{"id":"https://openalex.org/W7125366210","doi":"https://doi.org/10.1016/j.camwa.2026.01.019","title":"ITPINN : Integral-trainable physics informed neural network for solving high-dimensional evolution non-local partial differential equations","display_name":"ITPINN : Integral-trainable physics informed neural network for solving high-dimensional evolution non-local partial differential equations","publication_year":2026,"publication_date":"2026-01-21","ids":{"openalex":"https://openalex.org/W7125366210","doi":"https://doi.org/10.1016/j.camwa.2026.01.019"},"language":"en","primary_location":{"id":"doi:10.1016/j.camwa.2026.01.019","is_oa":false,"landing_page_url":"https://doi.org/10.1016/j.camwa.2026.01.019","pdf_url":null,"source":{"id":"https://openalex.org/S4210205691","display_name":"Computers & Mathematics with Applications","issn_l":"0898-1221","issn":["0898-1221","1873-7668"],"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":"Computers & Mathematics with Applications","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/A5123602972","display_name":"Wenkai Liu","orcid":null},"institutions":[{"id":"https://openalex.org/I2722730","display_name":"Inner Mongolia University","ror":"https://ror.org/0106qb496","country_code":"CN","type":"education","lineage":["https://openalex.org/I2722730"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Wenkai Liu","raw_affiliation_strings":["School of Mathematical Sciences, Inner Mongolia University, Hohhot, 010030, China"],"raw_orcid":"https://orcid.org/0000-0002-8708-3587","affiliations":[{"raw_affiliation_string":"School of Mathematical Sciences, Inner Mongolia University, Hohhot, 010030, China","institution_ids":["https://openalex.org/I2722730"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5052906032","display_name":"Fanhai Zeng","orcid":"https://orcid.org/0000-0003-4507-1278"},"institutions":[{"id":"https://openalex.org/I154099455","display_name":"Shandong University","ror":"https://ror.org/0207yh398","country_code":"CN","type":"education","lineage":["https://openalex.org/I154099455"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Fanhai Zeng","raw_affiliation_strings":["School of Mathematics, Shandong University, Jinan, 250100, China"],"raw_orcid":"https://orcid.org/0000-0003-4507-1278","affiliations":[{"raw_affiliation_string":"School of Mathematics, Shandong University, Jinan, 250100, China","institution_ids":["https://openalex.org/I154099455"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5123627283","display_name":"Hong Li","orcid":null},"institutions":[{"id":"https://openalex.org/I2722730","display_name":"Inner Mongolia University","ror":"https://ror.org/0106qb496","country_code":"CN","type":"education","lineage":["https://openalex.org/I2722730"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Hong Li","raw_affiliation_strings":["Inner Mongolia Key Laboratory of Mathematical Modeling and Scientific Computing, Hohhot, 010030, China","School of Mathematical Sciences, Inner Mongolia University, Hohhot, 010030, China"],"raw_orcid":"https://orcid.org/0000-0002-9832-0496","affiliations":[{"raw_affiliation_string":"Inner Mongolia Key Laboratory of Mathematical Modeling and Scientific Computing, Hohhot, 010030, China","institution_ids":["https://openalex.org/I2722730"]},{"raw_affiliation_string":"School of Mathematical Sciences, Inner Mongolia University, Hohhot, 010030, China","institution_ids":["https://openalex.org/I2722730"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5123626071","display_name":"Yang Liu","orcid":null},"institutions":[{"id":"https://openalex.org/I2722730","display_name":"Inner Mongolia University","ror":"https://ror.org/0106qb496","country_code":"CN","type":"education","lineage":["https://openalex.org/I2722730"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Yang Liu","raw_affiliation_strings":["Inner Mongolia Key Laboratory of Mathematical Modeling and Scientific Computing, Hohhot, 010030, China","School of Mathematical Sciences, Inner Mongolia University, Hohhot, 010030, China"],"raw_orcid":"https://orcid.org/0000-0001-8218-0196","affiliations":[{"raw_affiliation_string":"Inner Mongolia Key Laboratory of Mathematical Modeling and Scientific Computing, Hohhot, 010030, China","institution_ids":["https://openalex.org/I2722730"]},{"raw_affiliation_string":"School of Mathematical Sciences, Inner Mongolia University, Hohhot, 010030, China","institution_ids":["https://openalex.org/I2722730"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5123626071"],"corresponding_institution_ids":["https://openalex.org/I2722730"],"apc_list":{"value":3480,"currency":"USD","value_usd":3480},"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.15743643,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"205","issue":null,"first_page":"282","last_page":"317"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11206","display_name":"Model Reduction and Neural Networks","score":0.838100016117096,"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":0.838100016117096,"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/T10288","display_name":"Fractional Differential Equations Solutions","score":0.07400000095367432,"subfield":{"id":"https://openalex.org/subfields/2611","display_name":"Modeling and Simulation"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11416","display_name":"Numerical methods for differential equations","score":0.02590000070631504,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.5716999769210815},{"id":"https://openalex.org/keywords/fractional-calculus","display_name":"Fractional calculus","score":0.5658000111579895},{"id":"https://openalex.org/keywords/discretization","display_name":"Discretization","score":0.5552999973297119},{"id":"https://openalex.org/keywords/robustness","display_name":"Robustness (evolution)","score":0.5293999910354614},{"id":"https://openalex.org/keywords/partial-differential-equation","display_name":"Partial differential equation","score":0.5163000226020813},{"id":"https://openalex.org/keywords/integral-equation","display_name":"Integral equation","score":0.501800000667572},{"id":"https://openalex.org/keywords/numerical-integration","display_name":"Numerical integration","score":0.4341999888420105},{"id":"https://openalex.org/keywords/interpolation","display_name":"Interpolation (computer graphics)","score":0.43070000410079956},{"id":"https://openalex.org/keywords/functional-integration","display_name":"Functional integration","score":0.42170000076293945}],"concepts":[{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.5716999769210815},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.571399986743927},{"id":"https://openalex.org/C154249771","wikidata":"https://www.wikidata.org/wiki/Q1339058","display_name":"Fractional calculus","level":2,"score":0.5658000111579895},{"id":"https://openalex.org/C73000952","wikidata":"https://www.wikidata.org/wiki/Q17007827","display_name":"Discretization","level":2,"score":0.5552999973297119},{"id":"https://openalex.org/C63479239","wikidata":"https://www.wikidata.org/wiki/Q7353546","display_name":"Robustness (evolution)","level":3,"score":0.5293999910354614},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.5163000226020813},{"id":"https://openalex.org/C27016315","wikidata":"https://www.wikidata.org/wiki/Q580101","display_name":"Integral equation","level":2,"score":0.501800000667572},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.46939998865127563},{"id":"https://openalex.org/C127349201","wikidata":"https://www.wikidata.org/wiki/Q753445","display_name":"Numerical integration","level":2,"score":0.4341999888420105},{"id":"https://openalex.org/C137800194","wikidata":"https://www.wikidata.org/wiki/Q11713455","display_name":"Interpolation (computer graphics)","level":3,"score":0.43070000410079956},{"id":"https://openalex.org/C101812284","wikidata":"https://www.wikidata.org/wiki/Q2450135","display_name":"Functional integration","level":3,"score":0.42170000076293945},{"id":"https://openalex.org/C53846429","wikidata":"https://www.wikidata.org/wiki/Q186475","display_name":"Partial derivative","level":2,"score":0.40470001101493835},{"id":"https://openalex.org/C51577431","wikidata":"https://www.wikidata.org/wiki/Q1788036","display_name":"Volume integral","level":3,"score":0.3634999990463257},{"id":"https://openalex.org/C2776036281","wikidata":"https://www.wikidata.org/wiki/Q48769818","display_name":"Constraint (computer-aided design)","level":2,"score":0.3492000102996826},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.3481000065803528},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.32280001044273376},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.31029999256134033},{"id":"https://openalex.org/C2780801425","wikidata":"https://www.wikidata.org/wiki/Q5164392","display_name":"Construct (python library)","level":2,"score":0.2969000041484833},{"id":"https://openalex.org/C147168706","wikidata":"https://www.wikidata.org/wiki/Q1457734","display_name":"Recurrent neural network","level":3,"score":0.2939000129699707},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.29170000553131104},{"id":"https://openalex.org/C48753275","wikidata":"https://www.wikidata.org/wiki/Q11216","display_name":"Numerical analysis","level":2,"score":0.2734000086784363},{"id":"https://openalex.org/C75023562","wikidata":"https://www.wikidata.org/wiki/Q467699","display_name":"Line integral","level":3,"score":0.271699994802475},{"id":"https://openalex.org/C93226319","wikidata":"https://www.wikidata.org/wiki/Q193137","display_name":"Differential (mechanical device)","level":2,"score":0.27140000462532043},{"id":"https://openalex.org/C191832335","wikidata":"https://www.wikidata.org/wiki/Q4262191","display_name":"Fourier integral operator","level":3,"score":0.2711000144481659},{"id":"https://openalex.org/C116672817","wikidata":"https://www.wikidata.org/wiki/Q1454986","display_name":"Physical system","level":2,"score":0.25940001010894775},{"id":"https://openalex.org/C98045186","wikidata":"https://www.wikidata.org/wiki/Q205663","display_name":"Process (computing)","level":2,"score":0.2558000087738037},{"id":"https://openalex.org/C61797465","wikidata":"https://www.wikidata.org/wiki/Q1188986","display_name":"Term (time)","level":2,"score":0.25529998540878296}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1016/j.camwa.2026.01.019","is_oa":false,"landing_page_url":"https://doi.org/10.1016/j.camwa.2026.01.019","pdf_url":null,"source":{"id":"https://openalex.org/S4210205691","display_name":"Computers & Mathematics with Applications","issn_l":"0898-1221","issn":["0898-1221","1873-7668"],"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":"Computers & Mathematics with Applications","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"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":32,"referenced_works":["https://openalex.org/W2018698290","https://openalex.org/W2031022806","https://openalex.org/W2048626366","https://openalex.org/W2051434435","https://openalex.org/W2080946053","https://openalex.org/W2606385482","https://openalex.org/W2899283552","https://openalex.org/W2962996339","https://openalex.org/W2965419988","https://openalex.org/W2969381807","https://openalex.org/W3144964188","https://openalex.org/W3177788059","https://openalex.org/W4206264522","https://openalex.org/W4225995462","https://openalex.org/W4226109880","https://openalex.org/W4281624873","https://openalex.org/W4292440370","https://openalex.org/W4296060623","https://openalex.org/W4311155325","https://openalex.org/W4312056741","https://openalex.org/W4318540755","https://openalex.org/W4324092358","https://openalex.org/W4327571592","https://openalex.org/W4375858659","https://openalex.org/W4384404775","https://openalex.org/W4385520129","https://openalex.org/W4387613886","https://openalex.org/W4388616107","https://openalex.org/W4388839581","https://openalex.org/W4388904742","https://openalex.org/W4392138711","https://openalex.org/W4396894373"],"related_works":[],"abstract_inverted_index":null,"counts_by_year":[],"updated_date":"2026-01-23T23:24:52.574035","created_date":"2026-01-23T00:00:00"}