{"id":"https://openalex.org/W7220418017","doi":"https://doi.org/10.48550/arxiv.2610.02230","title":"Feature tracking in physics-informed neural networks via joint optimization of nonlinear deformation manifolds: application to shocks","display_name":"Feature tracking in physics-informed neural networks via joint optimization of nonlinear deformation manifolds: application to shocks","publication_year":2026,"publication_date":"2026-09-24","ids":{"openalex":"https://openalex.org/W7220418017","doi":"https://doi.org/10.48550/arxiv.2610.02230"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.02230","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.02230","pdf_url":"https://arxiv.org/pdf/2610.02230","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2610.02230","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5038545797","display_name":"Akshay Thakur","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I107639228","display_name":"University of Notre Dame","ror":"https://ror.org/00mkhxb43","country_code":"US","type":"education","lineage":["https://openalex.org/I107639228"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Thakur, Akshay","raw_affiliation_strings":["Department of Aerospace and Mechanical Engineering , University of Notre Dame , Notre Dame , IN 46556 , United States","Graduate Student , Department of Aerospace and Mechanical Engineering , University of Notre Dame"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Aerospace and Mechanical Engineering , University of Notre Dame , Notre Dame , IN 46556 , United States","institution_ids":["https://openalex.org/I107639228"]},{"raw_affiliation_string":"Graduate Student , Department of Aerospace and Mechanical Engineering , University of Notre Dame","institution_ids":["https://openalex.org/I107639228"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5088125725","display_name":"Matthew J. Zahr","orcid":"https://orcid.org/0000-0003-4066-981X","observed_orcids":["https://orcid.org/0000-0003-4066-981X"]},"institutions":[{"id":"https://openalex.org/I107639228","display_name":"University of Notre Dame","ror":"https://ror.org/00mkhxb43","country_code":"US","type":"education","lineage":["https://openalex.org/I107639228"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Zahr, Matthew","raw_affiliation_strings":["Associate Professor, Department of Aerospace and Mechanical Engineering , University of Notre Dame","Department of Aerospace and Mechanical Engineering , University of Notre Dame , Notre Dame , IN 46556 , United States"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Associate Professor, Department of Aerospace and Mechanical Engineering , University of Notre Dame","institution_ids":["https://openalex.org/I107639228"]},{"raw_affiliation_string":"Department of Aerospace and Mechanical Engineering , University of Notre Dame , Notre Dame , IN 46556 , United States","institution_ids":["https://openalex.org/I107639228"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I107639228"],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"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":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/T10173","display_name":"Computational Fluid Dynamics and Aerodynamics","score":0.0,"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"}},{"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"}}],"keywords":[{"id":"https://openalex.org/keywords/physics-informed-neural-networks","display_name":"physics-informed neural networks","score":0.9879999756813049},{"id":"https://openalex.org/keywords/feature-tracking","display_name":"feature tracking","score":0.8629999756813049},{"id":"https://openalex.org/keywords/shock-capturing","display_name":"shock capturing","score":0.699999988079071},{"id":"https://openalex.org/keywords/conservation-laws","display_name":"conservation laws","score":0.8299999833106995},{"id":"https://openalex.org/keywords/viscous-burgers-equation","display_name":"viscous Burgers equation","score":0.8320000171661377},{"id":"https://openalex.org/keywords/shock-wave-reflection","display_name":"shock wave reflection","score":0.9359999895095825},{"id":"https://openalex.org/keywords/boundary-preservation","display_name":"boundary preservation","score":0.656000018119812}],"concepts":[{"id":"https://openalex.org/C200331156","wikidata":"https://www.wikidata.org/wiki/Q506041","display_name":"Jacobian matrix and determinant","level":2,"score":0.6818000078201294},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.5781000256538391},{"id":"https://openalex.org/C80023036","wikidata":"https://www.wikidata.org/wiki/Q5147531","display_name":"Collocation (remote sensing)","level":2,"score":0.5361999869346619},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.4706000089645386},{"id":"https://openalex.org/C62354387","wikidata":"https://www.wikidata.org/wiki/Q875399","display_name":"Boundary (topology)","level":2,"score":0.46470001339912415},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.4399000108242035},{"id":"https://openalex.org/C2776401178","wikidata":"https://www.wikidata.org/wiki/Q12050496","display_name":"Feature (linguistics)","level":2,"score":0.4377000033855438},{"id":"https://openalex.org/C57493831","wikidata":"https://www.wikidata.org/wiki/Q3134666","display_name":"Projection (relational algebra)","level":2,"score":0.4278999865055084},{"id":"https://openalex.org/C2781300812","wikidata":"https://www.wikidata.org/wiki/Q178061","display_name":"Shock (circulatory)","level":2,"score":0.42579999566078186},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.4228000044822693},{"id":"https://openalex.org/C128107574","wikidata":"https://www.wikidata.org/wiki/Q182003","display_name":"Injective function","level":2,"score":0.4142000079154968},{"id":"https://openalex.org/C195065555","wikidata":"https://www.wikidata.org/wiki/Q214881","display_name":"Curvature","level":2,"score":0.4097000062465668},{"id":"https://openalex.org/C62884695","wikidata":"https://www.wikidata.org/wiki/Q184871","display_name":"Euler's formula","level":2,"score":0.4027999937534332},{"id":"https://openalex.org/C155512373","wikidata":"https://www.wikidata.org/wiki/Q287450","display_name":"Residual","level":2,"score":0.40209999680519104},{"id":"https://openalex.org/C36503486","wikidata":"https://www.wikidata.org/wiki/Q11235244","display_name":"Domain (mathematical analysis)","level":2,"score":0.3968999981880188},{"id":"https://openalex.org/C182310444","wikidata":"https://www.wikidata.org/wiki/Q1332643","display_name":"Boundary value problem","level":2,"score":0.39149999618530273},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.384799987077713},{"id":"https://openalex.org/C165464430","wikidata":"https://www.wikidata.org/wiki/Q1570441","display_name":"Parameterized complexity","level":2,"score":0.37549999356269836},{"id":"https://openalex.org/C38409319","wikidata":"https://www.wikidata.org/wiki/Q375175","display_name":"Euler equations","level":2,"score":0.37130001187324524},{"id":"https://openalex.org/C204366326","wikidata":"https://www.wikidata.org/wiki/Q3027650","display_name":"Deformation (meteorology)","level":2,"score":0.3702999949455261},{"id":"https://openalex.org/C47556283","wikidata":"https://www.wikidata.org/wiki/Q1058314","display_name":"Diffeomorphism","level":2,"score":0.36410000920295715},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.34529998898506165},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.34049999713897705},{"id":"https://openalex.org/C147764199","wikidata":"https://www.wikidata.org/wiki/Q6865248","display_name":"Minification","level":2,"score":0.33660000562667847},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3276999890804291},{"id":"https://openalex.org/C13662910","wikidata":"https://www.wikidata.org/wiki/Q193139","display_name":"Trajectory","level":2,"score":0.32710000872612},{"id":"https://openalex.org/C2776036281","wikidata":"https://www.wikidata.org/wiki/Q48769818","display_name":"Constraint (computer-aided design)","level":2,"score":0.3174999952316284},{"id":"https://openalex.org/C26955809","wikidata":"https://www.wikidata.org/wiki/Q10167591","display_name":"Initial value problem","level":2,"score":0.3041999936103821},{"id":"https://openalex.org/C47446073","wikidata":"https://www.wikidata.org/wiki/Q5165890","display_name":"Control theory (sociology)","level":3,"score":0.30230000615119934},{"id":"https://openalex.org/C39920418","wikidata":"https://www.wikidata.org/wiki/Q11476","display_name":"Kinematics","level":2,"score":0.296999990940094},{"id":"https://openalex.org/C202615002","wikidata":"https://www.wikidata.org/wiki/Q783507","display_name":"Differentiable function","level":2,"score":0.289900004863739},{"id":"https://openalex.org/C129747778","wikidata":"https://www.wikidata.org/wiki/Q1014918","display_name":"Burgers' equation","level":3,"score":0.28040000796318054},{"id":"https://openalex.org/C3445786","wikidata":"https://www.wikidata.org/wiki/Q205805","display_name":"Conservation law","level":2,"score":0.2784000039100647},{"id":"https://openalex.org/C2776545253","wikidata":"https://www.wikidata.org/wiki/Q5464292","display_name":"Folding (DSP implementation)","level":2,"score":0.27790001034736633},{"id":"https://openalex.org/C163681178","wikidata":"https://www.wikidata.org/wiki/Q1149279","display_name":"Neumann boundary condition","level":3,"score":0.2736000120639801},{"id":"https://openalex.org/C2779696439","wikidata":"https://www.wikidata.org/wiki/Q7512811","display_name":"Signature (topology)","level":2,"score":0.26579999923706055},{"id":"https://openalex.org/C2777693668","wikidata":"https://www.wikidata.org/wiki/Q25053743","display_name":"Deblurring","level":5,"score":0.2599000036716461},{"id":"https://openalex.org/C6180225","wikidata":"https://www.wikidata.org/wiki/Q3411771","display_name":"Penalty method","level":2,"score":0.25369998812675476},{"id":"https://openalex.org/C75380026","wikidata":"https://www.wikidata.org/wiki/Q868454","display_name":"Euler method","level":3,"score":0.2524999976158142}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.02230","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.02230","pdf_url":"https://arxiv.org/pdf/2610.02230","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":"doi:10.48550/arxiv.2610.02230","doi":"https://doi.org/10.48550/arxiv.2610.02230","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.02230","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2610.02230","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.02230","pdf_url":"https://arxiv.org/pdf/2610.02230","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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"},"sustainable_development_goals":[],"study_designs":[],"awards":[{"id":"https://openalex.org/G1509273647","display_name":null,"funder_award_id":"FA9550-22-1-0004","funder_id":"https://openalex.org/F4320338279","funder_display_name":"Air Force Office of Scientific Research"},{"id":"https://openalex.org/G2157809900","display_name":null,"funder_award_id":"CBET-2338843","funder_id":"https://openalex.org/F4320337390","funder_display_name":"Division of Chemical, Bioengineering, Environmental, and Transport Systems"},{"id":"https://openalex.org/G307930203","display_name":null,"funder_award_id":"N00014-22-1-2299","funder_id":"https://openalex.org/F4320337345","funder_display_name":"Office of Naval Research"},{"id":"https://openalex.org/G7035038375","display_name":"CAREER: Nonlinear Finite Element Manifolds for Improved Simulation of Shock-Dominated Turbulent Flows","funder_award_id":"2338843","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320337345","display_name":"Office of Naval Research","ror":"https://ror.org/00rk2pe57"},{"id":"https://openalex.org/F4320337390","display_name":"Division of Chemical, Bioengineering, Environmental, and Transport Systems","ror":"https://ror.org/0471zv972"},{"id":"https://openalex.org/F4320338279","display_name":"Air Force Office of Scientific Research","ror":"https://ror.org/011e9bt93"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7220418017.pdf","grobid_xml":"https://content.openalex.org/works/W7220418017.grobid-xml"},"referenced_works_count":3,"referenced_works":["https://openalex.org/W2964121744","https://openalex.org/W4307154444","https://openalex.org/W4385965970"],"related_works":["https://openalex.org/W7213979603","https://openalex.org/W7165886336","https://openalex.org/W4220869134","https://openalex.org/W7148358758","https://openalex.org/W7167923581","https://openalex.org/W4404570774","https://openalex.org/W4312340993","https://openalex.org/W4404415005","https://openalex.org/W4413987962","https://openalex.org/W4229368245"],"abstract_inverted_index":{"Physics-informed":[0],"neural":[1],"networks":[2],"(PINNs)":[3],"often":[4],"converge":[5],"to":[6,106,189],"inaccurate":[7],"solutions":[8],"for":[9],"conservation":[10],"laws":[11],"with":[12,55,142,168,176],"shocks,":[13,95,144],"because":[14],"uniformly":[15],"distributed":[16],"collocation":[17,81,171],"points":[18,82],"undersample":[19],"localized":[20],"features":[21,85],"and":[22,53,67,93,123,154,181],"let":[23],"the":[24,43,75,107,121,132,149,177,185],"residual":[25],"be":[26],"dominated":[27],"by":[28,73,127],"regions":[29],"that":[30],"are":[31,70],"already":[32],"well":[33],"resolved.":[34],"We":[35],"propose":[36],"a":[37,49,56,61,117,128,145,169],"feature-tracking":[38],"PINN":[39,175],"(FT-PINN)":[40],"in":[41],"which":[42],"solution":[44],"network":[45],"is":[46,104,113,125],"defined":[47],"on":[48,131],"fixed":[50],"reference":[51],"domain":[52],"composed":[54],"diffeomorphic":[57],"deformation":[58,66],"map":[59],"from":[60],"parameterized":[62],"nonlinear":[63],"manifold.":[64],"The":[65,102],"solution-network":[68],"parameters":[69],"trained":[71],"jointly":[72],"minimizing":[74],"pulled-back":[76],"conservation-law":[77],"residual.":[78],"This":[79],"lets":[80],"concentrate":[83],"along":[84],"of":[86,99,109,120],"essentially":[87],"arbitrary":[88],"geometry,":[89],"including":[90],"curved,":[91],"oblique,":[92],"merging":[94,143],"without":[96],"prior":[97],"knowledge":[98],"their":[100,165],"locations.":[101],"framework":[103],"agnostic":[105],"choice":[108],"parameterization.":[110],"Boundary":[111],"preservation":[112],"enforced":[114],"exactly":[115],"through":[116],"tangential":[118],"projection":[119],"displacement,":[122],"folding":[124],"discouraged":[126],"one-sided":[129],"penalty":[130],"Jacobian":[133],"determinant.":[134],"On":[135],"four":[136],"test":[137],"problems":[138],"(space-time":[139],"viscous":[140],"Burgers":[141,147],"decelerating":[146],"shock,":[148],"space-time":[150],"Euler":[151,157],"shock":[152,159],"tube,":[153],"steady":[155],"2D":[156],"regular":[158],"reflection),":[160],"FT-PINN":[161],"resolves":[162],"shocks":[163,186],"at":[164],"correct":[166],"locations":[167],"limited":[170],"budget.":[172],"A":[173],"vanilla":[174],"same":[178],"architecture,":[179],"budget,":[180],"training":[182],"either":[183],"misplaces":[184],"or":[187],"fails":[188],"form":[190],"them.":[191]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-10-06T00:00:00"}
