{"id":"https://openalex.org/W7131087710","doi":"https://doi.org/10.3389/frai.2026.1717117","title":"Implementing physics-informed neural networks with deep learning for differential equations","display_name":"Implementing physics-informed neural networks with deep learning for differential equations","publication_year":2026,"publication_date":"2026-02-23","ids":{"openalex":"https://openalex.org/W7131087710","doi":"https://doi.org/10.3389/frai.2026.1717117","pmid":"https://pubmed.ncbi.nlm.nih.gov/41809583"},"language":"en","primary_location":{"id":"doi:10.3389/frai.2026.1717117","is_oa":true,"landing_page_url":"https://doi.org/10.3389/frai.2026.1717117","pdf_url":"https://public-pages-files-2025.frontiersin.org/journals/artificial-intelligence/articles/10.3389/frai.2026.1717117/pdf","source":{"id":"https://openalex.org/S4210197006","display_name":"Frontiers in Artificial Intelligence","issn_l":"2624-8212","issn":["2624-8212"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310320527","host_organization_name":"Frontiers Media","host_organization_lineage":["https://openalex.org/P4310320527"],"host_organization_lineage_names":["Frontiers Media"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Frontiers in Artificial Intelligence","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj","pubmed"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://public-pages-files-2025.frontiersin.org/journals/artificial-intelligence/articles/10.3389/frai.2026.1717117/pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5057625030","display_name":"Frank Emmert\u2010Streib","orcid":"https://orcid.org/0000-0003-0745-5641"},"institutions":[{"id":"https://openalex.org/I176601375","display_name":"Khalifa University of Science and Technology","ror":"https://ror.org/05hffr360","country_code":"AE","type":"education","lineage":["https://openalex.org/I176601375"]},{"id":"https://openalex.org/I4210159872","display_name":"Predictive Science (United States)","ror":"https://ror.org/05canvq15","country_code":"US","type":"company","lineage":["https://openalex.org/I4210159872"]},{"id":"https://openalex.org/I4210164233","display_name":"Health & Life (Taiwan)","ror":"https://ror.org/05nnz8429","country_code":"TW","type":"company","lineage":["https://openalex.org/I4210164233"]}],"countries":["AE","TW","US"],"is_corresponding":false,"raw_author_name":"Frank Emmert-Streib","raw_affiliation_strings":["College of Health and Life Sciences, Hamad Bin Khalifa University","Predictive Society and Data Analytics Lab, Faculty of Information Technology and Communication Sciences, Tampere University"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"College of Health and Life Sciences, Hamad Bin Khalifa University","institution_ids":["https://openalex.org/I176601375","https://openalex.org/I4210164233"]},{"raw_affiliation_string":"Predictive Society and Data Analytics Lab, Faculty of Information Technology and Communication Sciences, Tampere University","institution_ids":["https://openalex.org/I4210159872"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5103047628","display_name":"Shailesh Tripathi","orcid":"https://orcid.org/0000-0002-3484-6368"},"institutions":[{"id":"https://openalex.org/I4210136249","display_name":"University of Applied Sciences Upper Austria","ror":"https://ror.org/03jqp6d56","country_code":"AT","type":"education","lineage":["https://openalex.org/I4210136249"]}],"countries":["AT"],"is_corresponding":true,"raw_author_name":"Shailesh Tripathi","raw_affiliation_strings":["Josef Ressel Centre for Data-Driven Business Model Innovation, University of Applied Sciences Upper Austria"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Josef Ressel Centre for Data-Driven Business Model Innovation, University of Applied Sciences Upper Austria","institution_ids":["https://openalex.org/I4210136249"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5111306296","display_name":"Amer Farea","orcid":null},"institutions":[{"id":"https://openalex.org/I4210159872","display_name":"Predictive Science (United States)","ror":"https://ror.org/05canvq15","country_code":"US","type":"company","lineage":["https://openalex.org/I4210159872"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Amer Farea","raw_affiliation_strings":["Predictive Society and Data Analytics Lab, Faculty of Information Technology and Communication Sciences, Tampere University"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Predictive Society and Data Analytics Lab, Faculty of Information Technology and Communication Sciences, Tampere University","institution_ids":["https://openalex.org/I4210159872"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5126640411","display_name":"Andreas Holzinger","orcid":null},"institutions":[{"id":"https://openalex.org/I4210096541","display_name":"Institute for Biodiversity","ror":"https://ror.org/01n8ggb71","country_code":"DE","type":"nonprofit","lineage":["https://openalex.org/I4210096541"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Andreas Holzinger","raw_affiliation_strings":["Human-Centered AI Lab, Institute of Forest Engineering, Department of Ecosystem Management, Climate and Biodiversity, BOKU University"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Human-Centered AI Lab, Institute of Forest Engineering, Department of Ecosystem Management, Climate and Biodiversity, BOKU University","institution_ids":["https://openalex.org/I4210096541"]}]}],"institutions":[],"countries_distinct_count":5,"institutions_distinct_count":5,"corresponding_author_ids":["https://openalex.org/A5103047628"],"corresponding_institution_ids":["https://openalex.org/I4210136249"],"apc_list":{"value":2125,"currency":"USD","value_usd":2125},"apc_paid":{"value":2125,"currency":"USD","value_usd":2125},"fwci":29.6255,"has_fulltext":true,"cited_by_count":4,"citation_normalized_percentile":{"value":0.99465948,"is_in_top_1_percent":true,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":98,"max":99},"biblio":{"volume":"9","issue":null,"first_page":"1717117","last_page":"1717117"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11206","display_name":"Model Reduction and Neural Networks","score":0.9692000150680542,"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":0.9692000150680542,"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/T11948","display_name":"Machine Learning in Materials Science","score":0.012500000186264515,"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"}},{"id":"https://openalex.org/T12611","display_name":"Neural Networks and Reservoir Computing","score":0.0026000000070780516,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/python","display_name":"Python (programming language)","score":0.6262000203132629},{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.5874000191688538},{"id":"https://openalex.org/keywords/deep-learning","display_name":"Deep learning","score":0.5766000151634216},{"id":"https://openalex.org/keywords/key","display_name":"Key (lock)","score":0.5364000201225281},{"id":"https://openalex.org/keywords/ordinary-differential-equation","display_name":"Ordinary differential equation","score":0.487199991941452},{"id":"https://openalex.org/keywords/partial-differential-equation","display_name":"Partial differential equation","score":0.3774999976158142}],"concepts":[{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.6793000102043152},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.6525999903678894},{"id":"https://openalex.org/C519991488","wikidata":"https://www.wikidata.org/wiki/Q28865","display_name":"Python (programming language)","level":2,"score":0.6262000203132629},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.5874000191688538},{"id":"https://openalex.org/C108583219","wikidata":"https://www.wikidata.org/wiki/Q197536","display_name":"Deep learning","level":2,"score":0.5766000151634216},{"id":"https://openalex.org/C26517878","wikidata":"https://www.wikidata.org/wiki/Q228039","display_name":"Key (lock)","level":2,"score":0.5364000201225281},{"id":"https://openalex.org/C119857082","wikidata":"https://www.wikidata.org/wiki/Q2539","display_name":"Machine learning","level":1,"score":0.5288000106811523},{"id":"https://openalex.org/C51544822","wikidata":"https://www.wikidata.org/wiki/Q465274","display_name":"Ordinary differential equation","level":3,"score":0.487199991941452},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.3774999976158142},{"id":"https://openalex.org/C2984842247","wikidata":"https://www.wikidata.org/wiki/Q197536","display_name":"Deep neural networks","level":3,"score":0.3682999908924103},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.3571999967098236},{"id":"https://openalex.org/C116672817","wikidata":"https://www.wikidata.org/wiki/Q1454986","display_name":"Physical system","level":2,"score":0.35600000619888306},{"id":"https://openalex.org/C135252773","wikidata":"https://www.wikidata.org/wiki/Q1567213","display_name":"Inverse problem","level":2,"score":0.3222000002861023},{"id":"https://openalex.org/C93226319","wikidata":"https://www.wikidata.org/wiki/Q193137","display_name":"Differential (mechanical device)","level":2,"score":0.3125999867916107}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.3389/frai.2026.1717117","is_oa":true,"landing_page_url":"https://doi.org/10.3389/frai.2026.1717117","pdf_url":"https://public-pages-files-2025.frontiersin.org/journals/artificial-intelligence/articles/10.3389/frai.2026.1717117/pdf","source":{"id":"https://openalex.org/S4210197006","display_name":"Frontiers in Artificial Intelligence","issn_l":"2624-8212","issn":["2624-8212"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310320527","host_organization_name":"Frontiers Media","host_organization_lineage":["https://openalex.org/P4310320527"],"host_organization_lineage_names":["Frontiers Media"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Frontiers in Artificial Intelligence","raw_type":"journal-article"},{"id":"pmid:41809583","is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/41809583","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":"Frontiers in artificial intelligence","raw_type":null},{"id":"pmh:oai:pubmedcentral.nih.gov:12968141","is_oa":true,"landing_page_url":"https://pmc.ncbi.nlm.nih.gov/articles/PMC12968141/","pdf_url":null,"source":{"id":"https://openalex.org/S2764455111","display_name":"PubMed Central","issn_l":null,"issn":null,"is_oa":true,"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Front Artif Intell","raw_type":"Text"},{"id":"pmh:oai:trepo.tuni.fi:10024/235666","is_oa":true,"landing_page_url":"https://trepo.tuni.fi/handle/10024/235666","pdf_url":null,"source":{"id":"https://openalex.org/S7407055260","display_name":"Trepo - Institutional Repository of Tampere University","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","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"publishedVersion"}],"best_oa_location":{"id":"doi:10.3389/frai.2026.1717117","is_oa":true,"landing_page_url":"https://doi.org/10.3389/frai.2026.1717117","pdf_url":"https://public-pages-files-2025.frontiersin.org/journals/artificial-intelligence/articles/10.3389/frai.2026.1717117/pdf","source":{"id":"https://openalex.org/S4210197006","display_name":"Frontiers in Artificial Intelligence","issn_l":"2624-8212","issn":["2624-8212"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310320527","host_organization_name":"Frontiers Media","host_organization_lineage":["https://openalex.org/P4310320527"],"host_organization_lineage_names":["Frontiers Media"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Frontiers in Artificial Intelligence","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7131087710.pdf","grobid_xml":"https://content.openalex.org/works/W7131087710.grobid-xml"},"referenced_works_count":47,"referenced_works":["https://openalex.org/W1968495411","https://openalex.org/W2010654495","https://openalex.org/W2011909181","https://openalex.org/W2054286320","https://openalex.org/W2160314966","https://openalex.org/W2163783967","https://openalex.org/W2335001735","https://openalex.org/W2797482001","https://openalex.org/W2897682710","https://openalex.org/W2944465249","https://openalex.org/W2980308966","https://openalex.org/W3014009018","https://openalex.org/W3029447859","https://openalex.org/W3126293391","https://openalex.org/W3134936618","https://openalex.org/W3136841107","https://openalex.org/W3151107810","https://openalex.org/W3163993681","https://openalex.org/W3207213471","https://openalex.org/W4220717841","https://openalex.org/W4281744908","https://openalex.org/W4283796342","https://openalex.org/W4288039037","https://openalex.org/W4309548460","https://openalex.org/W4313293844","https://openalex.org/W4366280606","https://openalex.org/W4367349439","https://openalex.org/W4367860144","https://openalex.org/W4376138024","https://openalex.org/W4380738843","https://openalex.org/W4381686075","https://openalex.org/W4381687077","https://openalex.org/W4385626908","https://openalex.org/W4388201041","https://openalex.org/W4388626883","https://openalex.org/W4391591764","https://openalex.org/W4394781925","https://openalex.org/W4396506372","https://openalex.org/W4397001969","https://openalex.org/W4401992148","https://openalex.org/W4402083191","https://openalex.org/W4402352902","https://openalex.org/W4402502606","https://openalex.org/W4404797973","https://openalex.org/W4405137115","https://openalex.org/W4414976502","https://openalex.org/W4416350488"],"related_works":[],"abstract_inverted_index":{"Physics-aware":[0],"machine":[1,8],"learning":[2,9,27],"integrates":[3],"domain-specific":[4],"physical":[5,22],"knowledge":[6],"into":[7,25,115],"models,":[10],"leading":[11],"to":[12,35,109],"the":[13,26,39,57,75,90,101],"development":[14],"of":[15,42,59,63,98],"physics-informed":[16],"neural":[17],"networks":[18],"(PINNs).":[19],"PINNs":[20,43,60],"embed":[21],"laws":[23],"directly":[24],"process,":[28],"enabling":[29],"interpretable":[30],"and":[31,46,100,137],"physically":[32],"consistent":[33],"solutions":[34],"complex":[36],"problems.":[37],"However,":[38],"practical":[40,113],"use":[41],"presents":[44],"challenges":[45,136],"their":[47],"applications":[48],"are":[49],"complex.":[50],"Therefore,":[51],"in":[52,144],"this":[53],"paper,":[54],"we":[55,117,133],"demonstrate":[56],"implementation":[58,126,146],"for":[61,141],"systems":[62],"ordinary":[64],"differential":[65,83],"equations":[66],"(ODEs),":[67],"an":[68,106],"area":[69],"that":[70],"is":[71],"often":[72],"overlooked":[73],"by":[74],"physics":[76],"community,":[77],"which":[78,93,104],"typically":[79],"focuses":[80],"on":[81,123,130],"partial":[82],"equations.":[84],"We":[85],"discuss":[86,134],"two":[87,119],"key":[88,135],"challenges:":[89],"inverse":[91],"problem,":[92,103],"involves":[94],"estimating":[95],"unknown":[96],"parameters":[97],"ODEs,":[99],"forward":[102],"provides":[105],"approximate":[107],"solution":[108],"ODEs.":[110],"To":[111],"provide":[112],"insights":[114],"PINNs,":[116],"present":[118],"case":[120],"studies":[121],"based":[122],"a":[124],"Python":[125],"using":[127],"DeepXDE.":[128],"Drawing":[129],"these":[131],"studies,":[132],"identify":[138],"promising":[139],"directions":[140],"future":[142],"research":[143],"PINN-based":[145],"frameworks.":[147]},"counts_by_year":[{"year":2026,"cited_by_count":4}],"updated_date":"2026-07-23T08:03:31.855105","created_date":"2026-02-24T00:00:00"}