{"id":"https://openalex.org/W7223683160","doi":"https://doi.org/10.3390/math14213896","title":"Physical-Admissibility Projection and Split-Conformal Calibration for Burgers Neural Operators","display_name":"Physical-Admissibility Projection and Split-Conformal Calibration for Burgers Neural Operators","publication_year":2026,"publication_date":"2026-10-26","ids":{"openalex":"https://openalex.org/W7223683160","doi":"https://doi.org/10.3390/math14213896"},"language":"en","primary_location":{"id":"doi:10.3390/math14213896","doi":"https://doi.org/10.3390/math14213896","is_oa":true,"landing_page_url":"https://doi.org/10.3390/math14213896","pdf_url":null,"source":{"id":"https://openalex.org/S4210192031","display_name":"Mathematics","issn_l":"2227-7390","issn":["2227-7390"],"is_oa":true,"is_in_doaj":true,"is_core":true,"listed_in":["anvur-scientific","cwts-core","doaj","poland-20","russia-white-list-1","vabb-shw"],"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics","raw_type":"journal-article"},"volume":null,"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://doi.org/10.3390/math14213896","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5061898095","display_name":"Ibrahim S. Elshazly","orcid":"https://orcid.org/0000-0001-6364-3327","observed_orcids":["https://orcid.org/0000-0001-6364-3327"]},"institutions":[{"id":"https://openalex.org/I28022161","display_name":"King Saud University","ror":"https://ror.org/02f81g417","country_code":"SA","type":"education","lineage":["https://openalex.org/I28022161"]}],"countries":["SA"],"is_corresponding":true,"raw_author_name":"Ibrahim S. Elshazly","raw_affiliation_strings":["Department of Basic Sciences, Common First Year Deanship, King Saud University, Riyadh 12373, Saudi Arabia"],"raw_orcid":"https://orcid.org/0000-0001-6364-3327","affiliations":[{"raw_affiliation_string":"Department of Basic Sciences, Common First Year Deanship, King Saud University, Riyadh 12373, Saudi Arabia","institution_ids":["https://openalex.org/I28022161"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5044630573","display_name":"Khaled Helmi Khashan","orcid":"https://orcid.org/0009-0009-8105-6367","observed_orcids":["https://orcid.org/0009-0009-8105-6367"]},"institutions":[{"id":"https://openalex.org/I28022161","display_name":"King Saud University","ror":"https://ror.org/02f81g417","country_code":"SA","type":"education","lineage":["https://openalex.org/I28022161"]}],"countries":["SA"],"is_corresponding":false,"raw_author_name":"Khaled H. Khashan","raw_affiliation_strings":["Department of Basic Sciences, Common First Year Deanship, King Saud University, Riyadh 12373, Saudi Arabia"],"raw_orcid":"https://orcid.org/0009-0009-8105-6367","affiliations":[{"raw_affiliation_string":"Department of Basic Sciences, Common First Year Deanship, King Saud University, Riyadh 12373, Saudi Arabia","institution_ids":["https://openalex.org/I28022161"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5114711345","display_name":"Amr M. Y. Abdelaty","orcid":"https://orcid.org/0009-0008-6151-5979","observed_orcids":["https://orcid.org/0009-0008-6151-5979"]},"institutions":[{"id":"https://openalex.org/I28022161","display_name":"King Saud University","ror":"https://ror.org/02f81g417","country_code":"SA","type":"education","lineage":["https://openalex.org/I28022161"]}],"countries":["SA"],"is_corresponding":true,"raw_author_name":"Amr M. Y. Abdelaty","raw_affiliation_strings":["Department of Basic Sciences, Common First Year Deanship, King Saud University, Riyadh 12373, Saudi Arabia"],"raw_orcid":"https://orcid.org/0009-0008-6151-5979","affiliations":[{"raw_affiliation_string":"Department of Basic Sciences, Common First Year Deanship, King Saud University, Riyadh 12373, Saudi Arabia","institution_ids":["https://openalex.org/I28022161"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5061898095","https://openalex.org/A5114711345"],"corresponding_institution_ids":["https://openalex.org/I28022161"],"apc_list":{"value":2600,"currency":"CHF","value_usd":3137},"apc_paid":{"value":2600,"currency":"CHF","value_usd":3137},"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.80059476,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"14","issue":"21","first_page":"3896","last_page":"3896"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11206","display_name":"Model Reduction and Neural Networks","score":0.9994999766349792,"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":0.9994999766349792,"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":9.999999747378752e-05,"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/T12814","display_name":"Gaussian Processes and Bayesian Inference","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/physical-admissibility","display_name":"physical admissibility","score":0.8920000195503235},{"id":"https://openalex.org/keywords/neural-operators","display_name":"neural operators","score":0.718999981880188},{"id":"https://openalex.org/keywords/uncertainty-quantification","display_name":"uncertainty quantification","score":0.6359999775886536},{"id":"https://openalex.org/keywords/cole-hopf-transform","display_name":"Cole-Hopf transform","score":0.8180000185966492},{"id":"https://openalex.org/keywords/distribution-shift","display_name":"distribution shift","score":0.8190000057220459}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7425000071525574},{"id":"https://openalex.org/C57493831","wikidata":"https://www.wikidata.org/wiki/Q3134666","display_name":"Projection (relational algebra)","level":2,"score":0.6802999973297119},{"id":"https://openalex.org/C155512373","wikidata":"https://www.wikidata.org/wiki/Q287450","display_name":"Residual","level":2,"score":0.6363999843597412},{"id":"https://openalex.org/C165838908","wikidata":"https://www.wikidata.org/wiki/Q736777","display_name":"Calibration","level":2,"score":0.6057999730110168},{"id":"https://openalex.org/C2776036281","wikidata":"https://www.wikidata.org/wiki/Q48769818","display_name":"Constraint (computer-aided design)","level":2,"score":0.5529999732971191},{"id":"https://openalex.org/C129782007","wikidata":"https://www.wikidata.org/wiki/Q162886","display_name":"Euclidean geometry","level":2,"score":0.4693000018596649},{"id":"https://openalex.org/C110121322","wikidata":"https://www.wikidata.org/wiki/Q865811","display_name":"Distribution (mathematics)","level":2,"score":0.39480000734329224},{"id":"https://openalex.org/C120174047","wikidata":"https://www.wikidata.org/wiki/Q847073","display_name":"Euclidean distance","level":2,"score":0.3765999972820282},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.3619999885559082},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.31940001249313354},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.2856000065803528},{"id":"https://openalex.org/C122383733","wikidata":"https://www.wikidata.org/wiki/Q865920","display_name":"Approximation error","level":2,"score":0.2809999883174896},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.28049999475479126},{"id":"https://openalex.org/C200941418","wikidata":"https://www.wikidata.org/wiki/Q273508","display_name":"Inflation (cosmology)","level":2,"score":0.27480000257492065},{"id":"https://openalex.org/C126780896","wikidata":"https://www.wikidata.org/wiki/Q899871","display_name":"Distortion (music)","level":4,"score":0.26910001039505005},{"id":"https://openalex.org/C2776502983","wikidata":"https://www.wikidata.org/wiki/Q690182","display_name":"Contrast (vision)","level":2,"score":0.2678999900817871},{"id":"https://openalex.org/C186450821","wikidata":"https://www.wikidata.org/wiki/Q17295","display_name":"Euclidean space","level":2,"score":0.26339998841285706},{"id":"https://openalex.org/C147764199","wikidata":"https://www.wikidata.org/wiki/Q6865248","display_name":"Minification","level":2,"score":0.2596000134944916},{"id":"https://openalex.org/C208081375","wikidata":"https://www.wikidata.org/wiki/Q274502","display_name":"Degrees of freedom (physics and chemistry)","level":2,"score":0.25920000672340393},{"id":"https://openalex.org/C202426404","wikidata":"https://www.wikidata.org/wiki/Q5318686","display_name":"Dykstra's projection algorithm","level":2,"score":0.25529998540878296},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.2526000142097473}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.3390/math14213896","doi":"https://doi.org/10.3390/math14213896","is_oa":true,"landing_page_url":"https://doi.org/10.3390/math14213896","pdf_url":null,"source":{"id":"https://openalex.org/S4210192031","display_name":"Mathematics","issn_l":"2227-7390","issn":["2227-7390"],"is_oa":true,"is_in_doaj":true,"is_core":true,"listed_in":["anvur-scientific","cwts-core","doaj","poland-20","russia-white-list-1","vabb-shw"],"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.3390/math14213896","doi":"https://doi.org/10.3390/math14213896","is_oa":true,"landing_page_url":"https://doi.org/10.3390/math14213896","pdf_url":null,"source":{"id":"https://openalex.org/S4210192031","display_name":"Mathematics","issn_l":"2227-7390","issn":["2227-7390"],"is_oa":true,"is_in_doaj":true,"is_core":true,"listed_in":["anvur-scientific","cwts-core","doaj","poland-20","russia-white-list-1","vabb-shw"],"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics","raw_type":"journal-article"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320321145","display_name":"King Saud University","ror":"https://ror.org/02f81g417"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":6,"referenced_works":["https://openalex.org/W2045384647","https://openalex.org/W2565432921","https://openalex.org/W2964060211","https://openalex.org/W2979786244","https://openalex.org/W4321648921","https://openalex.org/W4404626323"],"related_works":["https://openalex.org/W7213149935","https://openalex.org/W7164198731","https://openalex.org/W7204749452","https://openalex.org/W7220918689","https://openalex.org/W4416596107","https://openalex.org/W7164012527","https://openalex.org/W4416038180","https://openalex.org/W7214007541","https://openalex.org/W4404305005","https://openalex.org/W4416254969"],"abstract_inverted_index":{"Accurate":[0],"PDE":[1],"prediction":[2],"does":[3],"not":[4],"ensure":[5],"physical":[6],"admissibility":[7,178],"or":[8],"calibrated":[9],"uncertainty.":[10],"For":[11,43],"periodic":[12],"viscous":[13],"Burgers":[14],"dynamics,":[15],"we":[16],"combine":[17],"an":[18,56,79],"input-dependent":[19],"Euclidean":[20,53],"projection,":[21],"determined":[22],"by":[23],"the":[24,46,61,69,99,108,182],"initial":[25],"condition,":[26],"onto":[27],"discrete":[28,185],"mean":[29,102,126,152],"and":[30,50,88,94,113,124,130,165,189],"centered-energy":[31],"constraints":[32],"with":[33,78,84,158,179],"split-conformal":[34],"bands":[35,62,150],"based":[36],"on":[37],"one":[38],"maximum-coordinate":[39],"residual":[40],"per":[41],"function.":[42],"fixed":[44],"input,":[45],"projection":[47,119],"is":[48],"nonexpansive":[49],"cannot":[51],"increase":[52],"error":[54,127],"to":[55,134,146,156,172,181],"admissible":[57],"target.":[58],"Under":[59,161],"exchangeability,":[60],"provide":[63],"finite-sample":[64],"marginal":[65],"coverage":[66,142,159,191],"simultaneously":[67],"over":[68,115],"64-point":[70],"terminal":[71],"grid.":[72],"The":[73,175],"nonzero-mean":[74],"Cole\u2013Hopf":[75],"reference":[76],"agrees":[77],"independent":[80],"dealiased":[81],"Fourier\u2013RK4":[82],"solver,":[83],"maximum":[85],"relative":[86,103],"L2":[87,104],"scaled":[89],"L\u221e":[90],"discrepancies":[91],"of":[92],"3.62\u00d710\u22128":[93],"5.83\u00d710\u22128.":[95],"Projected":[96],"FNO":[97,149],"gives":[98],"lowest":[100],"ID":[101,140],"error,":[105],"0.0190,":[106],"among":[107],"four":[109],"implementations":[110],"at":[111],"N=64":[112],"T=0.5":[114],"five":[116],"initializations.":[117],"Full":[118],"eliminates":[120],"measured":[121],"constraint":[122],"violations":[123],"lowers":[125],"across":[128],"models":[129],"regimes":[131],"from":[132,144,154],"0.1984":[133],"0.1904.":[135],"At":[136],"nominal":[137],"90%,":[138],"constant-width":[139],"simultaneous":[141],"ranges":[143],"0.904":[145],"0.930;":[147],"normalized":[148],"reduce":[151],"width":[153,168],"0.108":[155],"0.085":[157],"0.901.":[160],"distribution":[162],"shift,":[163],"undercoverage":[164],"occasional":[166],"conservative":[167],"inflation":[169],"restrict":[170],"conclusions":[171],"empirical":[173],"diagnostics.":[174],"framework":[176],"separates":[177],"respect":[180],"two":[183],"selected":[184],"constraints,":[186],"predictive":[187],"accuracy,":[188],"statistical":[190],"without":[192],"a":[193],"new":[194],"architecture.":[195]},"counts_by_year":[],"updated_date":"2026-10-29T09:56:32.521200","created_date":"2026-10-27T00:00:00"}