{"id":"https://openalex.org/W4223456294","doi":"https://doi.org/10.1109/access.2022.3160186","title":"Neural Networks to Solve Partial Differential Equations: A Comparison With Finite Elements","display_name":"Neural Networks to Solve Partial Differential Equations: A Comparison With Finite Elements","publication_year":2022,"publication_date":"2022-01-01","ids":{"openalex":"https://openalex.org/W4223456294","doi":"https://doi.org/10.1109/access.2022.3160186"},"language":"en","primary_location":{"id":"doi:10.1109/access.2022.3160186","is_oa":true,"landing_page_url":"https://doi.org/10.1109/access.2022.3160186","pdf_url":"https://ieeexplore.ieee.org/ielx7/6287639/9668973/09737092.pdf","source":{"id":"https://openalex.org/S2485537415","display_name":"IEEE Access","issn_l":"2169-3536","issn":["2169-3536"],"is_oa":true,"is_in_doaj":true,"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Access","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://ieeexplore.ieee.org/ielx7/6287639/9668973/09737092.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5037604053","display_name":"A. Sacchetti","orcid":"https://orcid.org/0000-0002-5907-6603"},"institutions":[{"id":"https://openalex.org/I2972652528","display_name":"FHNW University of Applied Sciences and Arts Northwestern Switzerland","ror":"https://ror.org/04mq2g308","country_code":"CH","type":"education","lineage":["https://openalex.org/I2972652528"]}],"countries":["CH"],"is_corresponding":false,"raw_author_name":"Andrea Sacchetti","raw_affiliation_strings":["IMN, University of Applied Science and Arts Northwestern Switzerland, Windisch, Switzerland"],"raw_orcid":"https://orcid.org/0000-0002-5907-6603","affiliations":[{"raw_affiliation_string":"IMN, University of Applied Science and Arts Northwestern Switzerland, Windisch, Switzerland","institution_ids":["https://openalex.org/I2972652528"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5075387816","display_name":"Benjamin Bachmann","orcid":"https://orcid.org/0000-0002-3583-4895"},"institutions":[{"id":"https://openalex.org/I2972652528","display_name":"FHNW University of Applied Sciences and Arts Northwestern Switzerland","ror":"https://ror.org/04mq2g308","country_code":"CH","type":"education","lineage":["https://openalex.org/I2972652528"]}],"countries":["CH"],"is_corresponding":false,"raw_author_name":"Benjamin Bachmann","raw_affiliation_strings":["IPPE, University of Applied Science and Arts Northwestern Switzerland, Windisch, Switzerland"],"raw_orcid":"https://orcid.org/0000-0002-3583-4895","affiliations":[{"raw_affiliation_string":"IPPE, University of Applied Science and Arts Northwestern Switzerland, Windisch, Switzerland","institution_ids":["https://openalex.org/I2972652528"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5110524469","display_name":"Kaspar L\u00f6ffel","orcid":null},"institutions":[{"id":"https://openalex.org/I2972652528","display_name":"FHNW University of Applied Sciences and Arts Northwestern Switzerland","ror":"https://ror.org/04mq2g308","country_code":"CH","type":"education","lineage":["https://openalex.org/I2972652528"]}],"countries":["CH"],"is_corresponding":false,"raw_author_name":"Kaspar Loffel","raw_affiliation_strings":["IPPE, University of Applied Science and Arts Northwestern Switzerland, Windisch, Switzerland"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"IPPE, University of Applied Science and Arts Northwestern Switzerland, Windisch, Switzerland","institution_ids":["https://openalex.org/I2972652528"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5007795891","display_name":"Urs-Martin K\u00fcnzi","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Urs-Martin Kunzi","raw_affiliation_strings":["Laboratory of Web Science, Fernfachhhochschule FFHS, Zurich, Switzerland"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Laboratory of Web Science, Fernfachhhochschule FFHS, Zurich, Switzerland","institution_ids":[]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5012923372","display_name":"Beatrice Paoli","orcid":"https://orcid.org/0000-0001-5576-4110"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Beatrice Paoli","raw_affiliation_strings":["Laboratory of Web Science, Fernfachhhochschule FFHS, Zurich, Switzerland"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Laboratory of Web Science, Fernfachhhochschule FFHS, Zurich, Switzerland","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":5,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":{"value":1850,"currency":"USD","value_usd":1850},"apc_paid":{"value":1850,"currency":"USD","value_usd":1850},"fwci":1.7531,"has_fulltext":true,"cited_by_count":15,"citation_normalized_percentile":{"value":0.84381276,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":94,"max":98},"biblio":{"volume":"10","issue":null,"first_page":"32271","last_page":"32279"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11206","display_name":"Model Reduction and Neural Networks","score":1.0,"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":1.0,"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/T10998","display_name":"Heat Transfer and Optimization","score":0.9940999746322632,"subfield":{"id":"https://openalex.org/subfields/2210","display_name":"Mechanical Engineering"},"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/T10928","display_name":"Probabilistic and Robust Engineering Design","score":0.9916999936103821,"subfield":{"id":"https://openalex.org/subfields/1804","display_name":"Statistics, Probability and Uncertainty"},"field":{"id":"https://openalex.org/fields/18","display_name":"Decision Sciences"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/finite-element-method","display_name":"Finite element method","score":0.8161852955818176},{"id":"https://openalex.org/keywords/hyperparameter","display_name":"Hyperparameter","score":0.7514430284500122},{"id":"https://openalex.org/keywords/partial-differential-equation","display_name":"Partial differential equation","score":0.7000856399536133},{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.62360680103302},{"id":"https://openalex.org/keywords/computation","display_name":"Computation","score":0.5953441262245178},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.592929482460022},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.507675290107727},{"id":"https://openalex.org/keywords/differential-equation","display_name":"Differential equation","score":0.4710124731063843},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.46186524629592896},{"id":"https://openalex.org/keywords/point","display_name":"Point (geometry)","score":0.433050274848938},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.2967711091041565},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.21907144784927368},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.2097841203212738},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.0918039083480835},{"id":"https://openalex.org/keywords/structural-engineering","display_name":"Structural engineering","score":0.07118701934814453},{"id":"https://openalex.org/keywords/engineering","display_name":"Engineering","score":0.05930823087692261}],"concepts":[{"id":"https://openalex.org/C135628077","wikidata":"https://www.wikidata.org/wiki/Q220184","display_name":"Finite element method","level":2,"score":0.8161852955818176},{"id":"https://openalex.org/C8642999","wikidata":"https://www.wikidata.org/wiki/Q4171168","display_name":"Hyperparameter","level":2,"score":0.7514430284500122},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.7000856399536133},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.62360680103302},{"id":"https://openalex.org/C45374587","wikidata":"https://www.wikidata.org/wiki/Q12525525","display_name":"Computation","level":2,"score":0.5953441262245178},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.592929482460022},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.507675290107727},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.4710124731063843},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.46186524629592896},{"id":"https://openalex.org/C28719098","wikidata":"https://www.wikidata.org/wiki/Q44946","display_name":"Point (geometry)","level":2,"score":0.433050274848938},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.2967711091041565},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.21907144784927368},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.2097841203212738},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0918039083480835},{"id":"https://openalex.org/C66938386","wikidata":"https://www.wikidata.org/wiki/Q633538","display_name":"Structural engineering","level":1,"score":0.07118701934814453},{"id":"https://openalex.org/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","level":0,"score":0.05930823087692261}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1109/access.2022.3160186","is_oa":true,"landing_page_url":"https://doi.org/10.1109/access.2022.3160186","pdf_url":"https://ieeexplore.ieee.org/ielx7/6287639/9668973/09737092.pdf","source":{"id":"https://openalex.org/S2485537415","display_name":"IEEE Access","issn_l":"2169-3536","issn":["2169-3536"],"is_oa":true,"is_in_doaj":true,"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Access","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:2201.03269","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2201.03269","pdf_url":"https://arxiv.org/pdf/2201.03269","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"},{"id":"pmh:oai:doaj.org/article:d7860a7956d542cd8fb41835aef5323f","is_oa":true,"landing_page_url":"https://doaj.org/article/d7860a7956d542cd8fb41835aef5323f","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","issn_l":null,"issn":null,"is_oa":false,"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-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"IEEE Access, Vol 10, Pp 32271-32279 (2022)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.1109/access.2022.3160186","is_oa":true,"landing_page_url":"https://doi.org/10.1109/access.2022.3160186","pdf_url":"https://ieeexplore.ieee.org/ielx7/6287639/9668973/09737092.pdf","source":{"id":"https://openalex.org/S2485537415","display_name":"IEEE Access","issn_l":"2169-3536","issn":["2169-3536"],"is_oa":true,"is_in_doaj":true,"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Access","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320321942","display_name":"Hasler Stiftung","ror":"https://ror.org/04m3t9183"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4223456294.pdf","grobid_xml":"https://content.openalex.org/works/W4223456294.grobid-xml"},"referenced_works_count":22,"referenced_works":["https://openalex.org/W1577960745","https://openalex.org/W2036804930","https://openalex.org/W2101234009","https://openalex.org/W2125645174","https://openalex.org/W2168414272","https://openalex.org/W2749028154","https://openalex.org/W2770250658","https://openalex.org/W2803629276","https://openalex.org/W2805403880","https://openalex.org/W2962727772","https://openalex.org/W2998847955","https://openalex.org/W3008453832","https://openalex.org/W3014009018","https://openalex.org/W3093786597","https://openalex.org/W3112199937","https://openalex.org/W3179061817","https://openalex.org/W3196381478","https://openalex.org/W3201282075","https://openalex.org/W6638908295","https://openalex.org/W6675354045","https://openalex.org/W6766978945","https://openalex.org/W6798643404"],"related_works":["https://openalex.org/W4390421286","https://openalex.org/W4280563792","https://openalex.org/W2140186469","https://openalex.org/W4389724018","https://openalex.org/W4318719684","https://openalex.org/W2775233965","https://openalex.org/W4318559728","https://openalex.org/W4360995913","https://openalex.org/W2378760347","https://openalex.org/W2029287765"],"abstract_inverted_index":{"We":[0],"compare":[1],"the":[2,32,43,57,80,90,104,117,134,150,162,180,186,196,206],"Finite":[3],"Element":[4],"Method":[5],"(FEM)":[6],"simulation":[7],"of":[8,16,45,56,89,103,119,133,136,153,189,198],"a":[9,17,20,23,40,71,86,94,191,199],"standard":[10],"Partial":[11],"Differential":[12],"Equation":[13],"thermal":[14],"problem":[15],"plate":[18],"with":[19,22,50,65,74,161],"hole":[21],"Neural":[24],"Network":[25],"(NN)":[26],"simulation.":[27],"The":[28],"largest":[29],"deviation":[30],"from":[31,36],"true":[33],"solution":[34,41,207],"obtained":[35],"FEM":[37,98,121,200],"(0.015":[38],"for":[39,110,113,185],"on":[42,116,149],"order":[44,118],"unity)":[46],"is":[47,202],"easily":[48],"achieved":[49,160,175],"NN":[51,100,123],"too":[52],"without":[53],"much":[54],"tuning":[55],"hyperparameters.":[58],"Accuracies":[59,155],"below":[60,156,171],"0.01":[61,157],"instead":[62],"require":[63,124],"refinement":[64],"an":[66,114],"alternative":[67],"optimizer":[68],"to":[69],"reach":[70],"similar":[72],"performance":[73,197],"NN.":[75],"A":[76,130],"rough":[77],"comparison":[78],"between":[79,97],"Floating":[81],"Point":[82],"Operations":[83],"values,":[84],"as":[85,168],"machine-independent":[87],"quantification":[88],"computational":[91,144],"performance,":[92],"suggests":[93],"significant":[95],"difference":[96],"and":[99,122,126,143,165],"in":[101],"favour":[102],"former.":[105],"This":[106],"also":[107],"strongly":[108],"holds":[109],"computation":[111],"time:":[112],"accuracy":[115,142],"10<sup>&#x2212;5</sup>,":[120],"54":[125],"1100":[127],"seconds,":[128],"respectively.":[129],"detailed":[131],"analysis":[132],"effect":[135],"varying":[137],"different":[138],"hyperparameters":[139],"shows":[140,183],"that":[141,184],"time":[145],"only":[146],"weakly":[147],"depend":[148],"major":[151],"part":[152],"them.":[154],"cannot":[158,173],"be":[159,174],"&#x201C;adam&#x201D;":[163],"optimizers":[164],"it":[166],"looks":[167],"though":[169],"accuracies":[170],"10<sup>&#x2212;5</sup>":[172],"at":[176],"all.":[177],"In":[178],"conclusion,":[179],"present":[181],"work":[182],"concrete":[187],"case":[188],"solving":[190],"steady-state":[192],"2D":[193],"heat":[194],"equation,":[195],"algorithm":[201],"significantly":[203],"better":[204],"than":[205],"via":[208],"networks.":[209]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":3},{"year":2024,"cited_by_count":5},{"year":2023,"cited_by_count":4},{"year":2022,"cited_by_count":2}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2022-04-14T00:00:00"}
