{"id":"https://openalex.org/W3206631416","doi":"https://doi.org/10.1007/s10915-021-01650-5","title":"Numerical Solution and Bifurcation Analysis of Nonlinear Partial Differential Equations with Extreme Learning Machines","display_name":"Numerical Solution and Bifurcation Analysis of Nonlinear Partial Differential Equations with Extreme Learning Machines","publication_year":2021,"publication_date":"2021-04-13","ids":{"openalex":"https://openalex.org/W3206631416","doi":"https://doi.org/10.1007/s10915-021-01650-5","mag":"3206631416"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2104.06116","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2104.06116","pdf_url":"https://arxiv.org/pdf/2104.06116","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"type":"article","indexed_in":["arxiv"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2104.06116","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5043315604","display_name":"Gianluca Fabiani","orcid":"https://orcid.org/0000-0002-9799-451X"},"institutions":[{"id":"https://openalex.org/I4210149489","display_name":"Scuola Superiore Meridionale","ror":"https://ror.org/04swxte59","country_code":"IT","type":"education","lineage":["https://openalex.org/I4210149489"]},{"id":"https://openalex.org/I71267560","display_name":"University of Naples Federico II","ror":"https://ror.org/05290cv24","country_code":"IT","type":"education","lineage":["https://openalex.org/I71267560"]}],"countries":["IT"],"is_corresponding":false,"raw_author_name":"Fabiani, Gianluca","raw_affiliation_strings":["Scuola Superiore Meridionale, Universit\u00e0 degli Studi di Napoli Federico II, Naples, Italy"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Scuola Superiore Meridionale, Universit\u00e0 degli Studi di Napoli Federico II, Naples, Italy","institution_ids":["https://openalex.org/I4210149489","https://openalex.org/I71267560"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5041462050","display_name":"Francesco Calabr\u00f2","orcid":"https://orcid.org/0000-0002-1484-0217"},"institutions":[{"id":"https://openalex.org/I71267560","display_name":"University of Naples Federico II","ror":"https://ror.org/05290cv24","country_code":"IT","type":"education","lineage":["https://openalex.org/I71267560"]}],"countries":["IT"],"is_corresponding":false,"raw_author_name":"Calabr\u00f2, Francesco","raw_affiliation_strings":["Dipartimento di Matematica e Applicazioni \"Renato Caccioppoli\", Universit\u00e0 degli Studi di Napoli \"Federico II\", Naples, Italy"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Dipartimento di Matematica e Applicazioni \"Renato Caccioppoli\", Universit\u00e0 degli Studi di Napoli \"Federico II\", Naples, Italy","institution_ids":["https://openalex.org/I71267560"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5091118133","display_name":"Lucia Russo","orcid":"https://orcid.org/0000-0003-4888-467X"},"institutions":[{"id":"https://openalex.org/I4210109673","display_name":"Institute of Condensed Matter Chemistry and Technologies for Energy","ror":"https://ror.org/01rg40y89","country_code":"IT","type":"facility","lineage":["https://openalex.org/I4210109673","https://openalex.org/I4210155236"]}],"countries":["IT"],"is_corresponding":false,"raw_author_name":"Russo, Lucia","raw_affiliation_strings":["IIstituto di Scienze e Tecnologie per l'Energia e la Mobilit\u00e0 Sostenibili, Consiglio Nazionale delle Ricerche, Naples, Italy"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"IIstituto di Scienze e Tecnologie per l'Energia e la Mobilit\u00e0 Sostenibili, Consiglio Nazionale delle Ricerche, Naples, Italy","institution_ids":["https://openalex.org/I4210109673"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5090389995","display_name":"Constantinos Siettos","orcid":"https://orcid.org/0000-0002-9568-3355"},"institutions":[{"id":"https://openalex.org/I71267560","display_name":"University of Naples Federico II","ror":"https://ror.org/05290cv24","country_code":"IT","type":"education","lineage":["https://openalex.org/I71267560"]}],"countries":["IT"],"is_corresponding":false,"raw_author_name":"Siettos, Constantinos","raw_affiliation_strings":["Dipartimento di Matematica e Applicazioni \"Renato Caccioppoli\", Universit\u00e0 degli Studi di Napoli \"Federico II\", Naples, Italy"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Dipartimento di Matematica e Applicazioni \"Renato Caccioppoli\", Universit\u00e0 degli Studi di Napoli \"Federico II\", Naples, Italy","institution_ids":["https://openalex.org/I71267560"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":4,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":5.9071,"has_fulltext":true,"cited_by_count":83,"citation_normalized_percentile":{"value":0.97170353,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":94,"max":100},"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11206","display_name":"Model Reduction and Neural Networks","score":0.9994999766349792,"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":0.9994999766349792,"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/T12676","display_name":"Machine Learning and ELM","score":0.9983999729156494,"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/T11277","display_name":"Thermal properties of materials","score":0.9965999722480774,"subfield":{"id":"https://openalex.org/subfields/2505","display_name":"Materials Chemistry"},"field":{"id":"https://openalex.org/fields/25","display_name":"Materials Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.839665412902832},{"id":"https://openalex.org/keywords/partial-differential-equation","display_name":"Partial differential equation","score":0.6604119539260864},{"id":"https://openalex.org/keywords/dirichlet-boundary-condition","display_name":"Dirichlet boundary condition","score":0.6272580623626709},{"id":"https://openalex.org/keywords/finite-element-method","display_name":"Finite element method","score":0.5817447304725647},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.5557999610900879},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.47550898790359497},{"id":"https://openalex.org/keywords/boundary-value-problem","display_name":"Boundary value problem","score":0.47301438450813293},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.4562559723854065},{"id":"https://openalex.org/keywords/numerical-analysis","display_name":"Numerical analysis","score":0.44620954990386963},{"id":"https://openalex.org/keywords/bifurcation","display_name":"Bifurcation","score":0.42714864015579224}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.839665412902832},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.6604119539260864},{"id":"https://openalex.org/C110167270","wikidata":"https://www.wikidata.org/wiki/Q1193699","display_name":"Dirichlet boundary condition","level":3,"score":0.6272580623626709},{"id":"https://openalex.org/C135628077","wikidata":"https://www.wikidata.org/wiki/Q220184","display_name":"Finite element method","level":2,"score":0.5817447304725647},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.5557999610900879},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.47550898790359497},{"id":"https://openalex.org/C182310444","wikidata":"https://www.wikidata.org/wiki/Q1332643","display_name":"Boundary value problem","level":2,"score":0.47301438450813293},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.4562559723854065},{"id":"https://openalex.org/C48753275","wikidata":"https://www.wikidata.org/wiki/Q11216","display_name":"Numerical analysis","level":2,"score":0.44620954990386963},{"id":"https://openalex.org/C2781349735","wikidata":"https://www.wikidata.org/wiki/Q4904824","display_name":"Bifurcation","level":3,"score":0.42714864015579224},{"id":"https://openalex.org/C97355855","wikidata":"https://www.wikidata.org/wiki/Q11473","display_name":"Thermodynamics","level":1,"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/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"pmh:oai:arXiv.org:2104.06116","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2104.06116","pdf_url":"https://arxiv.org/pdf/2104.06116","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2104.06116","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2104.06116","pdf_url":"https://arxiv.org/pdf/2104.06116","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320309590","display_name":"Universit\u00e0 degli Studi di Napoli Federico II","ror":"https://ror.org/05290cv24"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":69,"referenced_works":["https://openalex.org/W21912675","https://openalex.org/W177905271","https://openalex.org/W601041187","https://openalex.org/W1136074057","https://openalex.org/W1530314303","https://openalex.org/W1582930900","https://openalex.org/W1596321532","https://openalex.org/W1967011851","https://openalex.org/W1968368569","https://openalex.org/W1972005403","https://openalex.org/W1980713635","https://openalex.org/W1985352344","https://openalex.org/W1996640396","https://openalex.org/W2026131661","https://openalex.org/W2028759731","https://openalex.org/W2029215818","https://openalex.org/W2029967456","https://openalex.org/W2041645196","https://openalex.org/W2068005555","https://openalex.org/W2079371292","https://openalex.org/W2096089659","https://openalex.org/W2102652546","https://openalex.org/W2109314821","https://openalex.org/W2111072639","https://openalex.org/W2113132555","https://openalex.org/W2118706537","https://openalex.org/W2121971770","https://openalex.org/W2134661792","https://openalex.org/W2142915558","https://openalex.org/W2150354929","https://openalex.org/W2155910151","https://openalex.org/W2157595416","https://openalex.org/W2158581396","https://openalex.org/W2158985775","https://openalex.org/W2163564294","https://openalex.org/W2249221522","https://openalex.org/W2280059822","https://openalex.org/W2301541953","https://openalex.org/W2566965042","https://openalex.org/W2605147767","https://openalex.org/W2734205292","https://openalex.org/W2773122281","https://openalex.org/W2799270168","https://openalex.org/W2802921497","https://openalex.org/W2803629276","https://openalex.org/W2883251900","https://openalex.org/W2885466093","https://openalex.org/W2899283552","https://openalex.org/W2925178532","https://openalex.org/W2964016242","https://openalex.org/W2979359507","https://openalex.org/W2994602700","https://openalex.org/W2996149946","https://openalex.org/W2997165591","https://openalex.org/W2998847955","https://openalex.org/W3008525407","https://openalex.org/W3024946036","https://openalex.org/W3048905115","https://openalex.org/W3101260193","https://openalex.org/W3101465594","https://openalex.org/W3106534923","https://openalex.org/W3111668711","https://openalex.org/W3112199937","https://openalex.org/W3136639013","https://openalex.org/W3153519416","https://openalex.org/W3202131078","https://openalex.org/W4249992205","https://openalex.org/W4292079189","https://openalex.org/W4366309888"],"related_works":["https://openalex.org/W2318061648","https://openalex.org/W1982022398","https://openalex.org/W2196719869","https://openalex.org/W1978830096","https://openalex.org/W1514481392","https://openalex.org/W2020502833","https://openalex.org/W892341692","https://openalex.org/W2100405323","https://openalex.org/W3155098563","https://openalex.org/W593077420"],"abstract_inverted_index":{"We":[0,136],"address":[1],"a":[2,8],"new":[3],"numerical":[4,100,105,119,142],"scheme":[5,122],"based":[6],"on":[7],"class":[9],"of":[10,28,34,39,109,113,116],"machine":[11,120],"learning":[12,121],"methods,":[13],"the":[14,26,32,55,70,82,99,104,114,117,139,161],"so-called":[15],"Extreme":[16],"Learning":[17],"Machines":[18],"with":[19,59,76,160],"both":[20,60,145],"sigmoidal":[21],"and":[22,31,63,72,85,93,111,130,147],"radial-basis":[23],"functions,":[24],"for":[25,95,150,163],"computation":[27],"steady-state":[29],"solutions":[30,90],"construction":[33],"(one":[35],"dimensional)":[36],"bifurcation":[37],"diagrams":[38],"nonlinear":[40],"partial":[41],"differential":[42],"equations":[43],"(PDEs).":[44],"For":[45,81],"our":[46],"illustrations,":[47],"we":[48],"considered":[49],"two":[50],"benchmark":[51],"problems,":[52],"namely":[53],"(a)":[54],"one-dimensional":[56,83],"viscous":[57],"Burgers":[58,84],"homogeneous":[61,77],"(Dirichlet)":[62],"non-homogeneous":[64],"mixed":[65],"boundary":[66,79],"conditions,":[67],"and,":[68],"(b)":[69],"one":[71],"two-dimensional":[73],"Liouville-Bratu-Gelfand":[74],"PDEs":[75],"Dirichlet":[78],"conditions.":[80],"Bratu":[86],"PDEs,":[87],"exact":[88],"analytical":[89],"are":[91],"available":[92],"used":[94],"comparison":[96],"purposes":[97],"against":[98,125],"derived":[101],"solutions.":[102],"Furthermore,":[103],"efficiency":[106],"(in":[107],"terms":[108],"accuracy":[110],"size":[112],"grid)":[115],"proposed":[118,140],"is":[123],"compared":[124],"central":[126],"finite":[127],"differences":[128],"(FD)":[129],"Galerkin":[131],"weighted-residuals":[132],"finite-element":[133],"methods":[134,149],"(FEM).":[135],"show":[137],"that":[138],"ELM":[141],"method":[143],"outperforms":[144],"FD":[146],"FEM":[148,162],"medium":[151,166],"to":[152,165],"large":[153],"sized":[154,167],"grids,":[155],"while":[156],"provides":[157],"equivalent":[158],"results":[159],"low":[164],"grids.":[168]},"counts_by_year":[{"year":2026,"cited_by_count":13},{"year":2025,"cited_by_count":25},{"year":2024,"cited_by_count":24},{"year":2023,"cited_by_count":14},{"year":2022,"cited_by_count":5},{"year":2021,"cited_by_count":2}],"updated_date":"2026-06-19T17:40:00.097472","created_date":"2021-10-25T00:00:00"}