{"id":"https://openalex.org/W4401994702","doi":"https://doi.org/10.3390/axioms13090588","title":"A Physics-Informed Neural Network Based on the Boltzmann Equation with Multiple-Relaxation-Time Collision Operators","display_name":"A Physics-Informed Neural Network Based on the Boltzmann Equation with Multiple-Relaxation-Time Collision Operators","publication_year":2024,"publication_date":"2024-08-29","ids":{"openalex":"https://openalex.org/W4401994702","doi":"https://doi.org/10.3390/axioms13090588"},"language":"en","primary_location":{"id":"doi:10.3390/axioms13090588","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms13090588","pdf_url":"https://www.mdpi.com/2075-1680/13/9/588/pdf?version=1724924180","source":{"id":"https://openalex.org/S4210173132","display_name":"Axioms","issn_l":"2075-1680","issn":["2075-1680"],"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":"Axioms","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2075-1680/13/9/588/pdf?version=1724924180","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5100743103","display_name":"Zhixiang Liu","orcid":"https://orcid.org/0000-0001-6554-9617"},"institutions":[{"id":"https://openalex.org/I44675526","display_name":"Shanghai Ocean University","ror":"https://ror.org/04n40zv07","country_code":"CN","type":"education","lineage":["https://openalex.org/I44675526"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Zhixiang Liu","raw_affiliation_strings":["College of Information Technology, Shanghai Ocean University, Shanghai 201306, China"],"raw_orcid":"https://orcid.org/0000-0001-6554-9617","affiliations":[{"raw_affiliation_string":"College of Information Technology, Shanghai Ocean University, Shanghai 201306, China","institution_ids":["https://openalex.org/I44675526"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5068648599","display_name":"Chenkai Zhang","orcid":"https://orcid.org/0009-0004-1583-0506"},"institutions":[{"id":"https://openalex.org/I44675526","display_name":"Shanghai Ocean University","ror":"https://ror.org/04n40zv07","country_code":"CN","type":"education","lineage":["https://openalex.org/I44675526"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Chenkai Zhang","raw_affiliation_strings":["College of Information Technology, Shanghai Ocean University, Shanghai 201306, China"],"raw_orcid":"https://orcid.org/0009-0004-1583-0506","affiliations":[{"raw_affiliation_string":"College of Information Technology, Shanghai Ocean University, Shanghai 201306, China","institution_ids":["https://openalex.org/I44675526"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5036943629","display_name":"Wenhao Zhu","orcid":"https://orcid.org/0000-0002-9656-9781"},"institutions":[{"id":"https://openalex.org/I141962983","display_name":"Shanghai University of Engineering Science","ror":"https://ror.org/0557b9y08","country_code":"CN","type":"education","lineage":["https://openalex.org/I141962983"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Wenhao Zhu","raw_affiliation_strings":["School of Computer Engineering and Science, Shanghai University, Shanghai 200444, China"],"raw_orcid":"https://orcid.org/0000-0002-9656-9781","affiliations":[{"raw_affiliation_string":"School of Computer Engineering and Science, Shanghai University, Shanghai 200444, China","institution_ids":["https://openalex.org/I141962983"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5101470402","display_name":"Dongmei Huang","orcid":"https://orcid.org/0000-0002-5698-8279"},"institutions":[{"id":"https://openalex.org/I23632641","display_name":"Shanghai University of Electric Power","ror":"https://ror.org/02w4tny03","country_code":"CN","type":"education","lineage":["https://openalex.org/I23632641"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Dongmei Huang","raw_affiliation_strings":["College of Electronic and Information Engineering, Shanghai University of Electric Power, Shanghai 201306, China"],"raw_orcid":"https://orcid.org/0000-0002-5698-8279","affiliations":[{"raw_affiliation_string":"College of Electronic and Information Engineering, Shanghai University of Electric Power, Shanghai 201306, China","institution_ids":["https://openalex.org/I23632641"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5036943629"],"corresponding_institution_ids":["https://openalex.org/I141962983"],"apc_list":{"value":1400,"currency":"CHF","value_usd":1782},"apc_paid":{"value":1400,"currency":"CHF","value_usd":1782},"fwci":0.8152,"has_fulltext":true,"cited_by_count":4,"citation_normalized_percentile":{"value":0.69253137,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":98,"max":99},"biblio":{"volume":"13","issue":"9","first_page":"588","last_page":"588"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11206","display_name":"Model Reduction and Neural Networks","score":0.9998999834060669,"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":0.9998999834060669,"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/T11751","display_name":"Lattice Boltzmann Simulation Studies","score":0.9987000226974487,"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/T10104","display_name":"Nanofluid Flow and Heat Transfer","score":0.9851999878883362,"subfield":{"id":"https://openalex.org/subfields/2204","display_name":"Biomedical Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/boltzmann-equation","display_name":"Boltzmann equation","score":0.8494349718093872},{"id":"https://openalex.org/keywords/collision","display_name":"Collision","score":0.5441956520080566},{"id":"https://openalex.org/keywords/lattice-boltzmann-methods","display_name":"Lattice Boltzmann methods","score":0.5393751263618469},{"id":"https://openalex.org/keywords/boltzmann-constant","display_name":"Boltzmann constant","score":0.5173668265342712},{"id":"https://openalex.org/keywords/relaxation","display_name":"Relaxation (psychology)","score":0.5106459259986877},{"id":"https://openalex.org/keywords/boltzmann-machine","display_name":"Boltzmann machine","score":0.5042315721511841},{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.4828563332557678},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.4592748284339905},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.41014519333839417},{"id":"https://openalex.org/keywords/statistical-physics","display_name":"Statistical physics","score":0.3899378180503845},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.38613778352737427},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.37475988268852234},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.3300206661224365},{"id":"https://openalex.org/keywords/mechanics","display_name":"Mechanics","score":0.14673826098442078},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.14630258083343506},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.09618791937828064}],"concepts":[{"id":"https://openalex.org/C165995430","wikidata":"https://www.wikidata.org/wiki/Q891653","display_name":"Boltzmann equation","level":2,"score":0.8494349718093872},{"id":"https://openalex.org/C121704057","wikidata":"https://www.wikidata.org/wiki/Q352070","display_name":"Collision","level":2,"score":0.5441956520080566},{"id":"https://openalex.org/C21821499","wikidata":"https://www.wikidata.org/wiki/Q1807064","display_name":"Lattice Boltzmann methods","level":2,"score":0.5393751263618469},{"id":"https://openalex.org/C35304006","wikidata":"https://www.wikidata.org/wiki/Q5962","display_name":"Boltzmann constant","level":2,"score":0.5173668265342712},{"id":"https://openalex.org/C2776029896","wikidata":"https://www.wikidata.org/wiki/Q3935810","display_name":"Relaxation (psychology)","level":2,"score":0.5106459259986877},{"id":"https://openalex.org/C192576344","wikidata":"https://www.wikidata.org/wiki/Q194706","display_name":"Boltzmann machine","level":3,"score":0.5042315721511841},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.4828563332557678},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4592748284339905},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.41014519333839417},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.3899378180503845},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.38613778352737427},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.37475988268852234},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.3300206661224365},{"id":"https://openalex.org/C57879066","wikidata":"https://www.wikidata.org/wiki/Q41217","display_name":"Mechanics","level":1,"score":0.14673826098442078},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.14630258083343506},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.09618791937828064},{"id":"https://openalex.org/C77805123","wikidata":"https://www.wikidata.org/wiki/Q161272","display_name":"Social psychology","level":1,"score":0.0},{"id":"https://openalex.org/C38652104","wikidata":"https://www.wikidata.org/wiki/Q3510521","display_name":"Computer security","level":1,"score":0.0},{"id":"https://openalex.org/C15744967","wikidata":"https://www.wikidata.org/wiki/Q9418","display_name":"Psychology","level":0,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/axioms13090588","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms13090588","pdf_url":"https://www.mdpi.com/2075-1680/13/9/588/pdf?version=1724924180","source":{"id":"https://openalex.org/S4210173132","display_name":"Axioms","issn_l":"2075-1680","issn":["2075-1680"],"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":"Axioms","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:5aa085d1b5d04fefb90b9950040682b4","is_oa":false,"landing_page_url":"https://doaj.org/article/5aa085d1b5d04fefb90b9950040682b4","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":"Axioms, Vol 13, Iss 9, p 588 (2024)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2075-1680/13/9/588/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/axioms13090588","pdf_url":null,"source":{"id":"https://openalex.org/S4306400947","display_name":"MDPI (MDPI AG)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210097602","host_organization_name":"Multidisciplinary Digital Publishing Institute (Switzerland)","host_organization_lineage":["https://openalex.org/I4210097602"],"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":"Axioms","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/axioms13090588","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms13090588","pdf_url":"https://www.mdpi.com/2075-1680/13/9/588/pdf?version=1724924180","source":{"id":"https://openalex.org/S4210173132","display_name":"Axioms","issn_l":"2075-1680","issn":["2075-1680"],"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":"Axioms","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G8278445047","display_name":null,"funder_award_id":"42376194","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":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4401994702.pdf","grobid_xml":"https://content.openalex.org/works/W4401994702.grobid-xml"},"referenced_works_count":47,"referenced_works":["https://openalex.org/W70668290","https://openalex.org/W285310959","https://openalex.org/W1977437918","https://openalex.org/W1979614071","https://openalex.org/W1991849166","https://openalex.org/W1999914547","https://openalex.org/W2060308631","https://openalex.org/W2135014481","https://openalex.org/W2512102211","https://openalex.org/W2794506364","https://openalex.org/W2892374370","https://openalex.org/W2899283552","https://openalex.org/W2940612399","https://openalex.org/W2963446664","https://openalex.org/W2964341462","https://openalex.org/W2978281981","https://openalex.org/W3004368638","https://openalex.org/W3006689658","https://openalex.org/W3027214383","https://openalex.org/W3041682155","https://openalex.org/W3042578835","https://openalex.org/W3045146186","https://openalex.org/W3081461774","https://openalex.org/W3081716925","https://openalex.org/W3083555822","https://openalex.org/W3098195931","https://openalex.org/W3115401021","https://openalex.org/W3152076024","https://openalex.org/W3166864375","https://openalex.org/W3170277033","https://openalex.org/W3197818013","https://openalex.org/W4225142395","https://openalex.org/W4226036541","https://openalex.org/W4244676802","https://openalex.org/W4287756134","https://openalex.org/W4312790010","https://openalex.org/W4321495620","https://openalex.org/W4366993162","https://openalex.org/W4378364402","https://openalex.org/W4382618549","https://openalex.org/W4387470797","https://openalex.org/W4399457481","https://openalex.org/W6779753539","https://openalex.org/W6782303003","https://openalex.org/W6793010855","https://openalex.org/W6854201698","https://openalex.org/W6969359468"],"related_works":["https://openalex.org/W44857498","https://openalex.org/W4234744498","https://openalex.org/W2007632118","https://openalex.org/W2062900651","https://openalex.org/W2028660548","https://openalex.org/W2133316674","https://openalex.org/W2560572760","https://openalex.org/W2179571827","https://openalex.org/W4387929093","https://openalex.org/W1641020902"],"abstract_inverted_index":{"The":[0,89,127,137,166,180],"Boltzmann":[1],"equation":[2,51,97],"with":[3,105],"multiple-relaxation-time":[4],"(MRT)":[5],"collision":[6,77,103],"operators":[7],"has":[8],"been":[9],"widely":[10],"employed":[11],"in":[12,81,94],"kinetic":[13],"theory":[14],"to":[15,45,47,100,117,133,143],"describe":[16],"the":[17,24,27,34,49,59,69,73,86,95,102,110,119,125,145,148,161,164,172,176,183,188],"behavior":[18],"of":[19,30,36,91,121,147,163,182,195],"gases":[20],"and":[21,33,175],"liquids":[22],"at":[23,153,197],"macro-level.":[25],"Given":[26],"successful":[28],"development":[29],"deep":[31],"learning":[32,135],"availability":[35],"data":[37],"analytic":[38],"tools,":[39],"it":[40],"is":[41,79,98,115,131,141],"a":[42,53,63,193],"feasible":[43],"idea":[44],"try":[46],"solve":[48],"Boltzmann-MRT":[50,70,87,96],"using":[52],"neural":[54,66],"network-based":[55,189],"method.":[56],"Based":[57],"on":[58],"canonical":[60],"polyadic":[61],"decomposition,":[62],"new":[64],"physics-informed":[65],"network":[67,74],"describing":[68],"equation,":[71],"named":[72],"for":[75,84,124,150],"MRT":[76],"(NMRT),":[78],"proposed":[80,142],"this":[82],"paper":[83],"solving":[85],"equation.":[88],"method":[90,130,190],"tensor":[92],"decomposition":[93,129],"utilized":[99],"combine":[101],"matrix":[104],"discrete":[106],"distribution":[107],"functions":[108],"within":[109],"moment":[111],"space.":[112],"Multiscale":[113],"modeling":[114],"adopted":[116],"accelerate":[118],"convergence":[120],"high":[122],"frequencies":[123],"equations.":[126],"micro\u2013macro":[128],"applied":[132],"improve":[134,160],"efficiency.":[136],"problem-dependent":[138],"loss":[139],"function":[140,149],"balance":[144],"weight":[146],"different":[151,154],"conditions":[152],"velocities.":[155],"These":[156],"strategies":[157],"will":[158],"greatly":[159],"accuracy":[162,196],"network.":[165],"numerical":[167,184],"experiments":[168],"are":[169],"tested,":[170],"including":[171],"advection\u2013diffusion":[173],"problem":[174],"wave":[177],"propagation":[178],"problem.":[179],"results":[181],"simulation":[185],"show":[186],"that":[187],"can":[191],"obtain":[192],"measure":[194],"O10\u22123.":[198]},"counts_by_year":[{"year":2026,"cited_by_count":4}],"updated_date":"2026-07-22T07:51:19.307946","created_date":"2025-10-10T00:00:00"}
