{"id":"https://openalex.org/W4416542666","doi":"https://doi.org/10.1016/j.camwa.2026.04.002","title":"Stochastic quadrature rules for solving PDEs using neural networks","display_name":"Stochastic quadrature rules for solving PDEs using neural networks","publication_year":2026,"publication_date":"2026-04-09","ids":{"openalex":"https://openalex.org/W4416542666","doi":"https://doi.org/10.1016/j.camwa.2026.04.002"},"language":"en","primary_location":{"id":"doi:10.1016/j.camwa.2026.04.002","is_oa":false,"landing_page_url":"https://doi.org/10.1016/j.camwa.2026.04.002","pdf_url":null,"source":{"id":"https://openalex.org/S4210205691","display_name":"Computers & Mathematics with Applications","issn_l":"0898-1221","issn":["0898-1221","1873-7668"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320990","host_organization_name":"Elsevier BV","host_organization_lineage":["https://openalex.org/P4310320990"],"host_organization_lineage_names":["Elsevier BV"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Computers &amp; Mathematics with Applications","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2504.11976","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5065057468","display_name":"Jamie M. Taylor","orcid":"https://orcid.org/0000-0002-5423-828X"},"institutions":[{"id":"https://openalex.org/I4403386682","display_name":"CUNEF Universidad","ror":"https://ror.org/02web8j86","country_code":null,"type":"education","lineage":["https://openalex.org/I4403386682"]}],"countries":["ES"],"is_corresponding":false,"raw_author_name":"Jamie M. Taylor","raw_affiliation_strings":["Department of Mathematics, CUNEF Universidad, Madrid, Spain"],"raw_orcid":"https://orcid.org/0000-0002-5423-828X","affiliations":[{"raw_affiliation_string":"Department of Mathematics, CUNEF Universidad, Madrid, Spain","institution_ids":["https://openalex.org/I4403386682"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5082696717","display_name":"David Pardo","orcid":"https://orcid.org/0000-0002-1101-2248"},"institutions":[{"id":"https://openalex.org/I110594554","display_name":"Ikerbasque","ror":"https://ror.org/01cc3fy72","country_code":"ES","type":"other","lineage":["https://openalex.org/I110594554"]},{"id":"https://openalex.org/I169108374","display_name":"University of the Basque Country","ror":"https://ror.org/000xsnr85","country_code":"ES","type":"education","lineage":["https://openalex.org/I169108374"]},{"id":"https://openalex.org/I2802176441","display_name":"Basque Center for Applied Mathematics","ror":"https://ror.org/03b21sh32","country_code":"ES","type":"education","lineage":["https://openalex.org/I2802176441"]}],"countries":["ES"],"is_corresponding":true,"raw_author_name":"David Pardo","raw_affiliation_strings":["Basque Center for Applied Mathematics (BCAM), Bilbao, Spain","Ikerbasque: Basque Foundation for Science, Bilbao, Spain","University of the Basque Country (UPV/EHU), Leioa, Spain"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Basque Center for Applied Mathematics (BCAM), Bilbao, Spain","institution_ids":["https://openalex.org/I2802176441"]},{"raw_affiliation_string":"Ikerbasque: Basque Foundation for Science, Bilbao, Spain","institution_ids":["https://openalex.org/I110594554"]},{"raw_affiliation_string":"University of the Basque Country (UPV/EHU), Leioa, Spain","institution_ids":["https://openalex.org/I169108374"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5082696717"],"corresponding_institution_ids":["https://openalex.org/I110594554","https://openalex.org/I169108374","https://openalex.org/I2802176441"],"apc_list":{"value":3480,"currency":"USD","value_usd":3480},"apc_paid":null,"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.01800594,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"214","issue":null,"first_page":"1","last_page":"26"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11206","display_name":"Model Reduction and Neural Networks","score":0.4169999957084656,"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":0.4169999957084656,"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/T11612","display_name":"Stochastic Gradient Optimization Techniques","score":0.12139999866485596,"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/T12404","display_name":"Mathematical Approximation and Integration","score":0.11580000072717667,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"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/numerical-integration","display_name":"Numerical integration","score":0.7838000059127808},{"id":"https://openalex.org/keywords/quadrature","display_name":"Quadrature (astronomy)","score":0.703499972820282},{"id":"https://openalex.org/keywords/bottleneck","display_name":"Bottleneck","score":0.6184999942779541},{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.5437999963760376},{"id":"https://openalex.org/keywords/gaussian-quadrature","display_name":"Gaussian quadrature","score":0.4875999987125397},{"id":"https://openalex.org/keywords/polygon-mesh","display_name":"Polygon mesh","score":0.42669999599456787},{"id":"https://openalex.org/keywords/adaptive-quadrature","display_name":"Adaptive quadrature","score":0.37040001153945923},{"id":"https://openalex.org/keywords/gauss\u2013hermite-quadrature","display_name":"Gauss\u2013Hermite quadrature","score":0.3644999861717224}],"concepts":[{"id":"https://openalex.org/C127349201","wikidata":"https://www.wikidata.org/wiki/Q753445","display_name":"Numerical integration","level":2,"score":0.7838000059127808},{"id":"https://openalex.org/C62869609","wikidata":"https://www.wikidata.org/wiki/Q28137","display_name":"Quadrature (astronomy)","level":2,"score":0.703499972820282},{"id":"https://openalex.org/C2780513914","wikidata":"https://www.wikidata.org/wiki/Q18210350","display_name":"Bottleneck","level":2,"score":0.6184999942779541},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.5437999963760376},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.5371000170707703},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5087000131607056},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.4884999990463257},{"id":"https://openalex.org/C167196314","wikidata":"https://www.wikidata.org/wiki/Q767680","display_name":"Gaussian quadrature","level":4,"score":0.4875999987125397},{"id":"https://openalex.org/C31487907","wikidata":"https://www.wikidata.org/wiki/Q1154597","display_name":"Polygon mesh","level":2,"score":0.42669999599456787},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.40119999647140503},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.3977999985218048},{"id":"https://openalex.org/C184599523","wikidata":"https://www.wikidata.org/wiki/Q4680758","display_name":"Adaptive quadrature","level":4,"score":0.37040001153945923},{"id":"https://openalex.org/C167590341","wikidata":"https://www.wikidata.org/wiki/Q5527850","display_name":"Gauss\u2013Hermite quadrature","level":5,"score":0.3644999861717224},{"id":"https://openalex.org/C162046254","wikidata":"https://www.wikidata.org/wiki/Q7683194","display_name":"Tanh-sinh quadrature","level":5,"score":0.3637000024318695},{"id":"https://openalex.org/C196083921","wikidata":"https://www.wikidata.org/wiki/Q7915758","display_name":"Variance (accounting)","level":2,"score":0.3425000011920929},{"id":"https://openalex.org/C51955184","wikidata":"https://www.wikidata.org/wiki/Q1545585","display_name":"Stochastic differential equation","level":2,"score":0.3416999876499176},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.3269999921321869},{"id":"https://openalex.org/C177606310","wikidata":"https://www.wikidata.org/wiki/Q5674297","display_name":"Adaptability","level":2,"score":0.3246000111103058},{"id":"https://openalex.org/C8272713","wikidata":"https://www.wikidata.org/wiki/Q176737","display_name":"Stochastic process","level":2,"score":0.30959999561309814},{"id":"https://openalex.org/C48753275","wikidata":"https://www.wikidata.org/wiki/Q11216","display_name":"Numerical analysis","level":2,"score":0.30649998784065247},{"id":"https://openalex.org/C21200559","wikidata":"https://www.wikidata.org/wiki/Q7451068","display_name":"Sensitivity (control systems)","level":2,"score":0.2978000044822693},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.27549999952316284},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.27239999175071716}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1016/j.camwa.2026.04.002","is_oa":false,"landing_page_url":"https://doi.org/10.1016/j.camwa.2026.04.002","pdf_url":null,"source":{"id":"https://openalex.org/S4210205691","display_name":"Computers & Mathematics with Applications","issn_l":"0898-1221","issn":["0898-1221","1873-7668"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320990","host_organization_name":"Elsevier BV","host_organization_lineage":["https://openalex.org/P4310320990"],"host_organization_lineage_names":["Elsevier BV"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Computers &amp; Mathematics with Applications","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:2504.11976","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2504.11976","pdf_url":"https://arxiv.org/pdf/2504.11976","source":{"id":"https://openalex.org/S4393918464","display_name":"ArXiv.org","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"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":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2504.11976","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2504.11976","pdf_url":null,"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2504.11976","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2504.11976","pdf_url":"https://arxiv.org/pdf/2504.11976","source":{"id":"https://openalex.org/S4393918464","display_name":"ArXiv.org","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"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":null,"raw_type":"text"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G2262748287","display_name":null,"funder_award_id":"501100011033","funder_id":"https://openalex.org/F4320335598","funder_display_name":"Agencia Estatal de Investigaci\u00f3n"},{"id":"https://openalex.org/G3081656539","display_name":null,"funder_award_id":"CEX2021-001142-S","funder_id":"https://openalex.org/F4320335598","funder_display_name":"Agencia Estatal de Investigaci\u00f3n"},{"id":"https://openalex.org/G3480869486","display_name":null,"funder_award_id":"13039","funder_id":"https://openalex.org/F4320335598","funder_display_name":"Agencia Estatal de Investigaci\u00f3n"},{"id":"https://openalex.org/G3681454997","display_name":null,"funder_award_id":"13039/501100011033","funder_id":"https://openalex.org/F4320320300","funder_display_name":"European Commission"},{"id":"https://openalex.org/G4042783231","display_name":null,"funder_award_id":"501100011033","funder_id":"https://openalex.org/F4320320300","funder_display_name":"European Commission"},{"id":"https://openalex.org/G4488079590","display_name":null,"funder_award_id":"BERC 2022-2025","funder_id":"https://openalex.org/F4320321705","funder_display_name":"Eusko Jaurlaritza"},{"id":"https://openalex.org/G6294024929","display_name":null,"funder_award_id":"CEX2021-001142-S","funder_id":"https://openalex.org/F4320328444","funder_display_name":"Basque Center for Applied Mathematics"},{"id":"https://openalex.org/G7084143925","display_name":null,"funder_award_id":"AEI/10","funder_id":"https://openalex.org/F4320335598","funder_display_name":"Agencia Estatal de Investigaci\u00f3n"},{"id":"https://openalex.org/G7266728691","display_name":null,"funder_award_id":"13039/501100011033","funder_id":"https://openalex.org/F4320335598","funder_display_name":"Agencia Estatal de Investigaci\u00f3n"},{"id":"https://openalex.org/G7436716198","display_name":null,"funder_award_id":"BERC 2022-2025","funder_id":"https://openalex.org/F4320328444","funder_display_name":"Basque Center for Applied Mathematics"},{"id":"https://openalex.org/G7713090867","display_name":null,"funder_award_id":"KK-2024/00086","funder_id":"https://openalex.org/F4320321705","funder_display_name":"Eusko Jaurlaritza"},{"id":"https://openalex.org/G7748812462","display_name":null,"funder_award_id":"2022-2025","funder_id":"https://openalex.org/F4320320300","funder_display_name":"European Commission"},{"id":"https://openalex.org/G7829631415","display_name":"Real-time inversion using self-explainable deep learning driven by expert knowledge","funder_award_id":"101119556","funder_id":"https://openalex.org/F4320320300","funder_display_name":"European Commission"},{"id":"https://openalex.org/G7852287701","display_name":null,"funder_award_id":"KK-2023/00012","funder_id":"https://openalex.org/F4320321705","funder_display_name":"Eusko Jaurlaritza"},{"id":"https://openalex.org/G8896920958","display_name":null,"funder_award_id":"IT1456-22","funder_id":"https://openalex.org/F4320321705","funder_display_name":"Eusko Jaurlaritza"},{"id":"https://openalex.org/G8939903727","display_name":null,"funder_award_id":"2022-2025","funder_id":"https://openalex.org/F4320321705","funder_display_name":"Eusko Jaurlaritza"}],"funders":[{"id":"https://openalex.org/F4320320300","display_name":"European Commission","ror":"https://ror.org/00k4n6c32"},{"id":"https://openalex.org/F4320321705","display_name":"Eusko Jaurlaritza","ror":"https://ror.org/00pz2fp31"},{"id":"https://openalex.org/F4320328444","display_name":"Basque Center for Applied Mathematics","ror":"https://ror.org/03b21sh32"},{"id":"https://openalex.org/F4320334322","display_name":"HORIZON EUROPE Framework Programme","ror":null},{"id":"https://openalex.org/F4320335598","display_name":"Agencia Estatal de Investigaci\u00f3n","ror":null}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4416542666.pdf","grobid_xml":"https://content.openalex.org/works/W4416542666.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":null,"counts_by_year":[],"updated_date":"2026-05-14T08:36:36.166977","created_date":"2025-10-10T00:00:00"}
