{"id":"https://openalex.org/W7223248994","doi":"https://doi.org/10.48550/arxiv.2610.29938","title":"A PDE-theory-guided neural operator and the long-time relaxation of the two-dimensional anisotropic Boussinesq system","display_name":"A PDE-theory-guided neural operator and the long-time relaxation of the two-dimensional anisotropic Boussinesq system","publication_year":2026,"publication_date":"2026-10-21","ids":{"openalex":"https://openalex.org/W7223248994","doi":"https://doi.org/10.48550/arxiv.2610.29938"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.29938","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.29938","pdf_url":"https://arxiv.org/pdf/2610.29938","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/2610.29938","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5143861656","display_name":"Akram Moustafa","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I1629065","display_name":"Central Michigan University","ror":"https://ror.org/02xawj266","country_code":"US","type":"education","lineage":["https://openalex.org/I1629065"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Moustafa, Akram","raw_affiliation_strings":["Department of Computer Science , Central Michigan University , Mount Pleasant , MI 48858"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Computer Science , Central Michigan University , Mount Pleasant , MI 48858","institution_ids":["https://openalex.org/I1629065"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5100368783","display_name":"Jiahong Wu","orcid":"https://orcid.org/0000-0001-9496-9709","observed_orcids":["https://orcid.org/0000-0001-9496-9709"]},"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":"Wu, Jiahong","raw_affiliation_strings":["Department of Mathematics , University of Notre Dame , Notre Dame , IN 46556"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics , University of Notre Dame , Notre Dame , IN 46556","institution_ids":["https://openalex.org/I107639228"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5159267774","display_name":"Xiaoming Zheng","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I1629065","display_name":"Central Michigan University","ror":"https://ror.org/02xawj266","country_code":"US","type":"education","lineage":["https://openalex.org/I1629065"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Zheng, Xiaoming","raw_affiliation_strings":["Department of Mathematics , Central Michigan University , Mount Pleasant , MI 48858"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics , Central Michigan University , Mount Pleasant , MI 48858","institution_ids":["https://openalex.org/I1629065"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"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":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/T10255","display_name":"Oceanographic and Atmospheric Processes","score":0.0,"subfield":{"id":"https://openalex.org/subfields/1910","display_name":"Oceanography"},"field":{"id":"https://openalex.org/fields/19","display_name":"Earth and Planetary Sciences"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/neural-operators","display_name":"neural operators","score":0.8529999852180481},{"id":"https://openalex.org/keywords/long-time-relaxation","display_name":"long-time relaxation","score":0.9589999914169312},{"id":"https://openalex.org/keywords/stratified-shear-flow","display_name":"stratified shear flow","score":0.8709999918937683},{"id":"https://openalex.org/keywords/vorticity","display_name":"vorticity","score":0.7979999780654907},{"id":"https://openalex.org/keywords/anisotropic-dissipation","display_name":"anisotropic dissipation","score":0.925000011920929},{"id":"https://openalex.org/keywords/two-dimensional-hydrodynamics","display_name":"two-dimensional fluid dynamics","score":0.5759999752044678}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6029999852180481},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.579800009727478},{"id":"https://openalex.org/C17020691","wikidata":"https://www.wikidata.org/wiki/Q139677","display_name":"Operator (biology)","level":5,"score":0.5386000275611877},{"id":"https://openalex.org/C200114574","wikidata":"https://www.wikidata.org/wiki/Q1143513","display_name":"Vorticity","level":3,"score":0.5311999917030334},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.5033000111579895},{"id":"https://openalex.org/C2778770139","wikidata":"https://www.wikidata.org/wiki/Q1966904","display_name":"Solver","level":2,"score":0.4505999982357025},{"id":"https://openalex.org/C177918212","wikidata":"https://www.wikidata.org/wiki/Q803623","display_name":"Perturbation (astronomy)","level":2,"score":0.44369998574256897},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.4011000096797943},{"id":"https://openalex.org/C134853933","wikidata":"https://www.wikidata.org/wiki/Q208641","display_name":"Hydrostatic equilibrium","level":2,"score":0.37529999017715454},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.3702000081539154},{"id":"https://openalex.org/C135402231","wikidata":"https://www.wikidata.org/wiki/Q898440","display_name":"Dissipation","level":2,"score":0.36160001158714294},{"id":"https://openalex.org/C85725439","wikidata":"https://www.wikidata.org/wiki/Q466686","display_name":"Anisotropy","level":2,"score":0.3587999939918518},{"id":"https://openalex.org/C57691317","wikidata":"https://www.wikidata.org/wiki/Q1289248","display_name":"Scalar (mathematics)","level":2,"score":0.33660000562667847},{"id":"https://openalex.org/C26955809","wikidata":"https://www.wikidata.org/wiki/Q10167591","display_name":"Initial value problem","level":2,"score":0.33640000224113464},{"id":"https://openalex.org/C180205008","wikidata":"https://www.wikidata.org/wiki/Q159190","display_name":"Amplitude","level":2,"score":0.33250001072883606},{"id":"https://openalex.org/C202615002","wikidata":"https://www.wikidata.org/wiki/Q783507","display_name":"Differentiable function","level":2,"score":0.3125999867916107},{"id":"https://openalex.org/C2776029896","wikidata":"https://www.wikidata.org/wiki/Q3935810","display_name":"Relaxation (psychology)","level":2,"score":0.3124000132083893},{"id":"https://openalex.org/C2778439541","wikidata":"https://www.wikidata.org/wiki/Q7106412","display_name":"Oscillation (cell signaling)","level":2,"score":0.30379998683929443},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.2840000092983246},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.2822999954223633},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.27639999985694885},{"id":"https://openalex.org/C178009071","wikidata":"https://www.wikidata.org/wiki/Q93344","display_name":"Trigonometric functions","level":2,"score":0.25859999656677246},{"id":"https://openalex.org/C84655787","wikidata":"https://www.wikidata.org/wiki/Q8067817","display_name":"Compressibility","level":2,"score":0.2558000087738037},{"id":"https://openalex.org/C38349280","wikidata":"https://www.wikidata.org/wiki/Q1434290","display_name":"Flow (mathematics)","level":2,"score":0.2542000114917755},{"id":"https://openalex.org/C190470478","wikidata":"https://www.wikidata.org/wiki/Q2370229","display_name":"Invariant (physics)","level":2,"score":0.2540999948978424},{"id":"https://openalex.org/C124584101","wikidata":"https://www.wikidata.org/wiki/Q1053266","display_name":"Multiplier (economics)","level":2,"score":0.2506999969482422},{"id":"https://openalex.org/C2776378722","wikidata":"https://www.wikidata.org/wiki/Q3454417","display_name":"Realizability","level":2,"score":0.25029999017715454}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.29938","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.29938","pdf_url":"https://arxiv.org/pdf/2610.29938","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.2610.29938","doi":"https://doi.org/10.48550/arxiv.2610.29938","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.29938","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","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2610.29938","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.29938","pdf_url":"https://arxiv.org/pdf/2610.29938","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/G3339921332","display_name":"Collaborative Research: Effective Numerical Schemes for Fundamental Problems Related to Incompressible Fluids","funder_award_id":"2309748","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G4649135849","display_name":"Collaborative Research: Effective Numerical Schemes for Fundamental Problems Related to Incompressible Fluids","funder_award_id":"2309747","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G4765031779","display_name":"Stabilizing Phenomenon for Incompressible Fluids","funder_award_id":"2104682","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"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7223248994.pdf","grobid_xml":"https://content.openalex.org/works/W7223248994.grobid-xml"},"referenced_works_count":9,"referenced_works":["https://openalex.org/W3014645608","https://openalex.org/W3089082087","https://openalex.org/W3159087135","https://openalex.org/W4225906496","https://openalex.org/W4387647828","https://openalex.org/W4409480452","https://openalex.org/W4416255847","https://openalex.org/W7117572898","https://openalex.org/W7166800084"],"related_works":["https://openalex.org/W2038081758","https://openalex.org/W7161091402","https://openalex.org/W4280623356","https://openalex.org/W4407296523","https://openalex.org/W3013825629","https://openalex.org/W4200006822","https://openalex.org/W4409461504","https://openalex.org/W4410475238","https://openalex.org/W4312073962","https://openalex.org/W7102391912"],"abstract_inverted_index":{"We":[0,45],"design":[1],"a":[2,19,29,42,52,125,138],"neural":[3],"operator":[4,27],"for":[5,179,197],"the":[6,14,39,48,57,70,78,101,109,131,135,151,154,161,172,174,186,190,194,198,201],"two-dimensional":[7],"Boussinesq":[8],"system":[9],"with":[10,56,148,167],"anisotropic":[11],"dissipation":[12],"on":[13,120],"periodic":[15],"box,":[16],"guided":[17],"by":[18,118,134,171,203],"rigorous":[20],"analysis":[21],"of":[22,32,35,64,74,127,141,163,200],"its":[23,65,89],"long-time":[24],"behavior.":[25],"The":[26],"maps":[28],"small":[30],"perturbation":[31],"hydrostatic":[33],"equilibrium,":[34],"amplitude":[36],"$\\varepsilon$,":[37,96],"to":[38,51,88,137,150,159],"vorticity":[40,60],"at":[41],"late":[43],"time.":[44],"prove":[46],"that":[47,204],"flow":[49],"relaxes":[50],"stratified":[53],"shear":[54],"flow,":[55],"horizontally":[58],"averaged":[59],"staying":[61],"within":[62],"$O(\\varepsilon^2)$":[63],"initial":[66,152,164,177],"value":[67,188],"$\\bar\u03c9_0$":[68,81],"and":[69,95,146,166],"oscillation":[71],"about":[72],"it":[73],"order":[75],"$\\varepsilon$.":[76],"Accordingly,":[77],"network":[79,115,155],"outputs":[80],"plus":[82],"these":[83],"two":[84],"parts,":[85],"each":[86],"scaled":[87],"size,":[90],"which":[91,180],"grows":[92],"as":[93],"$\\varepsilon^2$":[94,110],"respectively.":[97],"Without":[98],"this":[99,181],"guidance":[100],"training":[102],"either":[103],"does":[104],"not":[105],"converge":[106],"or":[107],"misses":[108],"growth.":[111],"With":[112],"it,":[113],"one":[114],"reproduces,":[116],"sample":[117,119],"$2\\times10^3$":[121],"test":[122,191],"runs":[123],"over":[124],"factor":[126],"$23$":[128],"in":[129],"amplitude,":[130],"quantity":[132,182],"bounded":[133],"theorem":[136,202],"median":[139],"error":[140],"$2.2$":[142],"percent.":[143],"Being":[144],"fast":[145],"differentiable":[147],"respect":[149],"data,":[153,165],"is":[156,183],"also":[157],"used":[158],"search":[160,175],"space":[162],"every":[168],"candidate":[169],"checked":[170],"solver":[173],"finds":[176],"data":[178],"$4.4$":[184],"times":[185],"largest":[187],"among":[189],"runs,":[192],"raising":[193],"lower":[195],"bound":[196],"constant":[199],"factor.":[205]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-10-23T00:00:00"}