"College of Physics and Optoelectronic Engineering, Ocean University of China, Qingdao 266100, China","institution_ids":["https://openalex.org/I59028903"]},{"raw_affiliation_string":"Engineering Research Center of Advanced Marine Physical Instruments and Equipment of Ministry of Education, Ocean University of China, Qingdao 266100, China","institution_ids":["https://openalex.org/I59028903"]},{"raw_affiliation_string":"Qingdao Key Laboratory of Optics and Optoelectronics, College of Physics and Optoelectronic Engineering, Ocean University of China, Qingdao 266100, China","institution_ids":["https://openalex.org/I4210098042","https://openalex.org/I59028903"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":4,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I4210098042","https://openalex.org/I59028903"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2227},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2227},"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.11118049,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"28","issue":"2","first_page":"133","last_page":"133"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11804","display_name":"Quantum many-body systems","score":0.8093000054359436,"subfield":{"id":"https://openalex.org/subfields/3107","display_name":"Atomic and Molecular Physics, and Optics"},"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/T11804","display_name":"Quantum many-body systems","score":0.8093000054359436,"subfield":{"id":"https://openalex.org/subfields/3107","display_name":"Atomic and Molecular Physics, and Optics"},"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/T11948","display_name":"Machine Learning in Materials Science","score":0.027799999341368675,"subfield":{"id":"https://openalex.org/subfields/2505","display_name":"Materials Chemistry"},"field":{"id":"https://openalex.org/fields/25","display_name":"Materials Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11177","display_name":"Spectroscopy and Quantum Chemical Studies","score":0.027300000190734863,"subfield":{"id":"https://openalex.org/subfields/3107","display_name":"Atomic and Molecular Physics, and Optics"},"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/quantum","display_name":"Quantum","score":0.5282999873161316},{"id":"https://openalex.org/keywords/generalization","display_name":"Generalization","score":0.49390000104904175},{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.45820000767707825},{"id":"https://openalex.org/keywords/dynamical-systems-theory","display_name":"Dynamical systems theory","score":0.43549999594688416},{"id":"https://openalex.org/keywords/density-matrix","display_name":"Density matrix","score":0.43070000410079956},{"id":"https://openalex.org/keywords/scalability","display_name":"Scalability","score":0.4277999997138977},{"id":"https://openalex.org/keywords/series","display_name":"Series (stratigraphy)","score":0.41620001196861267},{"id":"https://openalex.org/keywords/reservoir-computing","display_name":"Reservoir computing","score":0.3935999870300293},{"id":"https://openalex.org/keywords/quantum-system","display_name":"Quantum system","score":0.3869999945163727},{"id":"https://openalex.org/keywords/master-equation","display_name":"Master equation","score":0.36090001463890076}],"concepts":[{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.6381000280380249},{"id":"https://openalex.org/C84114770","wikidata":"https://www.wikidata.org/wiki/Q46344","display_name":"Quantum","level":2,"score":0.5282999873161316},{"id":"https://openalex.org/C177148314","wikidata":"https://www.wikidata.org/wiki/Q170084","display_name":"Generalization","level":2,"score":0.49390000104904175},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.45820000767707825},{"id":"https://openalex.org/C79379906","wikidata":"https://www.wikidata.org/wiki/Q3174497","display_name":"Dynamical systems theory","level":2,"score":0.43549999594688416},{"id":"https://openalex.org/C56911000","wikidata":"https://www.wikidata.org/wiki/Q831774","display_name":"Density matrix","level":3,"score":0.43070000410079956},{"id":"https://openalex.org/C48044578","wikidata":"https://www.wikidata.org/wiki/Q727490","display_name":"Scalability","level":2,"score":0.4277999997138977},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.42730000615119934},{"id":"https://openalex.org/C143724316","wikidata":"https://www.wikidata.org/wiki/Q312468","display_name":"Series (stratigraphy)","level":2,"score":0.41620001196861267},{"id":"https://openalex.org/C135796866","wikidata":"https://www.wikidata.org/wiki/Q7315328","display_name":"Reservoir computing","level":4,"score":0.3935999870300293},{"id":"https://openalex.org/C2778926657","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum system","level":3,"score":0.3869999945163727},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.3797000050544739},{"id":"https://openalex.org/C128805008","wikidata":"https://www.wikidata.org/wiki/Q722878","display_name":"Master equation","level":3,"score":0.36090001463890076},{"id":"https://openalex.org/C163677043","wikidata":"https://www.wikidata.org/wiki/Q2382874","display_name":"Open system (computing)","level":3,"score":0.34200000762939453},{"id":"https://openalex.org/C149441793","wikidata":"https://www.wikidata.org/wiki/Q200726","display_name":"Probability distribution","level":2,"score":0.3400000035762787},{"id":"https://openalex.org/C13662910","wikidata":"https://www.wikidata.org/wiki/Q193139","display_name":"Trajectory","level":2,"score":0.33660000562667847},{"id":"https://openalex.org/C190474826","wikidata":"https://www.wikidata.org/wiki/Q2495111","display_name":"Open quantum system","level":3,"score":0.33660000562667847},{"id":"https://openalex.org/C77405623","wikidata":"https://www.wikidata.org/wiki/Q598451","display_name":"System dynamics","level":2,"score":0.33649998903274536},{"id":"https://openalex.org/C2780009758","wikidata":"https://www.wikidata.org/wiki/Q6804172","display_name":"Measure (data warehouse)","level":2,"score":0.3319000005722046},{"id":"https://openalex.org/C106553664","wikidata":"https://www.wikidata.org/wiki/Q1156744","display_name":"Time evolution","level":2,"score":0.3255000114440918},{"id":"https://openalex.org/C47822265","wikidata":"https://www.wikidata.org/wiki/Q854457","display_name":"Complex system","level":2,"score":0.32109999656677246},{"id":"https://openalex.org/C147168706","wikidata":"https://www.wikidata.org/wiki/Q1457734","display_name":"Recurrent neural network","level":3,"score":0.3206000030040741},{"id":"https://openalex.org/C66322947","wikidata":"https://www.wikidata.org/wiki/Q11658","display_name":"Transformer","level":3,"score":0.3133000135421753},{"id":"https://openalex.org/C151406439","wikidata":"https://www.wikidata.org/wiki/Q186588","display_name":"Time series","level":2,"score":0.3107999861240387},{"id":"https://openalex.org/C116672817","wikidata":"https://www.wikidata.org/wiki/Q1454986","display_name":"Physical system","level":2,"score":0.3075999915599823},{"id":"https://openalex.org/C108583219","wikidata":"https://www.wikidata.org/wiki/Q197536","display_name":"Deep learning","level":2,"score":0.30720001459121704},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.30649998784065247},{"id":"https://openalex.org/C15184713","wikidata":"https://www.wikidata.org/wiki/Q7269042","display_name":"Quantum dynamics","level":3,"score":0.29440000653266907},{"id":"https://openalex.org/C131584629","wikidata":"https://www.wikidata.org/wiki/Q4308705","display_name":"Coupling (piping)","level":2,"score":0.29120001196861267},{"id":"https://openalex.org/C15706264","wikidata":"https://www.wikidata.org/wiki/Q230883","display_name":"Quantum state","level":3,"score":0.26910001039505005},{"id":"https://openalex.org/C58053490","wikidata":"https://www.wikidata.org/wiki/Q176555","display_name":"Quantum computer","level":3,"score":0.2655999958515167},{"id":"https://openalex.org/C169699857","wikidata":"https://www.wikidata.org/wiki/Q2122243","display_name":"Quantum information","level":3,"score":0.26030001044273376}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.3390/e28020133","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e28020133","pdf_url":"https://www.mdpi.com/1099-4300/28/2/133/pdf?version=1769169187","source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Entropy","raw_type":"journal-article"},{"id":"pmid:41751636","is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/41751636","pdf_url":null,"source":{"id":"https://openalex.org/S4306525036","display_name":"PubMed","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https://openalex.org/I1299303238"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Entropy (Basel, Switzerland)","raw_type":null},{"id":"pmh:oai:doaj.org/article:24eef29892da472aafcac00462d37eec","is_oa":false,"landing_page_url":"https://doaj.org/article/24eef29892da472aafcac00462d37eec","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Entropy, Vol 28, Iss 2, p 133 (2026)","raw_type":"article"},{"id":"pmh:oai:pubmedcentral.nih.gov:12939071","is_oa":true,"landing_page_url":"https://pmc.ncbi.nlm.nih.gov/articles/PMC12939071/","pdf_url":null,"source":{"id":"https://openalex.org/S2764455111","display_name":"PubMed Central","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https://openalex.org/I1299303238"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Entropy (Basel)","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/e28020133","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e28020133","pdf_url":"https://www.mdpi.com/1099-4300/28/2/133/pdf?version=1769169187","source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Entropy","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G2262748287","display_name":null,"funder_award_id":"501100011033","funder_id":"https://openalex.org/F4320335598","funder_display_name":"Agencia Estatal de Investigaci\u00f3n"},{"id":"https://openalex.org/G298511182","display_name":null,"funder_award_id":"PGC2018-101355-B-I00","funder_id":"https://openalex.org/F4320335598","funder_display_name":"Agencia Estatal de Investigaci\u00f3n"},{"id":"https://openalex.org/G3480869486","display_name":null,"funder_award_id":"13039","funder_id":"https://openalex.org/F4320335598","funder_display_name":"Agencia Estatal de Investigaci\u00f3n"},{"id":"https://openalex.org/G425668148","display_name":null,"funder_award_id":".13039/501100011033","funder_id":"https://openalex.org/F4320335598","funder_display_name":"Agencia Estatal de Investigaci\u00f3n"},{"id":"https://openalex.org/G4526231891","display_name":null,"funder_award_id":"MCIN / AEI/10","funder_id":"https://openalex.org/F4320335598","funder_display_name":"Agencia Estatal de Investigaci\u00f3n"},{"id":"https://openalex.org/G681035310","display_name":null,"funder_award_id":"ZR2021LLZ004","funder_id":"https://openalex.org/F4320324174","funder_display_name":"Natural Science Foundation of Shandong Province"},{"id":"https://openalex.org/G7084143925","display_name":null,"funder_award_id":"AEI/10","funder_id":"https://openalex.org/F4320335598","funder_display_name":"Agencia Estatal de Investigaci\u00f3n"},{"id":"https://openalex.org/G8510325072","display_name":null,"funder_award_id":"202364008","funder_id":"https://openalex.org/F4320335787","funder_display_name":"Fundamental Research Funds for the Central Universities"},{"id":"https://openalex.org/G8651036145","display_name":null,"funder_award_id":"ZR2024MA046","funder_id":"https://openalex.org/F4320324174","funder_display_name":"Natural Science Foundation of Shandong Province"}],"funders":[{"id":"https://openalex.org/F4320324174","display_name":"Natural Science Foundation of Shandong Province","ror":null},{"id":"https://openalex.org/F4320335598","display_name":"Agencia Estatal de Investigaci\u00f3n","ror":null},{"id":"https://openalex.org/F4320335787","display_name":"Fundamental Research Funds for the Central Universities","ror":null}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7125515233.pdf","grobid_xml":"https://content.openalex.org/works/W7125515233.grobid-xml"},"referenced_works_count":36,"referenced_works":["https://openalex.org/W2006452999","https://openalex.org/W2032839191","https://openalex.org/W2099265531","https://openalex.org/W2118492081","https://openalex.org/W2158904868","https://openalex.org/W2798593936","https://openalex.org/W2898121159","https://openalex.org/W2949449669","https://openalex.org/W2953495237","https://openalex.org/W2972664419","https://openalex.org/W2997289813","https://openalex.org/W3034599993","https://openalex.org/W3132033902","https://openalex.org/W3133983161","https://openalex.org/W3162246162","https://openalex.org/W3181231283","https://openalex.org/W3205926366","https://openalex.org/W3215598778","https://openalex.org/W3216923111","https://openalex.org/W4280535729","https://openalex.org/W4283396843","https://openalex.org/W4288049312","https://openalex.org/W4289316580","https://openalex.org/W4293198087","https://openalex.org/W4294786469","https://openalex.org/W4306248853","https://openalex.org/W4379390793","https://openalex.org/W4386142122","https://openalex.org/W4390546716","https://openalex.org/W4402047299","https://openalex.org/W4402271327","https://openalex.org/W4403983979","https://openalex.org/W4406610550","https://openalex.org/W4408455526","https://openalex.org/W4409505952","https://openalex.org/W4413381535"],"related_works":[],"abstract_inverted_index":{"The":[0],"dynamics":[1],"of":[2,62,79,104,122,151,160],"open":[3,38,63],"quantum":[4,11,39,64],"systems":[5],"play":[6],"a":[7,47,83,88],"crucial":[8],"role":[9],"in":[10,127],"information":[12],"science.":[13],"However,":[14],"obtaining":[15],"numerically":[16],"exact":[17],"solutions":[18],"for":[19,36],"the":[20,59,68,76,80,100,105,117,123,148,152,156,161],"Lindblad":[21],"master":[22],"equation":[23],"is":[24],"often":[25],"computationally":[26],"expensive.":[27],"Recently,":[28],"machine":[29],"learning":[30,49],"techniques":[31],"have":[32],"gained":[33],"considerable":[34],"attention":[35],"simulating":[37],"system":[40,81,106],"dynamics.":[41],"In":[42],"this":[43],"paper,":[44],"we":[45,145],"propose":[46],"deep":[48],"model":[50,90,97,118],"based":[51,91],"on":[52,92],"time":[53],"series":[54],"prediction":[55],"(TSP)":[56],"to":[57,74],"forecast":[58],"dynamical":[60],"evolution":[61,102,125],"systems.":[65],"We":[66],"employ":[67],"positive":[69],"operator-valued":[70],"measure":[71],"(POVM)":[72],"approach":[73],"convert":[75],"density":[77],"matrix":[78],"into":[82],"probability":[84],"distribution":[85],"and":[86,107,130,133,141,158],"construct":[87],"TSP":[89],"Transformer":[93],"neural":[94],"networks.":[95],"This":[96],"effectively":[98],"captures":[99],"historical":[101],"patterns":[103],"accurately":[108],"predicts":[109],"its":[110],"future":[111],"behavior.":[112],"Our":[113],"results":[114],"show":[115],"that":[116],"achieves":[119],"high-fidelity":[120],"predictions":[121],"system's":[124],"trajectory":[126],"both":[128],"short-":[129],"long-term":[131],"scenarios,":[132],"exhibits":[134],"robust":[135],"generalization":[136],"under":[137],"varying":[138],"initial":[139],"states":[140],"coupling":[142],"strengths.":[143],"Moreover,":[144],"successfully":[146],"predicted":[147],"steady-state":[149],"behavior":[150],"system,":[153],"further":[154],"proving":[155],"practicality":[157],"scalability":[159],"method.":[162]},"counts_by_year":[],"updated_date":"2026-08-01T09:00:35.917206","created_date":"2026-01-24T00:00:00"}
