{"id":"https://openalex.org/W2935339072","doi":"https://doi.org/10.32604/cmc.2019.06641","title":"Artificial Neural Network Methods for the Solution of Second Order Boundary Value Problems","display_name":"Artificial Neural Network Methods for the Solution of Second Order Boundary Value Problems","publication_year":2019,"publication_date":"2019-01-01","ids":{"openalex":"https://openalex.org/W2935339072","doi":"https://doi.org/10.32604/cmc.2019.06641","mag":"2935339072"},"language":"en","primary_location":{"id":"doi:10.32604/cmc.2019.06641","doi":"https://doi.org/10.32604/cmc.2019.06641","is_oa":true,"landing_page_url":"https://doi.org/10.32604/cmc.2019.06641","pdf_url":null,"source":{"id":"https://openalex.org/S4210191605","display_name":"Computers, materials & continua/Computers, materials & continua (Print)","issn_l":"1546-2218","issn":["1546-2218","1546-2226"],"is_oa":true,"is_in_doaj":false,"is_core":true,"listed_in":["cwts-core","dongbi-d","jufo-1","ki-jl-1","norway-1","poland-20","russia-white-list-1"],"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":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Computers, Materials &amp; Continua","raw_type":"journal-article"},"volume":null,"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"https://doi.org/10.32604/cmc.2019.06641","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5018353156","display_name":"Cosmin Anitescu","orcid":"https://orcid.org/0000-0002-3791-6085","observed_orcids":["https://orcid.org/0000-0002-3791-6085"]},"institutions":[{"id":"https://openalex.org/I28022161","display_name":"King Saud University","ror":"https://ror.org/02f81g417","country_code":"SA","type":"education","lineage":["https://openalex.org/I28022161"]},{"id":"https://openalex.org/I31746571","display_name":"UNSW Sydney","ror":"https://ror.org/03r8z3t63","country_code":"AU","type":"education","lineage":["https://openalex.org/I31746571"]},{"id":"https://openalex.org/I51441396","display_name":"Bauhaus-Universit\u00e4t Weimar","ror":"https://ror.org/033bb5z47","country_code":"DE","type":"education","lineage":["https://openalex.org/I51441396"]}],"countries":["AU","DE","SA"],"is_corresponding":false,"raw_author_name":"Cosmin Anitescu","raw_affiliation_strings":["Institute of Structural Mechanics, Bauhaus-Universit\u00e4t Weimar, 99423, Weimar, Germany. 2 School of Civil & Environmental Engineering, University of New South Wales, Sydney, Australia. 3 Department of Computer Engineering, College of Computer and Information Sciences, King Saud University, Riyadh, Saudi Arabia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institute of Structural Mechanics, Bauhaus-Universit\u00e4t Weimar, 99423, Weimar, Germany. 2 School of Civil & Environmental Engineering, University of New South Wales, Sydney, Australia. 3 Department of Computer Engineering, College of Computer and Information Sciences, King Saud University, Riyadh, Saudi Arabia","institution_ids":["https://openalex.org/I28022161","https://openalex.org/I31746571","https://openalex.org/I51441396"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5008984779","display_name":"Elena Atroshchenko","orcid":"https://orcid.org/0000-0002-7871-9926","observed_orcids":["https://orcid.org/0000-0002-7871-9926"]},"institutions":[{"id":"https://openalex.org/I28022161","display_name":"King Saud University","ror":"https://ror.org/02f81g417","country_code":"SA","type":"education","lineage":["https://openalex.org/I28022161"]},{"id":"https://openalex.org/I31746571","display_name":"UNSW Sydney","ror":"https://ror.org/03r8z3t63","country_code":"AU","type":"education","lineage":["https://openalex.org/I31746571"]},{"id":"https://openalex.org/I51441396","display_name":"Bauhaus-Universit\u00e4t Weimar","ror":"https://ror.org/033bb5z47","country_code":"DE","type":"education","lineage":["https://openalex.org/I51441396"]}],"countries":["AU","DE","SA"],"is_corresponding":false,"raw_author_name":"Elena Atroshchenko","raw_affiliation_strings":["Institute of Structural Mechanics, Bauhaus-Universit\u00e4t Weimar, 99423, Weimar, Germany. 2 School of Civil & Environmental Engineering, University of New South Wales, Sydney, Australia. 3 Department of Computer Engineering, College of Computer and Information Sciences, King Saud University, Riyadh, Saudi Arabia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institute of Structural Mechanics, Bauhaus-Universit\u00e4t Weimar, 99423, Weimar, Germany. 2 School of Civil & Environmental Engineering, University of New South Wales, Sydney, Australia. 3 Department of Computer Engineering, College of Computer and Information Sciences, King Saud University, Riyadh, Saudi Arabia","institution_ids":["https://openalex.org/I28022161","https://openalex.org/I31746571","https://openalex.org/I51441396"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5052594698","display_name":"Naif Alajlan","orcid":"https://orcid.org/0000-0003-1846-1131","observed_orcids":["https://orcid.org/0000-0003-1846-1131"]},"institutions":[{"id":"https://openalex.org/I28022161","display_name":"King Saud University","ror":"https://ror.org/02f81g417","country_code":"SA","type":"education","lineage":["https://openalex.org/I28022161"]},{"id":"https://openalex.org/I31746571","display_name":"UNSW Sydney","ror":"https://ror.org/03r8z3t63","country_code":"AU","type":"education","lineage":["https://openalex.org/I31746571"]},{"id":"https://openalex.org/I51441396","display_name":"Bauhaus-Universit\u00e4t Weimar","ror":"https://ror.org/033bb5z47","country_code":"DE","type":"education","lineage":["https://openalex.org/I51441396"]}],"countries":["AU","DE","SA"],"is_corresponding":false,"raw_author_name":"Naif Alajlan","raw_affiliation_strings":["Institute of Structural Mechanics, Bauhaus-Universit\u00e4t Weimar, 99423, Weimar, Germany. 2 School of Civil & Environmental Engineering, University of New South Wales, Sydney, Australia. 3 Department of Computer Engineering, College of Computer and Information Sciences, King Saud University, Riyadh, Saudi Arabia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institute of Structural Mechanics, Bauhaus-Universit\u00e4t Weimar, 99423, Weimar, Germany. 2 School of Civil & Environmental Engineering, University of New South Wales, Sydney, Australia. 3 Department of Computer Engineering, College of Computer and Information Sciences, King Saud University, Riyadh, Saudi Arabia","institution_ids":["https://openalex.org/I28022161","https://openalex.org/I31746571","https://openalex.org/I51441396"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5065190539","display_name":"Timon Rabczuk","orcid":"https://orcid.org/0000-0002-7150-296X","observed_orcids":["https://orcid.org/0000-0002-7150-296X"]},"institutions":[{"id":"https://openalex.org/I28022161","display_name":"King Saud University","ror":"https://ror.org/02f81g417","country_code":"SA","type":"education","lineage":["https://openalex.org/I28022161"]},{"id":"https://openalex.org/I31746571","display_name":"UNSW Sydney","ror":"https://ror.org/03r8z3t63","country_code":"AU","type":"education","lineage":["https://openalex.org/I31746571"]},{"id":"https://openalex.org/I51441396","display_name":"Bauhaus-Universit\u00e4t Weimar","ror":"https://ror.org/033bb5z47","country_code":"DE","type":"education","lineage":["https://openalex.org/I51441396"]}],"countries":["AU","DE","SA"],"is_corresponding":false,"raw_author_name":"Timon Rabczuk","raw_affiliation_strings":["Institute of Structural Mechanics, Bauhaus-Universit\u00e4t Weimar, 99423, Weimar, Germany. 2 School of Civil & Environmental Engineering, University of New South Wales, Sydney, Australia. 3 Department of Computer Engineering, College of Computer and Information Sciences, King Saud University, Riyadh, Saudi Arabia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institute of Structural Mechanics, Bauhaus-Universit\u00e4t Weimar, 99423, Weimar, Germany. 2 School of Civil & Environmental Engineering, University of New South Wales, Sydney, Australia. 3 Department of Computer Engineering, College of Computer and Information Sciences, King Saud University, Riyadh, Saudi Arabia","institution_ids":["https://openalex.org/I28022161","https://openalex.org/I31746571","https://openalex.org/I51441396"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":3,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":47.5799,"has_fulltext":false,"cited_by_count":668,"citation_normalized_percentile":{"value":0.9997912,"is_in_top_1_percent":true,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":99,"max":100},"biblio":{"volume":"59","issue":"1","first_page":"345","last_page":"359"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11206","display_name":"Model Reduction and Neural Networks","score":0.8047000169754028,"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":0.8047000169754028,"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/T10320","display_name":"Neural Networks and Applications","score":0.18569999933242798,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12589","display_name":"Differential Equations and Boundary Problems","score":0.0008999999845400453,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"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/neural-network","display_name":"artificial neural networks","score":0.9599999785423279},{"id":"https://openalex.org/keywords/second-order-boundary-value-problems","display_name":"second-order boundary value problems","score":0.9330000281333923},{"id":"https://openalex.org/keywords/partial-differential-equations","display_name":"partial differential equations","score":0.8360000252723694},{"id":"https://openalex.org/keywords/adaptive-collocation","display_name":"adaptive collocation","score":0.9819999933242798},{"id":"https://openalex.org/keywords/training-point-selection","display_name":"training point selection","score":0.5640000104904175},{"id":"https://openalex.org/keywords/nonsmooth-solutions","display_name":"non-smooth solutions","score":0.9649999737739563},{"id":"https://openalex.org/keywords/poisson-equation","display_name":"Poisson equation","score":0.9950000047683716},{"id":"https://openalex.org/keywords/helmholtz-equation","display_name":"Helmholtz equation","score":1.0}],"concepts":[{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.7402671575546265},{"id":"https://openalex.org/C18591234","wikidata":"https://www.wikidata.org/wiki/Q860615","display_name":"Helmholtz equation","level":3,"score":0.6299958229064941},{"id":"https://openalex.org/C137119250","wikidata":"https://www.wikidata.org/wiki/Q1413101","display_name":"Multigrid method","level":3,"score":0.6298841834068298},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.6054498553276062},{"id":"https://openalex.org/C63479239","wikidata":"https://www.wikidata.org/wiki/Q7353546","display_name":"Robustness (evolution)","level":3,"score":0.580410361289978},{"id":"https://openalex.org/C182310444","wikidata":"https://www.wikidata.org/wiki/Q1332643","display_name":"Boundary value problem","level":2,"score":0.5782767534255981},{"id":"https://openalex.org/C187691185","wikidata":"https://www.wikidata.org/wiki/Q2020720","display_name":"Grid","level":2,"score":0.5113884806632996},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.47150343656539917},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.46974924206733704},{"id":"https://openalex.org/C155512373","wikidata":"https://www.wikidata.org/wiki/Q287450","display_name":"Residual","level":2,"score":0.4610236585140228},{"id":"https://openalex.org/C135252773","wikidata":"https://www.wikidata.org/wiki/Q1567213","display_name":"Inverse problem","level":2,"score":0.4513775110244751},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.44793686270713806},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.4452727735042572},{"id":"https://openalex.org/C27592594","wikidata":"https://www.wikidata.org/wiki/Q865821","display_name":"Helmholtz free energy","level":2,"score":0.411469042301178},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.3203873634338379},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.2093067765235901},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.1803959310054779},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.09504517912864685},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.0},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C55493867","wikidata":"https://www.wikidata.org/wiki/Q7094","display_name":"Biochemistry","level":1,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C104317684","wikidata":"https://www.wikidata.org/wiki/Q7187","display_name":"Gene","level":2,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.32604/cmc.2019.06641","doi":"https://doi.org/10.32604/cmc.2019.06641","is_oa":true,"landing_page_url":"https://doi.org/10.32604/cmc.2019.06641","pdf_url":null,"source":{"id":"https://openalex.org/S4210191605","display_name":"Computers, materials & continua/Computers, materials & continua (Print)","issn_l":"1546-2218","issn":["1546-2218","1546-2226"],"is_oa":true,"is_in_doaj":false,"is_core":true,"listed_in":["cwts-core","dongbi-d","jufo-1","ki-jl-1","norway-1","poland-20","russia-white-list-1"],"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":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Computers, Materials &amp; Continua","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.32604/cmc.2019.06641","doi":"https://doi.org/10.32604/cmc.2019.06641","is_oa":true,"landing_page_url":"https://doi.org/10.32604/cmc.2019.06641","pdf_url":null,"source":{"id":"https://openalex.org/S4210191605","display_name":"Computers, materials & continua/Computers, materials & continua (Print)","issn_l":"1546-2218","issn":["1546-2218","1546-2226"],"is_oa":true,"is_in_doaj":false,"is_core":true,"listed_in":["cwts-core","dongbi-d","jufo-1","ki-jl-1","norway-1","poland-20","russia-white-list-1"],"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":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Computers, Materials &amp; Continua","raw_type":"journal-article"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W2158985775","https://openalex.org/W2020561998","https://openalex.org/W6948374085","https://openalex.org/W2125645174","https://openalex.org/W4221156459","https://openalex.org/W330122549","https://openalex.org/W1970454553","https://openalex.org/W4366994360","https://openalex.org/W3176613710","https://openalex.org/W7043990114"],"abstract_inverted_index":{"We":[0],"present":[1],"a":[2,21,50],"method":[3,57],"for":[4,83,86,97],"solving":[5],"partial":[6],"differential":[7],"equations":[8],"using":[9],"artificial":[10],"neural":[11,63],"networks":[12],"and":[13,66,91],"an":[14,98],"adaptive":[15],"collocation":[16],"strategy.":[17],"In":[18],"this":[19],"procedure,":[20],"coarse":[22],"grid":[23],"of":[24,46,53,61],"training":[25,32],"points":[26,36],"is":[27,77],"used":[28],"at":[29,39,49],"the":[30,44,47,59,62,75],"initial":[31],"stages,":[33],"while":[34],"more":[35],"are":[37,81],"added":[38],"later":[40],"stages":[41],"based":[42],"on":[43],"value":[45],"residual":[48],"larger":[51],"set":[52],"evaluation":[54],"points.":[55],"This":[56],"increases":[58],"robustness":[60],"network":[64],"approximation":[65],"can":[67],"result":[68],"in":[69],"significant":[70],"computational":[71],"savings,":[72],"particularly":[73],"when":[74],"solution":[76],"non-smooth.":[78],"Numerical":[79],"results":[80],"presented":[82],"benchmark":[84],"problems":[85],"scalar-valued":[87],"PDEs,":[88],"namely":[89],"Poisson":[90],"Helmholtz":[92],"equations,":[93],"as":[94,96],"well":[95],"inverse":[99],"acoustics":[100],"problem.":[101]},"counts_by_year":[{"year":2026,"cited_by_count":20},{"year":2025,"cited_by_count":66},{"year":2024,"cited_by_count":84},{"year":2023,"cited_by_count":87},{"year":2022,"cited_by_count":129},{"year":2021,"cited_by_count":134},{"year":2020,"cited_by_count":113},{"year":2019,"cited_by_count":35}],"updated_date":"2026-10-31T12:09:11.096373","created_date":"2019-04-11T00:00:00"}
