{"id":"https://openalex.org/W4288039037","doi":"https://doi.org/10.1007/s10915-022-01939-z","title":"Scientific Machine Learning Through Physics\u2013Informed Neural Networks: Where we are and What\u2019s Next","display_name":"Scientific Machine Learning Through Physics\u2013Informed Neural Networks: Where we are and What\u2019s Next","publication_year":2022,"publication_date":"2022-07-26","ids":{"openalex":"https://openalex.org/W4288039037","doi":"https://doi.org/10.1007/s10915-022-01939-z"},"language":"en","primary_location":{"id":"doi:10.1007/s10915-022-01939-z","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10915-022-01939-z","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10915-022-01939-z.pdf","source":{"id":"https://openalex.org/S151550017","display_name":"Journal of Scientific Computing","issn_l":"0885-7474","issn":["0885-7474","1573-7691"],"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":"journal"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Scientific Computing","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007/s10915-022-01939-z.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5002497813","display_name":"Salvatore Cuomo","orcid":"https://orcid.org/0000-0003-4128-2588"},"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":true,"raw_author_name":"Salvatore Cuomo","raw_affiliation_strings":["Department of Mathematics and Applications \"Renato Caccioppoli\", University of Naples Federico II, Napoli, 80126, Italy"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Applications \"Renato Caccioppoli\", University of Naples Federico II, Napoli, 80126, Italy","institution_ids":["https://openalex.org/I71267560"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5047698004","display_name":"Vincenzo Schiano Di Cola","orcid":null},"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":"Vincenzo Schiano Di Cola","raw_affiliation_strings":["Department of Electrical Engineering and Information Technology, University of Naples Federico II, Via Claudio, Napoli, 80125, Italy"],"affiliations":[{"raw_affiliation_string":"Department of Electrical Engineering and Information Technology, University of Naples Federico II, Via Claudio, Napoli, 80125, Italy","institution_ids":["https://openalex.org/I71267560"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5022404875","display_name":"Fabio Giampaolo","orcid":"https://orcid.org/0000-0001-5414-3435"},"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":"Fabio Giampaolo","raw_affiliation_strings":["Department of Mathematics and Applications \"Renato Caccioppoli\", University of Naples Federico II, Napoli, 80126, Italy"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Applications \"Renato Caccioppoli\", University of Naples Federico II, Napoli, 80126, Italy","institution_ids":["https://openalex.org/I71267560"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5061558433","display_name":"Gianluigi Rozza","orcid":"https://orcid.org/0000-0002-0810-8812"},"institutions":[{"id":"https://openalex.org/I138549579","display_name":"Scuola Internazionale Superiore di Studi Avanzati","ror":"https://ror.org/004fze387","country_code":"IT","type":"education","lineage":["https://openalex.org/I138549579"]}],"countries":["IT"],"is_corresponding":false,"raw_author_name":"Gianluigi Rozza","raw_affiliation_strings":["SISSA, International School for Advanced Studies, Mathematics Area, mathLab, via Bonomea 265, Trieste, 34136, Italy"],"affiliations":[{"raw_affiliation_string":"SISSA, International School for Advanced Studies, Mathematics Area, mathLab, via Bonomea 265, Trieste, 34136, Italy","institution_ids":["https://openalex.org/I138549579"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5012536010","display_name":"Maziar Raissi","orcid":"https://orcid.org/0000-0002-8467-4568"},"institutions":[{"id":"https://openalex.org/I188538660","display_name":"University of Colorado Boulder","ror":"https://ror.org/02ttsq026","country_code":"US","type":"education","lineage":["https://openalex.org/I188538660"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Maziar Raissi","raw_affiliation_strings":["Department of Applied Mathematics, University of Colorado Boulder, Boulder, CO, 80309, USA"],"affiliations":[{"raw_affiliation_string":"Department of Applied Mathematics, University of Colorado Boulder, Boulder, CO, 80309, USA","institution_ids":["https://openalex.org/I188538660"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5016445191","display_name":"Francesco Piccialli","orcid":"https://orcid.org/0000-0002-5179-2496"},"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":"Francesco Piccialli","raw_affiliation_strings":["Department of Mathematics and Applications \"Renato Caccioppoli\", University of Naples Federico II, Napoli, 80126, Italy"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Applications \"Renato Caccioppoli\", University of Naples Federico II, Napoli, 80126, Italy","institution_ids":["https://openalex.org/I71267560"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":6,"corresponding_author_ids":["https://openalex.org/A5002497813"],"corresponding_institution_ids":["https://openalex.org/I71267560"],"apc_list":{"value":2390,"currency":"EUR","value_usd":2990},"apc_paid":{"value":2390,"currency":"EUR","value_usd":2990},"fwci":222.4372,"has_fulltext":true,"cited_by_count":2043,"citation_normalized_percentile":{"value":1.0,"is_in_top_1_percent":true,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":99,"max":100},"biblio":{"volume":"92","issue":"3","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":0.9998999834060669,"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":0.9998999834060669,"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/T10104","display_name":"Nanofluid Flow and Heat Transfer","score":0.9761999845504761,"subfield":{"id":"https://openalex.org/subfields/2204","display_name":"Biomedical 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/T10998","display_name":"Heat Transfer and Optimization","score":0.9692999720573425,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.7395141124725342},{"id":"https://openalex.org/keywords/collocation","display_name":"Collocation (remote sensing)","score":0.5861973762512207},{"id":"https://openalex.org/keywords/partial-differential-equation","display_name":"Partial differential equation","score":0.543511688709259},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.5066474080085754},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.48861151933670044},{"id":"https://openalex.org/keywords/finite-element-method","display_name":"Finite element method","score":0.4674941599369049},{"id":"https://openalex.org/keywords/range","display_name":"Range (aeronautics)","score":0.46288132667541504},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.43295520544052124},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.3432826101779938},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.3359297215938568},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.3333073854446411},{"id":"https://openalex.org/keywords/machine-learning","display_name":"Machine learning","score":0.24750223755836487},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.20753055810928345},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.10703533887863159},{"id":"https://openalex.org/keywords/engineering","display_name":"Engineering","score":0.09233465790748596}],"concepts":[{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.7395141124725342},{"id":"https://openalex.org/C80023036","wikidata":"https://www.wikidata.org/wiki/Q5147531","display_name":"Collocation (remote sensing)","level":2,"score":0.5861973762512207},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.543511688709259},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.5066474080085754},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.48861151933670044},{"id":"https://openalex.org/C135628077","wikidata":"https://www.wikidata.org/wiki/Q220184","display_name":"Finite element method","level":2,"score":0.4674941599369049},{"id":"https://openalex.org/C204323151","wikidata":"https://www.wikidata.org/wiki/Q905424","display_name":"Range (aeronautics)","level":2,"score":0.46288132667541504},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.43295520544052124},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.3432826101779938},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.3359297215938568},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.3333073854446411},{"id":"https://openalex.org/C119857082","wikidata":"https://www.wikidata.org/wiki/Q2539","display_name":"Machine learning","level":1,"score":0.24750223755836487},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.20753055810928345},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.10703533887863159},{"id":"https://openalex.org/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","level":0,"score":0.09233465790748596},{"id":"https://openalex.org/C97355855","wikidata":"https://www.wikidata.org/wiki/Q11473","display_name":"Thermodynamics","level":1,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C78458016","wikidata":"https://www.wikidata.org/wiki/Q840400","display_name":"Evolutionary biology","level":1,"score":0.0},{"id":"https://openalex.org/C146978453","wikidata":"https://www.wikidata.org/wiki/Q3798668","display_name":"Aerospace engineering","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1007/s10915-022-01939-z","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10915-022-01939-z","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10915-022-01939-z.pdf","source":{"id":"https://openalex.org/S151550017","display_name":"Journal of Scientific Computing","issn_l":"0885-7474","issn":["0885-7474","1573-7691"],"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":"journal"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Scientific Computing","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.1007/s10915-022-01939-z","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10915-022-01939-z","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10915-022-01939-z.pdf","source":{"id":"https://openalex.org/S151550017","display_name":"Journal of Scientific Computing","issn_l":"0885-7474","issn":["0885-7474","1573-7691"],"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":"journal"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Scientific Computing","raw_type":"journal-article"},"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":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4288039037.pdf","grobid_xml":"https://content.openalex.org/works/W4288039037.grobid-xml"},"referenced_works_count":185,"referenced_works":["https://openalex.org/W568942680","https://openalex.org/W1600371943","https://openalex.org/W1856502440","https://openalex.org/W1968326286","https://openalex.org/W1970454553","https://openalex.org/W1978075626","https://openalex.org/W1993717606","https://openalex.org/W2031404277","https://openalex.org/W2103496339","https://openalex.org/W2137983211","https://openalex.org/W2148666448","https://openalex.org/W2158581396","https://openalex.org/W2158985775","https://openalex.org/W2185518952","https://openalex.org/W2261689926","https://openalex.org/W2507348356","https://openalex.org/W2528305538","https://openalex.org/W2573864470","https://openalex.org/W2605147767","https://openalex.org/W2745110207","https://openalex.org/W2749028154","https://openalex.org/W2760972773","https://openalex.org/W2770250658","https://openalex.org/W2790567079","https://openalex.org/W2790808809","https://openalex.org/W2792247102","https://openalex.org/W2887569307","https://openalex.org/W2890968382","https://openalex.org/W2893995718","https://openalex.org/W2899283552","https://openalex.org/W2900369848","https://openalex.org/W2908541468","https://openalex.org/W2914483840","https://openalex.org/W2919115771","https://openalex.org/W2919358988","https://openalex.org/W2929803724","https://openalex.org/W2930003055","https://openalex.org/W2942231644","https://openalex.org/W2946302218","https://openalex.org/W2947411064","https://openalex.org/W2948230027","https://openalex.org/W2948551291","https://openalex.org/W2951392159","https://openalex.org/W2963518130","https://openalex.org/W2969381807","https://openalex.org/W2973886134","https://openalex.org/W2979786244","https://openalex.org/W2981025625","https://openalex.org/W2990346675","https://openalex.org/W2993158195","https://openalex.org/W2993805018","https://openalex.org/W2997165591","https://openalex.org/W2997588930","https://openalex.org/W2997814214","https://openalex.org/W2998004632","https://openalex.org/W2998366519","https://openalex.org/W2999026783","https://openalex.org/W2999451223","https://openalex.org/W3000196868","https://openalex.org/W3003922491","https://openalex.org/W3004442222","https://openalex.org/W3004450693","https://openalex.org/W3008118574","https://openalex.org/W3008453832","https://openalex.org/W3010839048","https://openalex.org/W3010849941","https://openalex.org/W3011147100","https://openalex.org/W3011806874","https://openalex.org/W3012417314","https://openalex.org/W3014009018","https://openalex.org/W3014468003","https://openalex.org/W3015865829","https://openalex.org/W3017674692","https://openalex.org/W3020952481","https://openalex.org/W3021668893","https://openalex.org/W3022424628","https://openalex.org/W3023243091","https://openalex.org/W3025645353","https://openalex.org/W3031879091","https://openalex.org/W3035493266","https://openalex.org/W3037134996","https://openalex.org/W3039900661","https://openalex.org/W3041682155","https://openalex.org/W3042633958","https://openalex.org/W3084017095","https://openalex.org/W3088991477","https://openalex.org/W3091593783","https://openalex.org/W3092172785","https://openalex.org/W3092743582","https://openalex.org/W3093786597","https://openalex.org/W3099057226","https://openalex.org/W3100789280","https://openalex.org/W3104994177","https://openalex.org/W3106050203","https://openalex.org/W3109749236","https://openalex.org/W3109834426","https://openalex.org/W3111308046","https://openalex.org/W3111357769","https://openalex.org/W3111557793","https://openalex.org/W3111914315","https://openalex.org/W3114000978","https://openalex.org/W3115283966","https://openalex.org/W3115418185","https://openalex.org/W3117301071","https://openalex.org/W3119250864","https://openalex.org/W3119453977","https://openalex.org/W3119602513","https://openalex.org/W3120333448","https://openalex.org/W3121755381","https://openalex.org/W3123895780","https://openalex.org/W3124254264","https://openalex.org/W3126393581","https://openalex.org/W3128157635","https://openalex.org/W3128539297","https://openalex.org/W3129119946","https://openalex.org/W3133294546","https://openalex.org/W3133816032","https://openalex.org/W3134515731","https://openalex.org/W3135266120","https://openalex.org/W3135588948","https://openalex.org/W3136841107","https://openalex.org/W3137125108","https://openalex.org/W3137240924","https://openalex.org/W3137392741","https://openalex.org/W3152851547","https://openalex.org/W3153200540","https://openalex.org/W3154575637","https://openalex.org/W3159534089","https://openalex.org/W3161445736","https://openalex.org/W3162045419","https://openalex.org/W3163993681","https://openalex.org/W3165899507","https://openalex.org/W3166275749","https://openalex.org/W3168386838","https://openalex.org/W3170003932","https://openalex.org/W3171202467","https://openalex.org/W3173375283","https://openalex.org/W3175270682","https://openalex.org/W3176116059","https://openalex.org/W3176880480","https://openalex.org/W3180490621","https://openalex.org/W3184364599","https://openalex.org/W3186083829","https://openalex.org/W3186608048","https://openalex.org/W3188634983","https://openalex.org/W3193076486","https://openalex.org/W3194456427","https://openalex.org/W3194468165","https://openalex.org/W3194756493","https://openalex.org/W3197308578","https://openalex.org/W3197732874","https://openalex.org/W3198835738","https://openalex.org/W3200673624","https://openalex.org/W3201460085","https://openalex.org/W3203264955","https://openalex.org/W3205073660","https://openalex.org/W3206964070","https://openalex.org/W3207133153","https://openalex.org/W3207890637","https://openalex.org/W3208453486","https://openalex.org/W3208605770","https://openalex.org/W3208953217","https://openalex.org/W3212771327","https://openalex.org/W3216718982","https://openalex.org/W3217086243","https://openalex.org/W4212988227","https://openalex.org/W4213176644","https://openalex.org/W4220717841","https://openalex.org/W4225995462","https://openalex.org/W4226099344","https://openalex.org/W4226162729","https://openalex.org/W4226325816","https://openalex.org/W4237591687","https://openalex.org/W4247523314","https://openalex.org/W4247680473","https://openalex.org/W4255027698","https://openalex.org/W4280645905","https://openalex.org/W4281627930","https://openalex.org/W4286856901","https://openalex.org/W4300079339","https://openalex.org/W4301239768","https://openalex.org/W4317434745","https://openalex.org/W4385488719","https://openalex.org/W4389775319","https://openalex.org/W6756040250"],"related_works":["https://openalex.org/W2370840338","https://openalex.org/W2388641108","https://openalex.org/W2368317224","https://openalex.org/W2370651559","https://openalex.org/W2383177152","https://openalex.org/W3000893075","https://openalex.org/W2349694428","https://openalex.org/W2384630579","https://openalex.org/W2358919722","https://openalex.org/W3000494655"],"abstract_inverted_index":{"Abstract":[0],"Physics-Informed":[1],"Neural":[2],"Networks":[3],"(PINN)":[4],"are":[5,27,192],"neural":[6,23,107,124,152],"networks":[7,85,125],"(NNs)":[8],"that":[9,136,199],"encode":[10],"model":[11],"equations,":[12,34,36],"like":[13,186],"Partial":[14],"Differential":[15],"Equations":[16],"(PDE),":[17],"as":[18,45,115,117,122],"a":[19,46,52,60,66,100],"component":[20],"of":[21,69,78,103,163],"the":[22,70,75,79,112,143,160],"network":[24,153],"itself.":[25],"PINNs":[26,167],"nowadays":[28],"used":[29],"to":[30,82,96,175],"solve":[31],"PDEs,":[32],"fractional":[33],"integral-differential":[35],"and":[37,86,90,129,155],"stochastic":[38],"PDEs.":[39],"This":[40,63],"novel":[41],"methodology":[42],"has":[43,139],"arisen":[44],"multi-task":[47],"learning":[48],"framework":[49],"in":[50,179],"which":[51,109,166],"NN":[53],"must":[54],"fit":[55],"observed":[56],"data":[57],"while":[58,74],"reducing":[59],"PDE":[61],"residual.":[62],"article":[64],"provides":[65],"comprehensive":[67],"review":[68,93],"literature":[71],"on":[72,99,141],"PINNs:":[73],"primary":[76],"goal":[77],"study":[80,134],"was":[81],"characterize":[83],"these":[84],"their":[87,173],"related":[88],"advantages":[89],"disadvantages.":[91],"The":[92,133],"also":[94],"attempts":[95],"incorporate":[97],"publications":[98],"broader":[101],"range":[102,162],"collocation-based":[104],"physics":[105],"informed":[106],"networks,":[108],"stars":[110],"form":[111],"vanilla":[113],"PINN,":[114],"well":[116],"many":[118],"other":[119],"variants,":[120],"such":[121],"physics-constrained":[123],"(PCNN),":[126],"variational":[127],"hp-VPINN,":[128],"conservative":[130],"PINN":[131,144],"(CPINN).":[132],"indicates":[135],"most":[137,195],"research":[138],"focused":[140],"customizing":[142],"through":[145],"different":[146],"activation":[147],"functions,":[148],"gradient":[149],"optimization":[150],"techniques,":[151],"structures,":[154],"loss":[156],"function":[157],"structures.":[158],"Despite":[159],"wide":[161],"applications":[164],"for":[165],"have":[168],"been":[169],"used,":[170],"by":[171],"demonstrating":[172],"ability":[174],"be":[176],"more":[177],"feasible":[178],"some":[180],"contexts":[181],"than":[182],"classical":[183],"numerical":[184],"techniques":[185],"Finite":[187],"Element":[188],"Method":[189],"(FEM),":[190],"advancements":[191],"still":[193],"possible,":[194],"notably":[196],"theoretical":[197],"issues":[198],"remain":[200],"unresolved.":[201]},"counts_by_year":[{"year":2026,"cited_by_count":267},{"year":2025,"cited_by_count":865},{"year":2024,"cited_by_count":546},{"year":2023,"cited_by_count":314},{"year":2022,"cited_by_count":51}],"updated_date":"2026-04-06T07:47:59.780226","created_date":"2025-10-10T00:00:00"}
