{"id":"https://openalex.org/W7160925364","doi":"https://doi.org/10.48550/arxiv.2605.10277","title":"Generalization Error Bounds for Picard-Type Operator Learning in Nonlinear Parabolic PDEs","display_name":"Generalization Error Bounds for Picard-Type Operator Learning in Nonlinear Parabolic PDEs","publication_year":2026,"publication_date":"2026-05-11","ids":{"openalex":"https://openalex.org/W7160925364","doi":"https://doi.org/10.48550/arxiv.2605.10277"},"language":null,"primary_location":{"id":"doi:10.48550/arxiv.2605.10277","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2605.10277","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.2605.10277","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5100892512","display_name":"Koichi Taniguchi","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Taniguchi, Koichi","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5036537988","display_name":"Sho Sonoda","orcid":"https://orcid.org/0000-0001-7242-4740"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Sonoda, Sho","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":0.8500000238418579,"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":0.8500000238418579,"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/T11110","display_name":"Stability and Controllability of Differential Equations","score":0.04170000180602074,"subfield":{"id":"https://openalex.org/subfields/2207","display_name":"Control and Systems Engineering"},"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/T11236","display_name":"Control Systems and Identification","score":0.01810000091791153,"subfield":{"id":"https://openalex.org/subfields/2207","display_name":"Control and Systems Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.6445000171661377},{"id":"https://openalex.org/keywords/generalization","display_name":"Generalization","score":0.6315000057220459},{"id":"https://openalex.org/keywords/operator","display_name":"Operator (biology)","score":0.597000002861023},{"id":"https://openalex.org/keywords/partial-differential-equation","display_name":"Partial differential equation","score":0.4246000051498413},{"id":"https://openalex.org/keywords/truncation-error","display_name":"Truncation error","score":0.41029998660087585},{"id":"https://openalex.org/keywords/truncation","display_name":"Truncation (statistics)","score":0.35569998621940613},{"id":"https://openalex.org/keywords/operator-theory","display_name":"Operator theory","score":0.32749998569488525},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.3172000050544739}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7644000053405762},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.6445000171661377},{"id":"https://openalex.org/C177148314","wikidata":"https://www.wikidata.org/wiki/Q170084","display_name":"Generalization","level":2,"score":0.6315000057220459},{"id":"https://openalex.org/C17020691","wikidata":"https://www.wikidata.org/wiki/Q139677","display_name":"Operator (biology)","level":5,"score":0.597000002861023},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5566999912261963},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.4246000051498413},{"id":"https://openalex.org/C104942944","wikidata":"https://www.wikidata.org/wiki/Q3434686","display_name":"Truncation error","level":2,"score":0.41029998660087585},{"id":"https://openalex.org/C106195933","wikidata":"https://www.wikidata.org/wiki/Q7847935","display_name":"Truncation (statistics)","level":2,"score":0.35569998621940613},{"id":"https://openalex.org/C43929395","wikidata":"https://www.wikidata.org/wiki/Q1198874","display_name":"Operator theory","level":2,"score":0.32749998569488525},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.3172000050544739},{"id":"https://openalex.org/C202787564","wikidata":"https://www.wikidata.org/wiki/Q6510488","display_name":"Heat equation","level":2,"score":0.31610000133514404},{"id":"https://openalex.org/C61797465","wikidata":"https://www.wikidata.org/wiki/Q1188986","display_name":"Term (time)","level":2,"score":0.30790001153945923},{"id":"https://openalex.org/C202286095","wikidata":"https://www.wikidata.org/wiki/Q579262","display_name":"Error function","level":2,"score":0.3050999939441681},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.3043999969959259},{"id":"https://openalex.org/C102519508","wikidata":"https://www.wikidata.org/wiki/Q6520159","display_name":"Fourier transform","level":2,"score":0.2851000130176544},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.28439998626708984},{"id":"https://openalex.org/C3018824978","wikidata":"https://www.wikidata.org/wiki/Q2894891","display_name":"Error analysis","level":2,"score":0.28029999136924744},{"id":"https://openalex.org/C70915906","wikidata":"https://www.wikidata.org/wiki/Q1058681","display_name":"Differential operator","level":2,"score":0.27730000019073486},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.27570000290870667},{"id":"https://openalex.org/C49766605","wikidata":"https://www.wikidata.org/wiki/Q207643","display_name":"Linear map","level":2,"score":0.2728999853134155},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.26910001039505005},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.26649999618530273},{"id":"https://openalex.org/C207864730","wikidata":"https://www.wikidata.org/wiki/Q179467","display_name":"Fourier series","level":2,"score":0.25600001215934753},{"id":"https://openalex.org/C2777686260","wikidata":"https://www.wikidata.org/wiki/Q144037","display_name":"Calculus (dental)","level":2,"score":0.2556000053882599}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.48550/arxiv.2605.10277","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2605.10277","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.2605.10277","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2605.10277","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":{"Operator":[0],"learning":[1,58,66],"for":[2,25,67,92,170],"partial":[3],"differential":[4],"equations":[5,173],"(PDEs)":[6],"aims":[7],"to":[8,29,35,43,150],"learn":[9],"solution":[10,68],"operators":[11,69],"on":[12,75,174],"infinite-dimensional":[13],"function":[14],"spaces":[15],"from":[16,107],"finite-resolution":[17],"data.":[18],"In":[19,60],"this":[20,61],"setting,":[21],"it":[22],"is":[23,40,124],"important":[24],"the":[26,52,104,108,113,127,131,141,148,156,168,175],"learned":[27,158],"model":[28,53,86,116,160],"be":[30,49],"discretization-invariant,":[31],"or":[32,57],"resolution-robust,":[33],"and":[34,87],"reflect":[36],"PDE-specific":[37],"structure.":[38],"It":[39],"therefore":[41],"natural":[42],"ask":[44],"how":[45],"such":[46],"structure":[47],"should":[48],"encoded":[50],"in":[51],"architecture,":[54],"hypothesis":[55],"class,":[56],"procedure.":[59],"paper,":[62],"we":[63,166],"study":[64],"operator":[65,94,182],"of":[70,140],"nonlinear":[71,171],"parabolic":[72],"PDEs":[73],"based":[74],"Duhamel--Picard":[76],"iteration.":[77,120],"We":[78,96,145],"formulate":[79],"Picard":[80,119,128,132],"iteration":[81],"as":[82,183],"an":[83,137],"abstract":[84,114],"state-transition":[85,115],"present":[88],"a":[89,178,184],"theoretical":[90],"framework":[91],"Picard-type":[93,179],"learning.":[95],"derive":[97],"implementation-agnostic":[98],"generalization":[99],"error":[100,106,110,134],"bounds":[101],"that":[102,125],"separate":[103],"implementation":[105],"estimation":[109,143],"associated":[111],"with":[112],"induced":[117],"by":[118,153],"A":[121],"key":[122],"consequence":[123],"increasing":[126],"depth":[129],"reduces":[130],"truncation":[133],"without":[135],"causing":[136],"unbounded":[138],"growth":[139],"entropy-based":[142],"error.":[144],"also":[146],"extend":[147],"analysis":[149],"long-time":[151],"prediction":[152],"rolling":[154],"out":[155],"same":[157],"local":[159],"over":[161],"successive":[162],"time":[163],"blocks.":[164],"Finally,":[165],"illustrate":[167],"theory":[169],"heat":[172],"torus":[176],"using":[177],"Fourier":[180],"neural":[181],"concrete":[185],"implementation.":[186]},"counts_by_year":[],"updated_date":"2026-07-28T07:46:37.118299","created_date":"2026-05-13T00:00:00"}
