{"id":"https://openalex.org/W7155599753","doi":"https://doi.org/10.1016/j.neunet.2026.109040","title":"Multi-particle neural operator transformer for solving partial differential equations","display_name":"Multi-particle neural operator transformer for solving partial differential equations","publication_year":2026,"publication_date":"2026-04-25","ids":{"openalex":"https://openalex.org/W7155599753","doi":"https://doi.org/10.1016/j.neunet.2026.109040","pmid":"https://pubmed.ncbi.nlm.nih.gov/42096883"},"language":"en","primary_location":{"id":"doi:10.1016/j.neunet.2026.109040","is_oa":false,"landing_page_url":"https://doi.org/10.1016/j.neunet.2026.109040","pdf_url":null,"source":{"id":"https://openalex.org/S123019304","display_name":"Neural Networks","issn_l":"0893-6080","issn":["0893-6080","1879-2782"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320990","host_organization_name":"Elsevier BV","host_organization_lineage":["https://openalex.org/P4310320990"],"host_organization_lineage_names":["Elsevier BV"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Neural Networks","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","pubmed"],"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/A5100755583","display_name":"Shengjun Liu","orcid":"https://orcid.org/0000-0002-3222-5656"},"institutions":[{"id":"https://openalex.org/I139660479","display_name":"Central South University","ror":"https://ror.org/00f1zfq44","country_code":"CN","type":"education","lineage":["https://openalex.org/I139660479"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Shengjun Liu","raw_affiliation_strings":["School of Mathematics and Statistics, Central South University, Changsha, 410083, China"],"raw_orcid":"https://orcid.org/0000-0002-3222-5656","affiliations":[{"raw_affiliation_string":"School of Mathematics and Statistics, Central South University, Changsha, 410083, China","institution_ids":["https://openalex.org/I139660479"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5100417217","display_name":"Yu Yu","orcid":"https://orcid.org/0000-0002-2455-787X"},"institutions":[{"id":"https://openalex.org/I139660479","display_name":"Central South University","ror":"https://ror.org/00f1zfq44","country_code":"CN","type":"education","lineage":["https://openalex.org/I139660479"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Yu Yu","raw_affiliation_strings":["School of Mathematics and Statistics, Central South University, Changsha, 410083, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematics and Statistics, Central South University, Changsha, 410083, China","institution_ids":["https://openalex.org/I139660479"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5134592569","display_name":"Chenxiang Fan","orcid":null},"institutions":[{"id":"https://openalex.org/I139660479","display_name":"Central South University","ror":"https://ror.org/00f1zfq44","country_code":"CN","type":"education","lineage":["https://openalex.org/I139660479"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Chenxiang Fan","raw_affiliation_strings":["School of Mathematics and Statistics, Central South University, Changsha, 410083, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematics and Statistics, Central South University, Changsha, 410083, China","institution_ids":["https://openalex.org/I139660479"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5134572121","display_name":"Ting Zhang","orcid":null},"institutions":[{"id":"https://openalex.org/I139660479","display_name":"Central South University","ror":"https://ror.org/00f1zfq44","country_code":"CN","type":"education","lineage":["https://openalex.org/I139660479"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Ting Zhang","raw_affiliation_strings":["School of Mathematics and Statistics, Central South University, Changsha, 410083, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematics and Statistics, Central South University, Changsha, 410083, China","institution_ids":["https://openalex.org/I139660479"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5032177188","display_name":"Hanchao Liu","orcid":"https://orcid.org/0009-0000-6556-6222"},"institutions":[{"id":"https://openalex.org/I139660479","display_name":"Central South University","ror":"https://ror.org/00f1zfq44","country_code":"CN","type":"education","lineage":["https://openalex.org/I139660479"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Hanchao Liu","raw_affiliation_strings":["School of Mathematics and Statistics, Central South University, Changsha, 410083, China. Electronic address: hchliu@csu.edu.cn"],"raw_orcid":"https://orcid.org/0009-0000-6556-6222","affiliations":[{"raw_affiliation_string":"School of Mathematics and Statistics, Central South University, Changsha, 410083, China. Electronic address: hchliu@csu.edu.cn","institution_ids":["https://openalex.org/I139660479"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5134592500","display_name":"Xinru Liu","orcid":null},"institutions":[{"id":"https://openalex.org/I139660479","display_name":"Central South University","ror":"https://ror.org/00f1zfq44","country_code":"CN","type":"education","lineage":["https://openalex.org/I139660479"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Xinru Liu","raw_affiliation_strings":["School of Mathematics and Statistics, Central South University, Changsha, 410083, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematics and Statistics, Central South University, Changsha, 410083, China","institution_ids":["https://openalex.org/I139660479"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5032177188"],"corresponding_institution_ids":["https://openalex.org/I139660479"],"apc_list":{"value":3350,"currency":"USD","value_usd":3350},"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.50799324,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"202","issue":null,"first_page":"109040","last_page":"109040"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11206","display_name":"Model Reduction and Neural Networks","score":0.809499979019165,"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":0.809499979019165,"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/T11948","display_name":"Machine Learning in Materials Science","score":0.060600001364946365,"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/T11804","display_name":"Quantum many-body systems","score":0.0357000008225441,"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/interpretability","display_name":"Interpretability","score":0.649399995803833},{"id":"https://openalex.org/keywords/operator","display_name":"Operator (biology)","score":0.6134999990463257},{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.5529999732971191},{"id":"https://openalex.org/keywords/partial-differential-equation","display_name":"Partial differential equation","score":0.4964999854564667},{"id":"https://openalex.org/keywords/generalization","display_name":"Generalization","score":0.4341000020503998},{"id":"https://openalex.org/keywords/dynamical-systems-theory","display_name":"Dynamical systems theory","score":0.4334999918937683},{"id":"https://openalex.org/keywords/transformer","display_name":"Transformer","score":0.4307999908924103},{"id":"https://openalex.org/keywords/deep-learning","display_name":"Deep learning","score":0.40700000524520874}],"concepts":[{"id":"https://openalex.org/C2781067378","wikidata":"https://www.wikidata.org/wiki/Q17027399","display_name":"Interpretability","level":2,"score":0.649399995803833},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.6184999942779541},{"id":"https://openalex.org/C17020691","wikidata":"https://www.wikidata.org/wiki/Q139677","display_name":"Operator (biology)","level":5,"score":0.6134999990463257},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.5529999732971191},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.4964999854564667},{"id":"https://openalex.org/C177148314","wikidata":"https://www.wikidata.org/wiki/Q170084","display_name":"Generalization","level":2,"score":0.4341000020503998},{"id":"https://openalex.org/C79379906","wikidata":"https://www.wikidata.org/wiki/Q3174497","display_name":"Dynamical systems theory","level":2,"score":0.4334999918937683},{"id":"https://openalex.org/C66322947","wikidata":"https://www.wikidata.org/wiki/Q11658","display_name":"Transformer","level":3,"score":0.4307999908924103},{"id":"https://openalex.org/C108583219","wikidata":"https://www.wikidata.org/wiki/Q197536","display_name":"Deep learning","level":2,"score":0.40700000524520874},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.397599995136261},{"id":"https://openalex.org/C24902271","wikidata":"https://www.wikidata.org/wiki/Q7449297","display_name":"Semi-elliptic operator","level":3,"score":0.39149999618530273},{"id":"https://openalex.org/C91873725","wikidata":"https://www.wikidata.org/wiki/Q3445816","display_name":"Function approximation","level":3,"score":0.3747999966144562},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.37470000982284546},{"id":"https://openalex.org/C185798385","wikidata":"https://www.wikidata.org/wiki/Q1161707","display_name":"Benchmark (surveying)","level":2,"score":0.3686999976634979},{"id":"https://openalex.org/C53846429","wikidata":"https://www.wikidata.org/wiki/Q186475","display_name":"Partial derivative","level":2,"score":0.3479999899864197},{"id":"https://openalex.org/C70915906","wikidata":"https://www.wikidata.org/wiki/Q1058681","display_name":"Differential operator","level":2,"score":0.34790000319480896},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.310699999332428},{"id":"https://openalex.org/C134765980","wikidata":"https://www.wikidata.org/wiki/Q879126","display_name":"Bitwise operation","level":2,"score":0.30250000953674316},{"id":"https://openalex.org/C184720557","wikidata":"https://www.wikidata.org/wiki/Q7825049","display_name":"Topology (electrical circuits)","level":2,"score":0.2838999927043915},{"id":"https://openalex.org/C102519508","wikidata":"https://www.wikidata.org/wiki/Q6520159","display_name":"Fourier transform","level":2,"score":0.28029999136924744},{"id":"https://openalex.org/C13355873","wikidata":"https://www.wikidata.org/wiki/Q2920850","display_name":"Connection (principal bundle)","level":2,"score":0.27649998664855957},{"id":"https://openalex.org/C65630602","wikidata":"https://www.wikidata.org/wiki/Q2278680","display_name":"Shift operator","level":4,"score":0.27390000224113464},{"id":"https://openalex.org/C43929395","wikidata":"https://www.wikidata.org/wiki/Q1198874","display_name":"Operator theory","level":2,"score":0.26510000228881836},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.2623000144958496},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.2533000111579895}],"mesh":[{"descriptor_ui":"D000465","descriptor_name":"Algorithms","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D016571","descriptor_name":"Neural Networks, Computer","qualifier_ui":null,"qualifier_name":null,"is_major_topic":true}],"locations_count":2,"locations":[{"id":"doi:10.1016/j.neunet.2026.109040","is_oa":false,"landing_page_url":"https://doi.org/10.1016/j.neunet.2026.109040","pdf_url":null,"source":{"id":"https://openalex.org/S123019304","display_name":"Neural Networks","issn_l":"0893-6080","issn":["0893-6080","1879-2782"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320990","host_organization_name":"Elsevier BV","host_organization_lineage":["https://openalex.org/P4310320990"],"host_organization_lineage_names":["Elsevier BV"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Neural Networks","raw_type":"journal-article"},{"id":"pmid:42096883","is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/42096883","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":"Neural networks : the official journal of the International Neural Network Society","raw_type":"Journal Article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"},{"id":"https://openalex.org/F4320326217","display_name":"Hunan Provincial Innovation Foundation for Postgraduate","ror":null}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":12,"referenced_works":["https://openalex.org/W2021535288","https://openalex.org/W2137983211","https://openalex.org/W2139923370","https://openalex.org/W2239232218","https://openalex.org/W2603909116","https://openalex.org/W2612252966","https://openalex.org/W2974916071","https://openalex.org/W2979786244","https://openalex.org/W3201666041","https://openalex.org/W4221019179","https://openalex.org/W4321793797","https://openalex.org/W4411090585"],"related_works":[],"abstract_inverted_index":null,"counts_by_year":[],"updated_date":"2026-07-19T07:52:34.831488","created_date":"2026-04-26T00:00:00"}
