{"id":"https://openalex.org/W4400324089","doi":"https://doi.org/10.3390/e26070577","title":"A Best-Fitting B-Spline Neural Network Approach to the Prediction of Advection\u2013Diffusion Physical Fields with Absorption and Source Terms","display_name":"A Best-Fitting B-Spline Neural Network Approach to the Prediction of Advection\u2013Diffusion Physical Fields with Absorption and Source Terms","publication_year":2024,"publication_date":"2024-07-04","ids":{"openalex":"https://openalex.org/W4400324089","doi":"https://doi.org/10.3390/e26070577","pmid":"https://pubmed.ncbi.nlm.nih.gov/39056939"},"language":"en","primary_location":{"id":"doi:10.3390/e26070577","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e26070577","pdf_url":"https://www.mdpi.com/1099-4300/26/7/577/pdf?version=1720081639","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/1099-4300/26/7/577/pdf?version=1720081639","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5069431237","display_name":"Xuedong Zhu","orcid":"https://orcid.org/0009-0007-6622-1008"},"institutions":[{"id":"https://openalex.org/I125839683","display_name":"Beijing Institute of Technology","ror":"https://ror.org/01skt4w74","country_code":"CN","type":"education","lineage":["https://openalex.org/I125839683","https://openalex.org/I890469752"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Xuedong Zhu","raw_affiliation_strings":["School of Mechanical Engineering, Beijing Institute of Technology, Beijing 100081, China"],"affiliations":[{"raw_affiliation_string":"School of Mechanical Engineering, Beijing Institute of Technology, Beijing 100081, China","institution_ids":["https://openalex.org/I125839683"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5100405327","display_name":"Jianhua Liu","orcid":"https://orcid.org/0000-0002-4881-2193"},"institutions":[{"id":"https://openalex.org/I125839683","display_name":"Beijing Institute of Technology","ror":"https://ror.org/01skt4w74","country_code":"CN","type":"education","lineage":["https://openalex.org/I125839683","https://openalex.org/I890469752"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Jianhua Liu","raw_affiliation_strings":["Hebei Key Laboratory of Intelligent Assembly and Detection Technology, Tangshan Research Institute, Beijing Institute of Technology, Tangshan 063000, China","School of Mechanical Engineering, Beijing Institute of Technology, Beijing 100081, China"],"affiliations":[{"raw_affiliation_string":"Hebei Key Laboratory of Intelligent Assembly and Detection Technology, Tangshan Research Institute, Beijing Institute of Technology, Tangshan 063000, China","institution_ids":["https://openalex.org/I125839683"]},{"raw_affiliation_string":"School of Mechanical Engineering, Beijing Institute of Technology, Beijing 100081, China","institution_ids":["https://openalex.org/I125839683"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5069172214","display_name":"Xiaohui Ao","orcid":"https://orcid.org/0009-0008-8609-3688"},"institutions":[{"id":"https://openalex.org/I125839683","display_name":"Beijing Institute of Technology","ror":"https://ror.org/01skt4w74","country_code":"CN","type":"education","lineage":["https://openalex.org/I125839683","https://openalex.org/I890469752"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Xiaohui Ao","raw_affiliation_strings":["Hebei Key Laboratory of Intelligent Assembly and Detection Technology, Tangshan Research Institute, Beijing Institute of Technology, Tangshan 063000, China","School of Mechanical Engineering, Beijing Institute of Technology, Beijing 100081, China"],"affiliations":[{"raw_affiliation_string":"Hebei Key Laboratory of Intelligent Assembly and Detection Technology, Tangshan Research Institute, Beijing Institute of Technology, Tangshan 063000, China","institution_ids":["https://openalex.org/I125839683"]},{"raw_affiliation_string":"School of Mechanical Engineering, Beijing Institute of Technology, Beijing 100081, China","institution_ids":["https://openalex.org/I125839683"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5025364871","display_name":"Shaofan He","orcid":"https://orcid.org/0000-0001-8518-6403"},"institutions":[{"id":"https://openalex.org/I125839683","display_name":"Beijing Institute of Technology","ror":"https://ror.org/01skt4w74","country_code":"CN","type":"education","lineage":["https://openalex.org/I125839683","https://openalex.org/I890469752"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Sen He","raw_affiliation_strings":["School of Mechanical Engineering, Beijing Institute of Technology, Beijing 100081, China"],"affiliations":[{"raw_affiliation_string":"School of Mechanical Engineering, Beijing Institute of Technology, Beijing 100081, China","institution_ids":["https://openalex.org/I125839683"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5024129605","display_name":"Lei Tao","orcid":"https://orcid.org/0000-0002-8285-1356"},"institutions":[{"id":"https://openalex.org/I125839683","display_name":"Beijing Institute of Technology","ror":"https://ror.org/01skt4w74","country_code":"CN","type":"education","lineage":["https://openalex.org/I125839683","https://openalex.org/I890469752"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Lei Tao","raw_affiliation_strings":["School of Mechanical Engineering, Beijing Institute of Technology, Beijing 100081, China"],"affiliations":[{"raw_affiliation_string":"School of Mechanical Engineering, Beijing Institute of Technology, Beijing 100081, China","institution_ids":["https://openalex.org/I125839683"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5100316830","display_name":"Feng Gao","orcid":"https://orcid.org/0009-0003-0136-958X"},"institutions":[{"id":"https://openalex.org/I125839683","display_name":"Beijing Institute of Technology","ror":"https://ror.org/01skt4w74","country_code":"CN","type":"education","lineage":["https://openalex.org/I125839683","https://openalex.org/I890469752"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Feng Gao","raw_affiliation_strings":["School of Mechanical Engineering, Beijing Institute of Technology, Beijing 100081, China"],"affiliations":[{"raw_affiliation_string":"School of Mechanical Engineering, Beijing Institute of Technology, Beijing 100081, China","institution_ids":["https://openalex.org/I125839683"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":6,"corresponding_author_ids":["https://openalex.org/A5069172214"],"corresponding_institution_ids":["https://openalex.org/I125839683"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":0.5531,"has_fulltext":false,"cited_by_count":2,"citation_normalized_percentile":{"value":0.63163862,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":95},"biblio":{"volume":"26","issue":"7","first_page":"577","last_page":"577"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11206","display_name":"Model Reduction and Neural Networks","score":0.991100013256073,"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":0.991100013256073,"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/T10360","display_name":"Fluid Dynamics and Turbulent Flows","score":0.984499990940094,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"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/T10466","display_name":"Meteorological Phenomena and Simulations","score":0.9768999814987183,"subfield":{"id":"https://openalex.org/subfields/1902","display_name":"Atmospheric Science"},"field":{"id":"https://openalex.org/fields/19","display_name":"Earth and Planetary Sciences"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/advection","display_name":"Advection","score":0.7192054986953735},{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.6824110746383667},{"id":"https://openalex.org/keywords/diffusion","display_name":"Diffusion","score":0.5796890258789062},{"id":"https://openalex.org/keywords/absorption","display_name":"Absorption (acoustics)","score":0.5460600256919861},{"id":"https://openalex.org/keywords/spline","display_name":"Spline (mechanical)","score":0.5001862049102783},{"id":"https://openalex.org/keywords/b-spline","display_name":"B-spline","score":0.48774534463882446},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.45504701137542725},{"id":"https://openalex.org/keywords/statistical-physics","display_name":"Statistical physics","score":0.39663952589035034},{"id":"https://openalex.org/keywords/biological-system","display_name":"Biological system","score":0.37352806329727173},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.36310064792633057},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.354231059551239},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.2917212247848511},{"id":"https://openalex.org/keywords/machine-learning","display_name":"Machine learning","score":0.2837812006473541},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.2800501585006714},{"id":"https://openalex.org/keywords/thermodynamics","display_name":"Thermodynamics","score":0.22904443740844727},{"id":"https://openalex.org/keywords/optics","display_name":"Optics","score":0.1601146161556244}],"concepts":[{"id":"https://openalex.org/C5072599","wikidata":"https://www.wikidata.org/wiki/Q379788","display_name":"Advection","level":2,"score":0.7192054986953735},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.6824110746383667},{"id":"https://openalex.org/C69357855","wikidata":"https://www.wikidata.org/wiki/Q163214","display_name":"Diffusion","level":2,"score":0.5796890258789062},{"id":"https://openalex.org/C125287762","wikidata":"https://www.wikidata.org/wiki/Q1758948","display_name":"Absorption (acoustics)","level":2,"score":0.5460600256919861},{"id":"https://openalex.org/C10390562","wikidata":"https://www.wikidata.org/wiki/Q581809","display_name":"Spline (mechanical)","level":2,"score":0.5001862049102783},{"id":"https://openalex.org/C15945459","wikidata":"https://www.wikidata.org/wiki/Q2083109","display_name":"B-spline","level":2,"score":0.48774534463882446},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.45504701137542725},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.39663952589035034},{"id":"https://openalex.org/C186060115","wikidata":"https://www.wikidata.org/wiki/Q30336093","display_name":"Biological system","level":1,"score":0.37352806329727173},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.36310064792633057},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.354231059551239},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.2917212247848511},{"id":"https://openalex.org/C119857082","wikidata":"https://www.wikidata.org/wiki/Q2539","display_name":"Machine learning","level":1,"score":0.2837812006473541},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.2800501585006714},{"id":"https://openalex.org/C97355855","wikidata":"https://www.wikidata.org/wiki/Q11473","display_name":"Thermodynamics","level":1,"score":0.22904443740844727},{"id":"https://openalex.org/C120665830","wikidata":"https://www.wikidata.org/wiki/Q14620","display_name":"Optics","level":1,"score":0.1601146161556244},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.3390/e26070577","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e26070577","pdf_url":"https://www.mdpi.com/1099-4300/26/7/577/pdf?version=1720081639","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:39056939","is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/39056939","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:pubmedcentral.nih.gov:11275367","is_oa":true,"landing_page_url":"https://www.ncbi.nlm.nih.gov/pmc/articles/11275367","pdf_url":null,"source":{"id":"https://openalex.org/S2764455111","display_name":"PubMed Central","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":"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"},{"id":"pmh:oai:doaj.org/article:9b8b5165bcca4162b1638f2f69c540dd","is_oa":true,"landing_page_url":"https://doaj.org/article/9b8b5165bcca4162b1638f2f69c540dd","pdf_url":null,"source":{"id":"https://openalex.org/S112646816","display_name":"SHILAP Revista de lepidopterolog\u00eda","issn_l":"0300-5267","issn":["0300-5267","2340-4078"],"is_oa":true,"is_in_doaj":true,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Entropy, Vol 26, Iss 7, p 577 (2024)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.3390/e26070577","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e26070577","pdf_url":"https://www.mdpi.com/1099-4300/26/7/577/pdf?version=1720081639","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/G3133320703","display_name":null,"funder_award_id":"52105504","funder_id":"https://openalex.org/F4320335777","funder_display_name":"National Key Research and Development Program of China"},{"id":"https://openalex.org/G5055632298","display_name":null,"funder_award_id":"52105504","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G6021993448","display_name":null,"funder_award_id":"2022YFB3403800","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G8760886659","display_name":null,"funder_award_id":"2022YFB3403800","funder_id":"https://openalex.org/F4320335777","funder_display_name":"National Key Research and Development Program of China"}],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"},{"id":"https://openalex.org/F4320335777","display_name":"National Key Research and Development Program of China","ror":null}],"has_content":{"grobid_xml":false,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4400324089.pdf"},"referenced_works_count":45,"referenced_works":["https://openalex.org/W1972591580","https://openalex.org/W1972604265","https://openalex.org/W2006981280","https://openalex.org/W2066735467","https://openalex.org/W2080518109","https://openalex.org/W2130888122","https://openalex.org/W2134434931","https://openalex.org/W2166667156","https://openalex.org/W2203044942","https://openalex.org/W2574882494","https://openalex.org/W2608535648","https://openalex.org/W2736566689","https://openalex.org/W2910892140","https://openalex.org/W2946771678","https://openalex.org/W2950883932","https://openalex.org/W2989133617","https://openalex.org/W2992578100","https://openalex.org/W2997179704","https://openalex.org/W3000388076","https://openalex.org/W3008747420","https://openalex.org/W3014009018","https://openalex.org/W3036063661","https://openalex.org/W3039152077","https://openalex.org/W3041481721","https://openalex.org/W3093078230","https://openalex.org/W3096831136","https://openalex.org/W3099969702","https://openalex.org/W3119602513","https://openalex.org/W3123883114","https://openalex.org/W3124389259","https://openalex.org/W3131838898","https://openalex.org/W3159544519","https://openalex.org/W3194190144","https://openalex.org/W3211378683","https://openalex.org/W4205932592","https://openalex.org/W4213123144","https://openalex.org/W4220711990","https://openalex.org/W4225547518","https://openalex.org/W4226282825","https://openalex.org/W4283824472","https://openalex.org/W4306709449","https://openalex.org/W4309726528","https://openalex.org/W4310193759","https://openalex.org/W4313367698","https://openalex.org/W4321438589"],"related_works":["https://openalex.org/W3036101264","https://openalex.org/W2163151373","https://openalex.org/W1986676657","https://openalex.org/W2131662362","https://openalex.org/W2052425187","https://openalex.org/W2766141905","https://openalex.org/W3153287478","https://openalex.org/W3084529718","https://openalex.org/W1566260961","https://openalex.org/W2487825112"],"abstract_inverted_index":{"This":[0,88],"paper":[1],"proposed":[2,133,183],"a":[3,14,34,47,67,71,116,139,145,190],"two-dimensional":[4,96],"steady-state":[5],"field":[6,22,79,127,130],"prediction":[7,192],"approach":[8],"that":[9,153],"combines":[10],"B-spline":[11,36,86,123,155,175],"functions":[12],"and":[13,45,58,75,94,103,106,109,113,129,144,180,199],"fully":[15,48],"connected":[16,49],"neural":[17,50,63,147,156],"network.":[18],"In":[19],"this":[20],"approach,":[21],"data,":[23,198],"which":[24],"are":[25,31],"determined":[26],"by":[27,33,115,171],"corresponding":[28,41,78],"control":[29,59,73],"vectors,":[30,44],"fitted":[32],"selected":[35,85,174],"function":[37],"set,":[38],"yielding":[39],"the":[40,77,84,154,160,165,173,182,187],"best-fitting":[42],"weight":[43,56,68],"then":[46,76],"network":[51,64,142,148,157],"is":[52],"trained":[53,62],"using":[54,70],"those":[55],"vectors":[57],"vectors.":[60],"The":[61,132,150],"first":[65],"predicts":[66],"vector":[69],"given":[72],"vector,":[74],"can":[80,168],"be":[81,169],"restored":[82],"via":[83],"set.":[87,176],"method":[89,134,184],"was":[90,135],"applied":[91],"to":[92],"learn":[93],"predict":[95,159],"steady":[97],"advection-diffusion":[98],"physical":[99,162],"fields":[100,163],"with":[101,121,138,178],"absorption":[102],"source":[104],"terms,":[105],"its":[107],"accuracy":[108],"performance":[110],"were":[111],"tested":[112,161],"verified":[114],"series":[117],"of":[118,189],"numerical":[119],"experiments":[120],"different":[122],"sets,":[124],"boundary":[125],"conditions,":[126],"gradients,":[128],"states.":[131],"finally":[136],"compared":[137],"generative":[140],"adversarial":[141],"(GAN)":[143],"physics-informed":[146],"(PINN).":[149],"results":[151],"indicated":[152],"could":[158],"well;":[164],"overall":[166],"error":[167],"reduced":[170],"expanding":[172],"Compared":[177],"GAN":[179],"PINN,":[181],"also":[185],"presented":[186],"advantages":[188],"high":[191,200],"accuracy,":[193],"less":[194],"demand":[195],"for":[196],"training":[197,201],"efficiency.":[202]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":1}],"updated_date":"2026-04-09T08:11:56.329763","created_date":"2025-10-10T00:00:00"}