{"id":"https://openalex.org/W7163553579","doi":"https://doi.org/10.48550/arxiv.2606.04804","title":"The Right Measure for Physics-Constrained Generation: A Co-Area Correction for Posterior-Consistent PDE Inverse Problems","display_name":"The Right Measure for Physics-Constrained Generation: A Co-Area Correction for Posterior-Consistent PDE Inverse Problems","publication_year":2026,"publication_date":"2026-06-03","ids":{"openalex":"https://openalex.org/W7163553579","doi":"https://doi.org/10.48550/arxiv.2606.04804"},"language":null,"primary_location":{"id":"doi:10.48550/arxiv.2606.04804","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2606.04804","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","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.2606.04804","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5137811770","display_name":"Jian Xu","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Xu, Jian","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5137811354","display_name":"Delu Zeng","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Wu, Yanning","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5137874785","display_name":"John Paisley","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Zeng, Delu","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5137870201","display_name":"Qibin Zhao","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Paisley, John","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":null,"display_name":"Zhao, Qibin","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Zhao, Qibin","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/T12056","display_name":"Markov Chains and Monte Carlo Methods","score":0.2554999887943268,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T12056","display_name":"Markov Chains and Monte Carlo Methods","score":0.2554999887943268,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10775","display_name":"Generative Adversarial Networks and Image Synthesis","score":0.2296999990940094,"subfield":{"id":"https://openalex.org/subfields/1707","display_name":"Computer Vision and Pattern Recognition"},"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/T11206","display_name":"Model Reduction and Neural Networks","score":0.18520000576972961,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/constraint","display_name":"Constraint (computer-aided design)","score":0.617900013923645},{"id":"https://openalex.org/keywords/inverse-problem","display_name":"Inverse problem","score":0.582099974155426},{"id":"https://openalex.org/keywords/measure","display_name":"Measure (data warehouse)","score":0.5796999931335449},{"id":"https://openalex.org/keywords/bayesian-probability","display_name":"Bayesian probability","score":0.5763999819755554},{"id":"https://openalex.org/keywords/projection","display_name":"Projection (relational algebra)","score":0.5457000136375427},{"id":"https://openalex.org/keywords/jacobian-matrix-and-determinant","display_name":"Jacobian matrix and determinant","score":0.49149999022483826},{"id":"https://openalex.org/keywords/matching","display_name":"Matching (statistics)","score":0.4745999872684479},{"id":"https://openalex.org/keywords/inverse","display_name":"Inverse","score":0.4447999894618988},{"id":"https://openalex.org/keywords/scalar","display_name":"Scalar (mathematics)","score":0.4392000138759613},{"id":"https://openalex.org/keywords/generative-model","display_name":"Generative model","score":0.41620001196861267}],"concepts":[{"id":"https://openalex.org/C2776036281","wikidata":"https://www.wikidata.org/wiki/Q48769818","display_name":"Constraint (computer-aided design)","level":2,"score":0.617900013923645},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6051999926567078},{"id":"https://openalex.org/C135252773","wikidata":"https://www.wikidata.org/wiki/Q1567213","display_name":"Inverse problem","level":2,"score":0.582099974155426},{"id":"https://openalex.org/C2780009758","wikidata":"https://www.wikidata.org/wiki/Q6804172","display_name":"Measure (data warehouse)","level":2,"score":0.5796999931335449},{"id":"https://openalex.org/C107673813","wikidata":"https://www.wikidata.org/wiki/Q812534","display_name":"Bayesian probability","level":2,"score":0.5763999819755554},{"id":"https://openalex.org/C57493831","wikidata":"https://www.wikidata.org/wiki/Q3134666","display_name":"Projection (relational algebra)","level":2,"score":0.5457000136375427},{"id":"https://openalex.org/C200331156","wikidata":"https://www.wikidata.org/wiki/Q506041","display_name":"Jacobian matrix and determinant","level":2,"score":0.49149999022483826},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.48820000886917114},{"id":"https://openalex.org/C165064840","wikidata":"https://www.wikidata.org/wiki/Q1321061","display_name":"Matching (statistics)","level":2,"score":0.4745999872684479},{"id":"https://openalex.org/C207467116","wikidata":"https://www.wikidata.org/wiki/Q4385666","display_name":"Inverse","level":2,"score":0.4447999894618988},{"id":"https://openalex.org/C57691317","wikidata":"https://www.wikidata.org/wiki/Q1289248","display_name":"Scalar (mathematics)","level":2,"score":0.4392000138759613},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.42800000309944153},{"id":"https://openalex.org/C167966045","wikidata":"https://www.wikidata.org/wiki/Q5532625","display_name":"Generative model","level":3,"score":0.41620001196861267},{"id":"https://openalex.org/C529865628","wikidata":"https://www.wikidata.org/wiki/Q1790740","display_name":"Manifold (fluid mechanics)","level":2,"score":0.41029998660087585},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.40880000591278076},{"id":"https://openalex.org/C57830394","wikidata":"https://www.wikidata.org/wiki/Q278079","display_name":"Posterior probability","level":3,"score":0.40389999747276306},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.39340001344680786},{"id":"https://openalex.org/C176217482","wikidata":"https://www.wikidata.org/wiki/Q860554","display_name":"Metric (unit)","level":2,"score":0.39149999618530273},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.35350000858306885},{"id":"https://openalex.org/C2777634741","wikidata":"https://www.wikidata.org/wiki/Q768993","display_name":"Wasserstein metric","level":2,"score":0.3400999903678894},{"id":"https://openalex.org/C140779682","wikidata":"https://www.wikidata.org/wiki/Q210868","display_name":"Sampling (signal processing)","level":3,"score":0.33649998903274536},{"id":"https://openalex.org/C52740198","wikidata":"https://www.wikidata.org/wiki/Q1539564","display_name":"Importance sampling","level":3,"score":0.33009999990463257},{"id":"https://openalex.org/C38349280","wikidata":"https://www.wikidata.org/wiki/Q1434290","display_name":"Flow (mathematics)","level":2,"score":0.3271999955177307},{"id":"https://openalex.org/C160234255","wikidata":"https://www.wikidata.org/wiki/Q812535","display_name":"Bayesian inference","level":3,"score":0.31779998540878296},{"id":"https://openalex.org/C177769412","wikidata":"https://www.wikidata.org/wiki/Q278090","display_name":"Prior probability","level":3,"score":0.31520000100135803},{"id":"https://openalex.org/C21031990","wikidata":"https://www.wikidata.org/wiki/Q355020","display_name":"Probability measure","level":2,"score":0.30309998989105225},{"id":"https://openalex.org/C51955184","wikidata":"https://www.wikidata.org/wiki/Q1545585","display_name":"Stochastic differential equation","level":2,"score":0.30140000581741333},{"id":"https://openalex.org/C2777686260","wikidata":"https://www.wikidata.org/wiki/Q144037","display_name":"Calculus (dental)","level":2,"score":0.27799999713897705},{"id":"https://openalex.org/C49937458","wikidata":"https://www.wikidata.org/wiki/Q2599292","display_name":"Probabilistic logic","level":2,"score":0.2777999937534332},{"id":"https://openalex.org/C22324862","wikidata":"https://www.wikidata.org/wiki/Q652707","display_name":"Lipschitz continuity","level":2,"score":0.2590000033378601},{"id":"https://openalex.org/C138777275","wikidata":"https://www.wikidata.org/wiki/Q6884054","display_name":"Mixing (physics)","level":2,"score":0.2590000033378601},{"id":"https://openalex.org/C2776135515","wikidata":"https://www.wikidata.org/wiki/Q17143721","display_name":"Regularization (linguistics)","level":2,"score":0.2574000060558319},{"id":"https://openalex.org/C68022304","wikidata":"https://www.wikidata.org/wiki/Q842217","display_name":"Bayes estimator","level":3,"score":0.2535000145435333},{"id":"https://openalex.org/C185429906","wikidata":"https://www.wikidata.org/wiki/Q1130160","display_name":"Estimator","level":2,"score":0.25060001015663147}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.48550/arxiv.2606.04804","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2606.04804","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"doi:10.48550/arxiv.2606.04804","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2606.04804","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","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":{"Generative":[0],"models":[1],"--":[2,7,69],"diffusion":[3],"and":[4,31,79,96,117,158,180,207],"flow":[5],"matching":[6],"are":[8],"increasingly":[9],"used":[10],"to":[11,142,188],"solve":[12],"partial":[13],"differential":[14],"equation":[15],"(PDE)":[16],"inverse":[17],"problems,":[18],"enforcing":[19],"the":[20,33,51,84,103,111,114,139,144,156,176,185,197,201,205],"governing":[21],"physics":[22],"as":[23,36,203],"a":[24,37,55,59,66,88,133,159,170,209],"\\emph{hard":[25],"constraint}":[26],"(via":[27],"projection":[28,148],"or":[29],"guidance)":[30],"reporting":[32],"resulting":[34],"samples":[35,50],"Bayesian":[38],"posterior":[39,140,187],"with":[40,110],"calibrated":[41],"uncertainty.":[42],"We":[43,101,167],"show":[44,106,181],"that":[45,72,94,107,174,182,195],"this":[46],"widely":[47],"adopted":[48],"recipe":[49],"wrong":[52],"distribution.":[53],"Conditioning":[54],"generative":[56],"prior":[57],"on":[58,65,120],"hard":[60],"PDE":[61],"constraint":[62,115],"is":[63,73,131,152,199],"conditioning":[64],"measure-zero":[67],"manifold":[68],"an":[70,124],"operation":[71],"intrinsically":[74],"ambiguous":[75],"(the":[76],"Borel--Kolmogorov":[77],"paradox)":[78],"whose":[80],"physically":[81],"correct":[82,177],"resolution,":[83],"small-residual-noise":[85],"limit,":[86],"carries":[87],"co-area":[89,178],"(Fixman)":[90],"Jacobian":[91],"factor":[92,130],"$[det(JJ^{\\top})]^{-1/2}$":[93],"projection-":[95],"guidance-based":[97],"methods":[98],"silently":[99],"omit.":[100],"make":[102],"bias":[104],"precise,":[105],"it":[108,119,137,183],"grows":[109],"heterogeneity":[112],"of":[113],"sensitivity,":[116],"validate":[118],"controlled":[121],"problems":[122],"against":[123],"\\emph{i.i.d.}":[125],"ground-truth":[126],"arbiter.":[127],"The":[128],"omitted":[129],"not":[132,164,200],"second-order":[134],"detail:":[135],"removing":[136],"inflates":[138],"error":[141],"$20\\times$":[143],"sampling-noise":[145],"floor;":[146,157],"minimal-displacement":[147],"(as":[149],"in":[150],"PCFM)":[151],"biased":[153],"at":[154],"$9\\times$":[155],"naive":[160],"scalar":[161],"reweighting":[162],"does":[163],"fix":[165],"it.":[166],"introduce":[168],"\\textbf{CoCoS},":[169],"measure-aware":[171],"constrained":[172],"sampler":[173],"targets":[175],"posterior,":[179],"matches":[184],"gold-standard":[186],"within":[189],"sampling":[190],"noise.":[191],"Our":[192],"results":[193],"imply":[194],"``satisfying":[196],"physics''":[198],"same":[202],"``sampling":[204],"posterior,''":[206],"give":[208],"principled":[210],"correction":[211],"for":[212],"uncertainty-aware":[213],"scientific":[214],"inference.":[215]},"counts_by_year":[],"updated_date":"2026-07-01T06:00:48.157686","created_date":"2026-06-05T00:00:00"}