{"id":"https://openalex.org/W2125645174","doi":"https://doi.org/10.1109/tnn.2009.2020735","title":"Artificial Neural Network Method for Solution of Boundary Value Problems With Exact Satisfaction of Arbitrary Boundary Conditions","display_name":"Artificial Neural Network Method for Solution of Boundary Value Problems With Exact Satisfaction of Arbitrary Boundary Conditions","publication_year":2009,"publication_date":"2009-06-02","ids":{"openalex":"https://openalex.org/W2125645174","doi":"https://doi.org/10.1109/tnn.2009.2020735","mag":"2125645174","pmid":"https://pubmed.ncbi.nlm.nih.gov/19497815"},"language":"en","primary_location":{"id":"doi:10.1109/tnn.2009.2020735","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tnn.2009.2020735","pdf_url":null,"source":{"id":"https://openalex.org/S42080949","display_name":"IEEE Transactions on Neural Networks","issn_l":"1045-9227","issn":["1045-9227","1941-0093"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on 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/A5001407866","display_name":"Kevin McFall","orcid":"https://orcid.org/0000-0003-4085-0109"},"institutions":[{"id":"https://openalex.org/I130769515","display_name":"Pennsylvania State University","ror":"https://ror.org/04p491231","country_code":"US","type":"education","lineage":["https://openalex.org/I130769515"]},{"id":"https://openalex.org/I221180497","display_name":"University of Valley Forge","ror":"https://ror.org/003js3p07","country_code":"US","type":"education","lineage":["https://openalex.org/I221180497"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"K.S. McFall","raw_affiliation_strings":["The Pennsylvania State University, Lehigh Valley Campus, Fogelsville, PA 18051 USA. kevin.mcfall@psu.edu","Lehigh Valley Campus, Pennsylvania State Univ., Fogelsville, PA, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"The Pennsylvania State University, Lehigh Valley Campus, Fogelsville, PA 18051 USA. kevin.mcfall@psu.edu","institution_ids":["https://openalex.org/I221180497"]},{"raw_affiliation_string":"Lehigh Valley Campus, Pennsylvania State Univ., Fogelsville, PA, USA","institution_ids":["https://openalex.org/I130769515"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5109965274","display_name":"J. Robert Mahan","orcid":null},"institutions":[{"id":"https://openalex.org/I130701444","display_name":"Georgia Institute of Technology","ror":"https://ror.org/01zkghx44","country_code":"US","type":"education","lineage":["https://openalex.org/I130701444"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"J.R. Mahan","raw_affiliation_strings":["George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, USA","George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Metz, France#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, USA","institution_ids":["https://openalex.org/I130701444"]},{"raw_affiliation_string":"George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Metz, France#TAB#","institution_ids":["https://openalex.org/I130701444"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":2.0138,"has_fulltext":false,"cited_by_count":212,"citation_normalized_percentile":{"value":0.87504779,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":96,"max":100},"biblio":{"volume":"20","issue":"8","first_page":"1221","last_page":"1233"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11206","display_name":"Model Reduction and Neural Networks","score":0.9987000226974487,"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":0.9987000226974487,"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/T10514","display_name":"Numerical methods in engineering","score":0.9958999752998352,"subfield":{"id":"https://openalex.org/subfields/2211","display_name":"Mechanics of Materials"},"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/T11263","display_name":"Electromagnetic Simulation and Numerical Methods","score":0.9912999868392944,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic 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/artificial-neural-network","display_name":"Artificial neural network","score":0.7407380938529968},{"id":"https://openalex.org/keywords/boundary-value-problem","display_name":"Boundary value problem","score":0.6764389276504517},{"id":"https://openalex.org/keywords/domain","display_name":"Domain (mathematical analysis)","score":0.544409453868866},{"id":"https://openalex.org/keywords/dirichlet-distribution","display_name":"Dirichlet distribution","score":0.5126413702964783},{"id":"https://openalex.org/keywords/boundary","display_name":"Boundary (topology)","score":0.5001707077026367},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.48405948281288147},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.4588748514652252},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.4479587972164154},{"id":"https://openalex.org/keywords/value","display_name":"Value (mathematics)","score":0.4474053680896759},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.42745834589004517},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.3553776144981384},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.32557082176208496},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.2373899519443512},{"id":"https://openalex.org/keywords/machine-learning","display_name":"Machine learning","score":0.1540764570236206}],"concepts":[{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.7407380938529968},{"id":"https://openalex.org/C182310444","wikidata":"https://www.wikidata.org/wiki/Q1332643","display_name":"Boundary value problem","level":2,"score":0.6764389276504517},{"id":"https://openalex.org/C36503486","wikidata":"https://www.wikidata.org/wiki/Q11235244","display_name":"Domain (mathematical analysis)","level":2,"score":0.544409453868866},{"id":"https://openalex.org/C169214877","wikidata":"https://www.wikidata.org/wiki/Q981016","display_name":"Dirichlet distribution","level":3,"score":0.5126413702964783},{"id":"https://openalex.org/C62354387","wikidata":"https://www.wikidata.org/wiki/Q875399","display_name":"Boundary (topology)","level":2,"score":0.5001707077026367},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.48405948281288147},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4588748514652252},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.4479587972164154},{"id":"https://openalex.org/C2776291640","wikidata":"https://www.wikidata.org/wiki/Q2912517","display_name":"Value (mathematics)","level":2,"score":0.4474053680896759},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.42745834589004517},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.3553776144981384},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.32557082176208496},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.2373899519443512},{"id":"https://openalex.org/C119857082","wikidata":"https://www.wikidata.org/wiki/Q2539","display_name":"Machine learning","level":1,"score":0.1540764570236206},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1109/tnn.2009.2020735","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tnn.2009.2020735","pdf_url":null,"source":{"id":"https://openalex.org/S42080949","display_name":"IEEE Transactions on Neural Networks","issn_l":"1045-9227","issn":["1045-9227","1941-0093"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Neural Networks","raw_type":"journal-article"},{"id":"pmid:19497815","is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/19497815","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":"IEEE transactions on neural networks","raw_type":null}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":23,"referenced_works":["https://openalex.org/W1512069430","https://openalex.org/W1634261337","https://openalex.org/W1978075626","https://openalex.org/W2022910713","https://openalex.org/W2026656818","https://openalex.org/W2034650523","https://openalex.org/W2034903716","https://openalex.org/W2059548105","https://openalex.org/W2078193948","https://openalex.org/W2078886696","https://openalex.org/W2103414828","https://openalex.org/W2103756561","https://openalex.org/W2137983211","https://openalex.org/W2148666448","https://openalex.org/W2155598194","https://openalex.org/W2158985775","https://openalex.org/W2163789949","https://openalex.org/W2181681702","https://openalex.org/W2490615762","https://openalex.org/W3101260193","https://openalex.org/W3132178687","https://openalex.org/W3146803896","https://openalex.org/W4247533770"],"related_works":["https://openalex.org/W2391251536","https://openalex.org/W2465918047","https://openalex.org/W2563901864","https://openalex.org/W2953252165","https://openalex.org/W3168009835","https://openalex.org/W4299072667","https://openalex.org/W2048047226","https://openalex.org/W2964287305","https://openalex.org/W2133690079","https://openalex.org/W2111215486"],"abstract_inverted_index":{"A":[0],"method":[1,42],"for":[2,15,49,70,97,124],"solving":[3,50],"boundary":[4,22],"value":[5],"problems":[6],"(BVPs)":[7],"is":[8,43,80],"introduced":[9],"using":[10],"artificial":[11],"neural":[12],"networks":[13],"(ANNs)":[14],"irregular":[16],"domain":[17],"boundaries":[18],"with":[19],"mixed":[20],"Dirichlet/Neumann":[21],"conditions":[23],"(BCs).":[24],"The":[25],"approximate":[26,68,113],"ANN":[27,47],"solution":[28,69,94,114],"automatically":[29],"satisfies":[30],"BCs":[31,63,79],"at":[32],"all":[33,125],"stages":[34],"of":[35,61,73,78,85,92,118],"training,":[36],"including":[37],"before":[38],"training":[39],"commences.":[40],"This":[41],"simpler":[44],"than":[45],"other":[46],"methods":[48],"BVPs":[51],"due":[52],"to":[53,120],"its":[54],"unconstrained":[55],"nature":[56],"and":[57,100,106],"because":[58],"automatic":[59],"satisfaction":[60,77],"Dirichlet":[62],"provides":[64],"a":[65],"good":[66],"starting":[67],"significant":[71],"portions":[72],"the":[74,83,112,116],"domain.":[75],"Automatic":[76],"accomplished":[81],"by":[82],"introduction":[84],"an":[86],"innovative":[87],"length":[88],"factor.":[89],"Several":[90],"examples":[91],"BVP":[93],"are":[95,122],"presented":[96],"both":[98],"linear":[99],"nonlinear":[101],"differential":[102],"equations":[103],"in":[104,111],"two":[105],"three":[107],"dimensions.":[108],"Error":[109],"norms":[110],"on":[115],"order":[117],"10(-4)":[119],"10(-5)":[121],"reported":[123],"example":[126],"problems.":[127]},"counts_by_year":[{"year":2026,"cited_by_count":13},{"year":2025,"cited_by_count":24},{"year":2024,"cited_by_count":23},{"year":2023,"cited_by_count":27},{"year":2022,"cited_by_count":23},{"year":2021,"cited_by_count":22},{"year":2020,"cited_by_count":23},{"year":2019,"cited_by_count":10},{"year":2018,"cited_by_count":3},{"year":2017,"cited_by_count":3},{"year":2016,"cited_by_count":11},{"year":2015,"cited_by_count":6},{"year":2014,"cited_by_count":9},{"year":2013,"cited_by_count":5},{"year":2012,"cited_by_count":7}],"updated_date":"2026-07-17T09:13:05.818461","created_date":"2025-10-10T00:00:00"}