{"id":"https://openalex.org/W4285200085","doi":"https://doi.org/10.1007/978-3-031-08754-7_41","title":"Deep Neural Networks and\u00a0Smooth Approximation of\u00a0PDEs","display_name":"Deep Neural Networks and\u00a0Smooth Approximation of\u00a0PDEs","publication_year":2022,"publication_date":"2022-01-01","ids":{"openalex":"https://openalex.org/W4285200085","doi":"https://doi.org/10.1007/978-3-031-08754-7_41"},"language":"en","primary_location":{"id":"doi:10.1007/978-3-031-08754-7_41","is_oa":false,"landing_page_url":"https://doi.org/10.1007/978-3-031-08754-7_41","pdf_url":null,"source":{"id":"https://openalex.org/S106296714","display_name":"Lecture notes in computer science","issn_l":"0302-9743","issn":["0302-9743","1611-3349"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"book series"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Lecture Notes in Computer Science","raw_type":"book-chapter"},"type":"book-chapter","indexed_in":["crossref"],"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/A5047028796","display_name":"Kamil Doleg\u0142o","orcid":null},"institutions":[{"id":"https://openalex.org/I686019","display_name":"AGH University of Krakow","ror":"https://ror.org/00bas1c41","country_code":"PL","type":"education","lineage":["https://openalex.org/I686019"]}],"countries":["PL"],"is_corresponding":false,"raw_author_name":"Kamil Doleg\u0142o","raw_affiliation_strings":["AGH University of Science and Technology, Krak\u00f3w, Poland"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"AGH University of Science and Technology, Krak\u00f3w, Poland","institution_ids":["https://openalex.org/I686019"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5075191779","display_name":"Maciej Paszy\u0144ski","orcid":"https://orcid.org/0000-0001-7766-6052"},"institutions":[{"id":"https://openalex.org/I686019","display_name":"AGH University of Krakow","ror":"https://ror.org/00bas1c41","country_code":"PL","type":"education","lineage":["https://openalex.org/I686019"]}],"countries":["PL"],"is_corresponding":false,"raw_author_name":"Maciej Paszy\u0144ski","raw_affiliation_strings":["AGH University of Science and Technology, Krakow, Poland","AGH University of Science and Technology, Krak\u00f3w, Poland"],"raw_orcid":"https://orcid.org/0000-0001-7766-6052","affiliations":[{"raw_affiliation_string":"AGH University of Science and Technology, Krakow, Poland","institution_ids":["https://openalex.org/I686019"]},{"raw_affiliation_string":"AGH University of Science and Technology, Krak\u00f3w, Poland","institution_ids":["https://openalex.org/I686019"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5082045365","display_name":"Leszek Demkowicz","orcid":"https://orcid.org/0000-0001-7839-8037"},"institutions":[{"id":"https://openalex.org/I86519309","display_name":"The University of Texas at Austin","ror":"https://ror.org/00hj54h04","country_code":"US","type":"education","lineage":["https://openalex.org/I86519309"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Leszek Demkowicz","raw_affiliation_strings":["Oden Institute, The University of Texas in Austin, Austin, USA"],"raw_orcid":"https://orcid.org/0000-0001-7839-8037","affiliations":[{"raw_affiliation_string":"Oden Institute, The University of Texas in Austin, Austin, USA","institution_ids":["https://openalex.org/I86519309"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":{"value":5000,"currency":"EUR","value_usd":5392},"apc_paid":null,"fwci":6.1045,"has_fulltext":false,"cited_by_count":5,"citation_normalized_percentile":{"value":0.96242775,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":90,"max":96},"biblio":{"volume":null,"issue":null,"first_page":"328","last_page":"332"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11206","display_name":"Model Reduction and Neural Networks","score":0.9962999820709229,"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":0.9962999820709229,"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/T11245","display_name":"Advanced Numerical Analysis Techniques","score":0.9934999942779541,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/concatenation","display_name":"Concatenation (mathematics)","score":0.7226883172988892},{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.6015526056289673},{"id":"https://openalex.org/keywords/partial-differential-equation","display_name":"Partial differential equation","score":0.5591607689857483},{"id":"https://openalex.org/keywords/linear-differential-equation","display_name":"Linear differential equation","score":0.5290364623069763},{"id":"https://openalex.org/keywords/computation","display_name":"Computation","score":0.5286912322044373},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5199088454246521},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.5126070380210876},{"id":"https://openalex.org/keywords/isogeometric-analysis","display_name":"Isogeometric analysis","score":0.5101185441017151},{"id":"https://openalex.org/keywords/domain","display_name":"Domain (mathematical analysis)","score":0.4694152772426605},{"id":"https://openalex.org/keywords/activation-function","display_name":"Activation function","score":0.4533911347389221},{"id":"https://openalex.org/keywords/deep-learning","display_name":"Deep learning","score":0.4474591016769409},{"id":"https://openalex.org/keywords/focus","display_name":"Focus (optics)","score":0.4107182025909424},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.3815303146839142},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.37177324295043945},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.35004788637161255},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.25716033577919006},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.22612795233726501},{"id":"https://openalex.org/keywords/finite-element-method","display_name":"Finite element method","score":0.09683674573898315},{"id":"https://openalex.org/keywords/arithmetic","display_name":"Arithmetic","score":0.07927539944648743}],"concepts":[{"id":"https://openalex.org/C87619178","wikidata":"https://www.wikidata.org/wiki/Q126002","display_name":"Concatenation (mathematics)","level":2,"score":0.7226883172988892},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.6015526056289673},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.5591607689857483},{"id":"https://openalex.org/C13982400","wikidata":"https://www.wikidata.org/wiki/Q1129902","display_name":"Linear differential equation","level":3,"score":0.5290364623069763},{"id":"https://openalex.org/C45374587","wikidata":"https://www.wikidata.org/wiki/Q12525525","display_name":"Computation","level":2,"score":0.5286912322044373},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5199088454246521},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5126070380210876},{"id":"https://openalex.org/C2780737243","wikidata":"https://www.wikidata.org/wiki/Q17082433","display_name":"Isogeometric analysis","level":3,"score":0.5101185441017151},{"id":"https://openalex.org/C36503486","wikidata":"https://www.wikidata.org/wiki/Q11235244","display_name":"Domain (mathematical analysis)","level":2,"score":0.4694152772426605},{"id":"https://openalex.org/C38365724","wikidata":"https://www.wikidata.org/wiki/Q4677469","display_name":"Activation function","level":3,"score":0.4533911347389221},{"id":"https://openalex.org/C108583219","wikidata":"https://www.wikidata.org/wiki/Q197536","display_name":"Deep learning","level":2,"score":0.4474591016769409},{"id":"https://openalex.org/C192209626","wikidata":"https://www.wikidata.org/wiki/Q190909","display_name":"Focus (optics)","level":2,"score":0.4107182025909424},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.3815303146839142},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.37177324295043945},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.35004788637161255},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.25716033577919006},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.22612795233726501},{"id":"https://openalex.org/C135628077","wikidata":"https://www.wikidata.org/wiki/Q220184","display_name":"Finite element method","level":2,"score":0.09683674573898315},{"id":"https://openalex.org/C94375191","wikidata":"https://www.wikidata.org/wiki/Q11205","display_name":"Arithmetic","level":1,"score":0.07927539944648743},{"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/C120665830","wikidata":"https://www.wikidata.org/wiki/Q14620","display_name":"Optics","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1007/978-3-031-08754-7_41","is_oa":false,"landing_page_url":"https://doi.org/10.1007/978-3-031-08754-7_41","pdf_url":null,"source":{"id":"https://openalex.org/S106296714","display_name":"Lecture notes in computer science","issn_l":"0302-9743","issn":["0302-9743","1611-3349"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"book series"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Lecture Notes in Computer Science","raw_type":"book-chapter"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":3,"referenced_works":["https://openalex.org/W2899283552","https://openalex.org/W4221067122","https://openalex.org/W4237713727"],"related_works":["https://openalex.org/W2373577936","https://openalex.org/W4221148444","https://openalex.org/W1940709639","https://openalex.org/W4387678054","https://openalex.org/W2389596151","https://openalex.org/W3095575180","https://openalex.org/W4306784355","https://openalex.org/W1924025565","https://openalex.org/W2159994350","https://openalex.org/W2962758781"],"abstract_inverted_index":null,"counts_by_year":[{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":1}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}
