{"id":"https://openalex.org/W7214760627","doi":"https://doi.org/10.17759/mda.2026160308","title":"Embedding Physics-Informed Neural Networks into Numerical Schemes for Modeling Dynamical Systems","display_name":"Embedding Physics-Informed Neural Networks into Numerical Schemes for Modeling Dynamical Systems","publication_year":2026,"publication_date":"2026-09-29","ids":{"openalex":"https://openalex.org/W7214760627","doi":"https://doi.org/10.17759/mda.2026160308"},"language":"en","primary_location":{"id":"doi:10.17759/mda.2026160308","doi":"https://doi.org/10.17759/mda.2026160308","is_oa":true,"landing_page_url":"https://doi.org/10.17759/mda.2026160308","pdf_url":"https://psyjournals.ru/journals/mda/archive/2026_n3/mda_2026_n3_Velikorechanin_Lazovskaya_Tarkhov.pdf","source":{"id":"https://openalex.org/S4210188866","display_name":"Modelling and Data Analysis","issn_l":"2219-3758","issn":["2219-3758","2311-9454"],"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":["russia-white-list-3"],"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"journal"},"license":"cc-by-nc","license_id":"https://openalex.org/licenses/cc-by-nc","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Modelling and Data Analysis","raw_type":"journal-article"},"volume":null,"type":"article","is_traditional":true,"indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://psyjournals.ru/journals/mda/archive/2026_n3/mda_2026_n3_Velikorechanin_Lazovskaya_Tarkhov.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5153509575","display_name":"I.A. Velikorechanin","orcid":"https://orcid.org/0009-0007-7889-6811","observed_orcids":["https://orcid.org/0009-0007-7889-6811"]},"institutions":[{"id":"https://openalex.org/I153845743","display_name":"Moscow Institute of Physics and Technology","ror":"https://ror.org/00v0z9322","country_code":"RU","type":"education","lineage":["https://openalex.org/I153845743"]}],"countries":["RU"],"is_corresponding":false,"raw_author_name":"I.A. Velikorechanin","raw_affiliation_strings":["Moscow Institute of Physics and Technology"],"raw_orcid":"https://orcid.org/0009-0007-7889-6811","affiliations":[{"raw_affiliation_string":"Moscow Institute of Physics and Technology","institution_ids":["https://openalex.org/I153845743"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5056034703","display_name":"Tatiana V. Lazovskaya","orcid":"https://orcid.org/0000-0002-3324-6213","observed_orcids":["https://orcid.org/0000-0002-3324-6213"]},"institutions":[{"id":"https://openalex.org/I212220629","display_name":"Peter the Great St. Petersburg Polytechnic University","ror":"https://ror.org/02x91aj62","country_code":"RU","type":"education","lineage":["https://openalex.org/I212220629"]}],"countries":["RU"],"is_corresponding":false,"raw_author_name":"T.V. Lazovskaya","raw_affiliation_strings":["Peter the Great Saint Petersburg Polytechnic University (SPbPU)"],"raw_orcid":"https://orcid.org/0000-0002-3324-6213","affiliations":[{"raw_affiliation_string":"Peter the Great Saint Petersburg Polytechnic University (SPbPU)","institution_ids":["https://openalex.org/I212220629"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5023559475","display_name":"Dmitriy A. Tarkhov","orcid":"https://orcid.org/0000-0002-9431-8241","observed_orcids":["https://orcid.org/0000-0002-9431-8241"]},"institutions":[{"id":"https://openalex.org/I212220629","display_name":"Peter the Great St. Petersburg Polytechnic University","ror":"https://ror.org/02x91aj62","country_code":"RU","type":"education","lineage":["https://openalex.org/I212220629"]}],"countries":["RU"],"is_corresponding":false,"raw_author_name":"D.A. Tarkhov","raw_affiliation_strings":["Peter the Great Saint Petersburg Polytechnic University (SPbPU)"],"raw_orcid":"https://orcid.org/0000-0002-9431-8241","affiliations":[{"raw_affiliation_string":"Peter the Great Saint Petersburg Polytechnic University (SPbPU)","institution_ids":["https://openalex.org/I212220629"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.0,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"16","issue":"3","first_page":"169","last_page":"185"},"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/T10320","display_name":"Neural Networks and Applications","score":0.0,"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/T11416","display_name":"Numerical Solvers for Ordinary Differential Equations","score":0.0,"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/physics-informed-neural-networks","display_name":"physics-informed neural networks","score":1.0},{"id":"https://openalex.org/keywords/trapezoidal-method","display_name":"trapezoidal method","score":0.9929999709129333},{"id":"https://openalex.org/keywords/compact-neural-network-architecture","display_name":"compact neural network architecture","score":0.6919999718666077},{"id":"https://openalex.org/keywords/computational-efficiency","display_name":"computational efficiency","score":0.6830000281333923},{"id":"https://openalex.org/keywords/surrogate-modeling","display_name":"surrogate models","score":0.6980000138282776},{"id":"https://openalex.org/keywords/dynamical-systems","display_name":"dynamical systems","score":0.7300000190734863},{"id":"https://openalex.org/keywords/heat-equation","display_name":"heat equation","score":0.7730000019073486},{"id":"https://openalex.org/keywords/wave-equation","display_name":"wave equation","score":0.9990000128746033},{"id":"https://openalex.org/keywords/reaction-diffusion-systems","display_name":"reaction-diffusion systems","score":0.9990000128746033},{"id":"https://openalex.org/keywords/burgers-equation","display_name":"Burgers equation","score":0.9309999942779541},{"id":"https://openalex.org/keywords/differential-equations","display_name":"differential equations","score":0.46700000762939453}],"concepts":[{"id":"https://openalex.org/C79379906","wikidata":"https://www.wikidata.org/wiki/Q3174497","display_name":"Dynamical systems theory","level":2,"score":0.6937000155448914},{"id":"https://openalex.org/C41608201","wikidata":"https://www.wikidata.org/wiki/Q980509","display_name":"Embedding","level":2,"score":0.621399998664856},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.6146000027656555},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.5526999831199646},{"id":"https://openalex.org/C2781067378","wikidata":"https://www.wikidata.org/wiki/Q17027399","display_name":"Interpretability","level":2,"score":0.5076000094413757},{"id":"https://openalex.org/C117251300","wikidata":"https://www.wikidata.org/wiki/Q1849855","display_name":"Parametric statistics","level":2,"score":0.47699999809265137},{"id":"https://openalex.org/C33962884","wikidata":"https://www.wikidata.org/wiki/Q378637","display_name":"Dynamical system (definition)","level":3,"score":0.46790000796318054},{"id":"https://openalex.org/C2779343474","wikidata":"https://www.wikidata.org/wiki/Q3109175","display_name":"Context (archaeology)","level":2,"score":0.46369999647140503},{"id":"https://openalex.org/C12426560","wikidata":"https://www.wikidata.org/wiki/Q189569","display_name":"Basis (linear algebra)","level":2,"score":0.3970000147819519},{"id":"https://openalex.org/C45374587","wikidata":"https://www.wikidata.org/wiki/Q12525525","display_name":"Computation","level":2,"score":0.38760000467300415},{"id":"https://openalex.org/C50897621","wikidata":"https://www.wikidata.org/wiki/Q2665508","display_name":"Hybrid system","level":2,"score":0.37610000371932983},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.3573000133037567},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.3555000126361847},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.3172000050544739},{"id":"https://openalex.org/C48753275","wikidata":"https://www.wikidata.org/wiki/Q11216","display_name":"Numerical analysis","level":2,"score":0.31700000166893005},{"id":"https://openalex.org/C2776359362","wikidata":"https://www.wikidata.org/wiki/Q2145286","display_name":"Representation (politics)","level":3,"score":0.3125999867916107},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.3093999922275543},{"id":"https://openalex.org/C193415008","wikidata":"https://www.wikidata.org/wiki/Q639681","display_name":"Network architecture","level":2,"score":0.3018999993801117},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.2985000014305115},{"id":"https://openalex.org/C24574437","wikidata":"https://www.wikidata.org/wiki/Q7135228","display_name":"Parametric model","level":3,"score":0.2833999991416931},{"id":"https://openalex.org/C177148314","wikidata":"https://www.wikidata.org/wiki/Q170084","display_name":"Generalization","level":2,"score":0.2694000005722046},{"id":"https://openalex.org/C80444323","wikidata":"https://www.wikidata.org/wiki/Q2878974","display_name":"Theoretical computer science","level":1,"score":0.2583000063896179},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.25619998574256897},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.25540000200271606},{"id":"https://openalex.org/C86582703","wikidata":"https://www.wikidata.org/wiki/Q7617824","display_name":"Stochastic neural network","level":4,"score":0.25380000472068787},{"id":"https://openalex.org/C155032097","wikidata":"https://www.wikidata.org/wiki/Q798503","display_name":"Backpropagation","level":3,"score":0.251800000667572},{"id":"https://openalex.org/C184720557","wikidata":"https://www.wikidata.org/wiki/Q7825049","display_name":"Topology (electrical circuits)","level":2,"score":0.250900000333786}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.17759/mda.2026160308","doi":"https://doi.org/10.17759/mda.2026160308","is_oa":true,"landing_page_url":"https://doi.org/10.17759/mda.2026160308","pdf_url":"https://psyjournals.ru/journals/mda/archive/2026_n3/mda_2026_n3_Velikorechanin_Lazovskaya_Tarkhov.pdf","source":{"id":"https://openalex.org/S4210188866","display_name":"Modelling and Data Analysis","issn_l":"2219-3758","issn":["2219-3758","2311-9454"],"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":["russia-white-list-3"],"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"journal"},"license":"cc-by-nc","license_id":"https://openalex.org/licenses/cc-by-nc","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Modelling and Data Analysis","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.17759/mda.2026160308","doi":"https://doi.org/10.17759/mda.2026160308","is_oa":true,"landing_page_url":"https://doi.org/10.17759/mda.2026160308","pdf_url":"https://psyjournals.ru/journals/mda/archive/2026_n3/mda_2026_n3_Velikorechanin_Lazovskaya_Tarkhov.pdf","source":{"id":"https://openalex.org/S4210188866","display_name":"Modelling and Data Analysis","issn_l":"2219-3758","issn":["2219-3758","2311-9454"],"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":["russia-white-list-3"],"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"journal"},"license":"cc-by-nc","license_id":"https://openalex.org/licenses/cc-by-nc","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Modelling and Data Analysis","raw_type":"journal-article"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7214760627.pdf","grobid_xml":"https://content.openalex.org/works/W7214760627.grobid-xml"},"referenced_works_count":17,"referenced_works":["https://openalex.org/W2565178437","https://openalex.org/W2749231211","https://openalex.org/W2892219708","https://openalex.org/W2899283552","https://openalex.org/W2946469657","https://openalex.org/W2948551291","https://openalex.org/W2983440239","https://openalex.org/W3009038870","https://openalex.org/W3163993681","https://openalex.org/W3173233222","https://openalex.org/W3198903284","https://openalex.org/W3200673624","https://openalex.org/W4255754126","https://openalex.org/W4288039037","https://openalex.org/W4313894417","https://openalex.org/W4366830443","https://openalex.org/W7087444019"],"related_works":["https://openalex.org/W7221642851","https://openalex.org/W4390730230","https://openalex.org/W6946397848","https://openalex.org/W7058191303","https://openalex.org/W7106684211","https://openalex.org/W4410493704","https://openalex.org/W4387232549","https://openalex.org/W4415935344","https://openalex.org/W7154478580","https://openalex.org/W4321195233"],"abstract_inverted_index":{"Context":[0],"and":[1,38,48,98,131,165,218],"relevance.":[2],"Physics-Informed":[3],"Neural":[4],"Networks":[5],"(PINNs)":[6],"are":[7],"considered":[8,234],"a":[9,105,173,188,235],"promising":[10],"tool":[11],"for":[12,237],"mathematical":[13,54],"modeling":[14,43,55,88,243],"of":[15,29,56,76,85,138,150,154,206,213,221],"dynamical":[16,57,86,156,215],"systems":[17,58],"described":[18],"by":[19],"differential":[20,151],"equations.":[21],"However,":[22],"classical":[23,66,94,190],"PINN":[24,95,113],"approaches":[25],"require":[26],"repeated":[27],"computation":[28],"high-order":[30],"derivatives,":[31],"which":[32],"leads":[33],"to":[34,53,93],"significant":[35],"computational":[36,83],"costs":[37],"limits":[39],"their":[40],"applicability":[41],"in":[42,241],"tasks.":[44,244],"Objective.":[45],"To":[46],"develop":[47],"experimentally":[49],"validate":[50],"an":[51],"approach":[52,231],"based":[59],"on":[60,147],"embedding":[61],"physics-informed":[62,77,207],"neural":[63,78,101,208],"networks":[64,79,209],"into":[65,73,111,203],"numerical":[67,71,133,201,227],"schemes.":[68],"Hypothesis.":[69],"Embedding":[70],"schemes":[72,134,202],"the":[74,82,112,127,139,158,170,178,204,211,222,225],"architecture":[75,103,114,205,223],"can":[80,232],"improve":[81],"efficiency":[84,212],"system":[87,216],"while":[89,192],"maintaining":[90],"accuracy":[91,186],"comparable":[92,183],"approaches.":[96],"Methods":[97],"materials.":[99],"A":[100,117],"network":[102,120],"integrating":[104],"generalized":[106],"\u03b8-scheme":[107],"(trapezoidal":[108],"method)":[109],"directly":[110],"is":[115],"proposed.":[116],"compact":[118,239],"parametric":[119],"with":[121],"13":[122],"trainable":[123],"parameters":[124],"adaptively":[125],"selects":[126],"balance":[128],"between":[129],"explicit":[130],"implicit":[132],"at":[135],"different":[136],"points":[137],"space\u2013time":[140],"domain.":[141],"The":[142,229],"experimental":[143],"study":[144],"was":[145],"conducted":[146],"four":[148],"types":[149],"equations":[152],"characteristic":[153],"various":[155],"processes:":[157],"heat":[159],"equation,":[160,162,164],"wave":[161],"reaction\u2013diffusion":[163],"Burgers":[166],"equation.":[167],"Results.":[168],"In":[169],"experiments,":[171],"under":[172],"fixed":[174],"training":[175],"time":[176],"budget,":[177],"proposed":[179,230],"13-parameter":[180],"model":[181],"achieves":[182],"or":[184],"higher":[185],"than":[187],"67-parameter":[189],"PINN,":[191],"exhibiting":[193],"significantly":[194],"lower":[195],"variance":[196],"across":[197],"runs.":[198],"Conclusions.":[199],"Integrating":[200],"improves":[210],"constructing":[214],"models":[217,240],"provides":[219],"interpretability":[220],"through":[224],"underlying":[226],"methods.":[228],"be":[233],"basis":[236],"building":[238],"surrogate":[242]},"counts_by_year":[],"updated_date":"2026-10-28T10:33:24.094632","created_date":"2026-09-30T00:00:00"}
