{"id":"https://openalex.org/W7155106655","doi":"https://doi.org/10.48550/arxiv.2604.17156","title":"Uncertainty Quantification in PINNs for Turbulent Flows: Bayesian Inference and Repulsive Ensembles","display_name":"Uncertainty Quantification in PINNs for Turbulent Flows: Bayesian Inference and Repulsive Ensembles","publication_year":2026,"publication_date":"2026-04-18","ids":{"openalex":"https://openalex.org/W7155106655","doi":"https://doi.org/10.48550/arxiv.2604.17156"},"language":null,"primary_location":{"id":"doi:10.48550/arxiv.2604.17156","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2604.17156","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,"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":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},"type":"preprint","indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.48550/arxiv.2604.17156","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5134128594","display_name":"Khemraj Shukla","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Shukla, Khemraj","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5021341531","display_name":"Zongren Zou","orcid":"https://orcid.org/0009-0000-0095-3539"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Zou, Zongren","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5134142587","display_name":"Theo Kaeufer","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Kaeufer, Theo","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5005867844","display_name":"Michael Triantafyllou","orcid":"https://orcid.org/0000-0001-5206-3108"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Triantafyllou, Michael","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5134157435","display_name":"George Em Karniadakis","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Karniadakis, George Em","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":false,"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":0.992900013923645,"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":0.992900013923645,"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/T10928","display_name":"Probabilistic and Robust Engineering Design","score":0.0024999999441206455,"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"}},{"id":"https://openalex.org/T11948","display_name":"Machine Learning in Materials Science","score":0.0008999999845400453,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/uncertainty-quantification","display_name":"Uncertainty quantification","score":0.8044999837875366},{"id":"https://openalex.org/keywords/monte-carlo-method","display_name":"Monte Carlo method","score":0.5781999826431274},{"id":"https://openalex.org/keywords/inverse-problem","display_name":"Inverse problem","score":0.5171999931335449},{"id":"https://openalex.org/keywords/bayesian-inference","display_name":"Bayesian inference","score":0.49140000343322754},{"id":"https://openalex.org/keywords/probabilistic-logic","display_name":"Probabilistic logic","score":0.46950000524520874},{"id":"https://openalex.org/keywords/bayesian-probability","display_name":"Bayesian probability","score":0.46540001034736633},{"id":"https://openalex.org/keywords/calibration","display_name":"Calibration","score":0.4447999894618988},{"id":"https://openalex.org/keywords/turbulence","display_name":"Turbulence","score":0.41519999504089355},{"id":"https://openalex.org/keywords/inference","display_name":"Inference","score":0.3873000144958496}],"concepts":[{"id":"https://openalex.org/C32230216","wikidata":"https://www.wikidata.org/wiki/Q7882499","display_name":"Uncertainty quantification","level":2,"score":0.8044999837875366},{"id":"https://openalex.org/C19499675","wikidata":"https://www.wikidata.org/wiki/Q232207","display_name":"Monte Carlo method","level":2,"score":0.5781999826431274},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.5315999984741211},{"id":"https://openalex.org/C135252773","wikidata":"https://www.wikidata.org/wiki/Q1567213","display_name":"Inverse problem","level":2,"score":0.5171999931335449},{"id":"https://openalex.org/C160234255","wikidata":"https://www.wikidata.org/wiki/Q812535","display_name":"Bayesian inference","level":3,"score":0.49140000343322754},{"id":"https://openalex.org/C49937458","wikidata":"https://www.wikidata.org/wiki/Q2599292","display_name":"Probabilistic logic","level":2,"score":0.46950000524520874},{"id":"https://openalex.org/C107673813","wikidata":"https://www.wikidata.org/wiki/Q812534","display_name":"Bayesian probability","level":2,"score":0.46540001034736633},{"id":"https://openalex.org/C165838908","wikidata":"https://www.wikidata.org/wiki/Q736777","display_name":"Calibration","level":2,"score":0.4447999894618988},{"id":"https://openalex.org/C196558001","wikidata":"https://www.wikidata.org/wiki/Q190132","display_name":"Turbulence","level":2,"score":0.41519999504089355},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.4113999903202057},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.3873000144958496},{"id":"https://openalex.org/C2776214188","wikidata":"https://www.wikidata.org/wiki/Q408386","display_name":"Inference","level":2,"score":0.3873000144958496},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.37450000643730164},{"id":"https://openalex.org/C13153151","wikidata":"https://www.wikidata.org/wiki/Q1639846","display_name":"Hybrid Monte Carlo","level":4,"score":0.3582000136375427},{"id":"https://openalex.org/C137209882","wikidata":"https://www.wikidata.org/wiki/Q1403517","display_name":"Measurement uncertainty","level":2,"score":0.3562000095844269},{"id":"https://openalex.org/C38349280","wikidata":"https://www.wikidata.org/wiki/Q1434290","display_name":"Flow (mathematics)","level":2,"score":0.34850001335144043},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.34630000591278076},{"id":"https://openalex.org/C176147448","wikidata":"https://www.wikidata.org/wiki/Q1889114","display_name":"Sensitivity analysis","level":3,"score":0.33469998836517334},{"id":"https://openalex.org/C177803969","wikidata":"https://www.wikidata.org/wiki/Q29205","display_name":"Uncertainty analysis","level":2,"score":0.32839998602867126},{"id":"https://openalex.org/C149441793","wikidata":"https://www.wikidata.org/wiki/Q200726","display_name":"Probability distribution","level":2,"score":0.32339999079704285},{"id":"https://openalex.org/C131675550","wikidata":"https://www.wikidata.org/wiki/Q7646884","display_name":"Surrogate model","level":2,"score":0.32190001010894775},{"id":"https://openalex.org/C207467116","wikidata":"https://www.wikidata.org/wiki/Q4385666","display_name":"Inverse","level":2,"score":0.3179999887943268},{"id":"https://openalex.org/C140779682","wikidata":"https://www.wikidata.org/wiki/Q210868","display_name":"Sampling (signal processing)","level":3,"score":0.31119999289512634},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.2896000146865845},{"id":"https://openalex.org/C134261354","wikidata":"https://www.wikidata.org/wiki/Q938438","display_name":"Statistical inference","level":2,"score":0.287200003862381},{"id":"https://openalex.org/C52740198","wikidata":"https://www.wikidata.org/wiki/Q1539564","display_name":"Importance sampling","level":3,"score":0.27869999408721924},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.2639999985694885},{"id":"https://openalex.org/C207201462","wikidata":"https://www.wikidata.org/wiki/Q182505","display_name":"Bayes' theorem","level":3,"score":0.25929999351501465},{"id":"https://openalex.org/C57830394","wikidata":"https://www.wikidata.org/wiki/Q278079","display_name":"Posterior probability","level":3,"score":0.25769999623298645},{"id":"https://openalex.org/C148043351","wikidata":"https://www.wikidata.org/wiki/Q4456944","display_name":"Current (fluid)","level":2,"score":0.2563999891281128},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.25200000405311584}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.48550/arxiv.2604.17156","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2604.17156","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,"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":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"doi:10.48550/arxiv.2604.17156","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2604.17156","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,"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":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"Physics-informed":[0],"neural":[1],"networks":[2],"(PINNs)":[3],"have":[4],"emerged":[5],"as":[6,53],"a":[7,67,93,138,153,194],"promising":[8],"framework":[9,82],"for":[10,49,74,129,201,227],"solving":[11],"inverse":[12,131],"problems":[13,51],"governed":[14],"by":[15],"partial":[16],"differential":[17],"equations":[18],"(PDEs),":[19],"including":[20,143],"the":[21,117,144,180,211],"reconstruction":[22],"of":[23,43,69,72,119,140],"turbulent":[24,150],"flow":[25,151,203],"fields":[26],"from":[27],"sparse":[28],"data.":[29],"However,":[30],"most":[31,181],"existing":[32],"PINN":[33],"formulations":[34],"are":[35,135],"deterministic":[36],"and":[37,64,92,101,122,149,164,217,223],"do":[38],"not":[39],"provide":[40,179,207],"reliable":[41],"quantification":[42,76,229],"epistemic":[44],"uncertainty,":[45],"which":[46],"is":[47,114],"critical":[48],"ill-posed":[50],"such":[52],"data-driven":[54,231],"Reynolds-averaged":[55],"Navier-Stokes":[56],"(RANS)":[57],"modeling.":[58,79,233],"In":[59],"this":[60],"work,":[61],"we":[62],"develop":[63],"systematically":[65],"evaluate":[66],"set":[68],"probabilistic":[70],"extensions":[71],"PINNs":[73,86,178],"uncertainty":[75,127,183,218,228],"in":[77,109,125,220,230],"turbulence":[78,232],"The":[80,133,173],"proposed":[81],"combines":[83],"(i)":[84],"Bayesian":[85,177],"with":[87,198],"Hamiltonian":[88],"Monte":[89,98],"Carlo":[90,99],"sampling":[91],"tempered":[94],"multi-component":[95],"likelihood,":[96],"(ii)":[97],"dropout,":[100],"(iii)":[102],"repulsive":[103,191],"deep":[104],"ensembles":[105,192],"that":[106,176],"enforce":[107],"diversity":[108,121],"function":[110],"space.":[111],"Particular":[112],"emphasis":[113],"placed":[115],"on":[116,137],"role":[118],"ensemble":[120],"likelihood":[123],"tempering":[124],"improving":[126],"calibration":[128,219],"PDE-constrained":[130],"problems.":[132],"methods":[134],"assessed":[136],"hierarchy":[139],"test":[141],"cases,":[142],"Van":[145],"der":[146],"Pol":[147],"oscillator":[148],"past":[152],"circular":[154],"cylinder":[155],"at":[156],"Reynolds":[157],"numbers":[158],"Re=3,900":[159],"(direct":[160],"numerical":[161],"simulation":[162],"data)":[163],"Re":[165],"=":[166],"10,000":[167],"(experimental":[168],"particle":[169],"image":[170],"velocimetry":[171],"data).":[172],"results":[174],"demonstrate":[175],"consistent":[182],"estimates":[184],"across":[185],"all":[186],"inferred":[187],"quantities,":[188],"while":[189],"function-space":[190],"offer":[193,224],"computationally":[195],"efficient":[196],"approximation":[197],"competitive":[199],"accuracy":[200],"primary":[202],"variables.":[204],"These":[205],"findings":[206],"quantitative":[208],"insight":[209],"into":[210],"trade-offs":[212],"between":[213],"accuracy,":[214],"computational":[215],"cost,":[216],"physics-informed":[221],"learning,":[222],"practical":[225],"guidance":[226]},"counts_by_year":[],"updated_date":"2026-07-28T07:46:37.118299","created_date":"2026-04-22T00:00:00"}
