{"id":"https://openalex.org/W7214578138","doi":"https://doi.org/10.48550/arxiv.2609.30819","title":"Learning Provable Neural Network Observer for Uncertain Dynamical Systems","display_name":"Learning Provable Neural Network Observer for Uncertain Dynamical Systems","publication_year":2026,"publication_date":"2026-09-25","ids":{"openalex":"https://openalex.org/W7214578138","doi":"https://doi.org/10.48550/arxiv.2609.30819"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.30819","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.30819","pdf_url":"https://arxiv.org/pdf/2609.30819","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":"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/2609.30819","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5101680716","display_name":"Zhangyi Wang","orcid":"https://orcid.org/0000-0003-1494-7476","observed_orcids":["https://orcid.org/0000-0003-1494-7476"]},"institutions":[{"id":"https://openalex.org/I76130692","display_name":"Zhejiang University","ror":"https://ror.org/00a2xv884","country_code":"CN","type":"education","lineage":["https://openalex.org/I76130692"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Wang, Zhangyi","raw_affiliation_strings":["College of Control Science and Engineering , Zhejiang University , China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"College of Control Science and Engineering , Zhejiang University , China","institution_ids":["https://openalex.org/I76130692"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5140607164","display_name":"Jiaxu Liu","orcid":"https://orcid.org/0009-0006-4779-7389","observed_orcids":["https://orcid.org/0009-0006-4779-7389"]},"institutions":[{"id":"https://openalex.org/I3018263800","display_name":"Huzhou Normal University","ror":"https://ror.org/04mvpxy20","country_code":"CN","type":"education","lineage":["https://openalex.org/I3018263800"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Liu, Jiaxu","raw_affiliation_strings":["School of Science , Huzhou Normal University , China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Science , Huzhou Normal University , China","institution_ids":["https://openalex.org/I3018263800"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5153286334","display_name":"Chen Song","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I165932596","display_name":"National University of Singapore","ror":"https://ror.org/01tgyzw49","country_code":"SG","type":"education","lineage":["https://openalex.org/I165932596"]}],"countries":["SG"],"is_corresponding":false,"raw_author_name":"Chen, Song","raw_affiliation_strings":["Department of Mathematics , National University of Singapore , Singapore"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics , National University of Singapore , Singapore","institution_ids":["https://openalex.org/I165932596"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5153299017","display_name":"Chao Xu","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I76130692","display_name":"Zhejiang University","ror":"https://ror.org/00a2xv884","country_code":"CN","type":"education","lineage":["https://openalex.org/I76130692"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Xu, Chao","raw_affiliation_strings":["College of Control Science and Engineering , Zhejiang University , China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"College of Control Science and Engineering , Zhejiang University , China","institution_ids":["https://openalex.org/I76130692"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5140944662","display_name":"Shengze Cai","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I76130692","display_name":"Zhejiang University","ror":"https://ror.org/00a2xv884","country_code":"CN","type":"education","lineage":["https://openalex.org/I76130692"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Cai, Shengze","raw_affiliation_strings":["College of Control Science and Engineering , Zhejiang University , China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"College of Control Science and Engineering , Zhejiang University , China","institution_ids":["https://openalex.org/I76130692"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"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/T10046","display_name":"Stability and Control of Uncertain Systems","score":0.4323999881744385,"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"}},"topics":[{"id":"https://openalex.org/T10046","display_name":"Stability and Control of Uncertain Systems","score":0.4323999881744385,"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/T10040","display_name":"Adaptive Control of Nonlinear Systems","score":0.2953999936580658,"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":0.13330000638961792,"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/neural-network-observer","display_name":"neural network observer","score":0.996999979019165},{"id":"https://openalex.org/keywords/lyapunov-stability","display_name":"Lyapunov stability","score":0.8960000276565552},{"id":"https://openalex.org/keywords/linear-matrix-inequalities","display_name":"linear matrix inequalities","score":0.902999997138977},{"id":"https://openalex.org/keywords/uncertain-dynamical-systems","display_name":"uncertain dynamical systems","score":0.8420000076293945},{"id":"https://openalex.org/keywords/state-and-disturbance-estimation","display_name":"state and disturbance estimation","score":0.7699999809265137},{"id":"https://openalex.org/keywords/two-stage-training","display_name":"two-stage training","score":0.824999988079071},{"id":"https://openalex.org/keywords/probabilistic-coverage","display_name":"probabilistic coverage","score":0.7540000081062317},{"id":"https://openalex.org/keywords/safety-critical-control","display_name":"safety-critical control","score":0.7559999823570251}],"concepts":[{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.566100001335144},{"id":"https://openalex.org/C2780704645","wikidata":"https://www.wikidata.org/wiki/Q9251458","display_name":"Observer (physics)","level":2,"score":0.5507000088691711},{"id":"https://openalex.org/C47446073","wikidata":"https://www.wikidata.org/wiki/Q5165890","display_name":"Control theory (sociology)","level":3,"score":0.5400999784469604},{"id":"https://openalex.org/C49937458","wikidata":"https://www.wikidata.org/wiki/Q2599292","display_name":"Probabilistic logic","level":2,"score":0.524399995803833},{"id":"https://openalex.org/C2776829284","wikidata":"https://www.wikidata.org/wiki/Q1341651","display_name":"Lyapunov stability","level":3,"score":0.5099999904632568},{"id":"https://openalex.org/C60640748","wikidata":"https://www.wikidata.org/wiki/Q2337858","display_name":"Lyapunov function","level":3,"score":0.4943000078201294},{"id":"https://openalex.org/C201829737","wikidata":"https://www.wikidata.org/wiki/Q2022760","display_name":"Linear matrix inequality","level":2,"score":0.49239999055862427},{"id":"https://openalex.org/C112972136","wikidata":"https://www.wikidata.org/wiki/Q7595718","display_name":"Stability (learning theory)","level":2,"score":0.47609999775886536},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.46650001406669617},{"id":"https://openalex.org/C79379906","wikidata":"https://www.wikidata.org/wiki/Q3174497","display_name":"Dynamical systems theory","level":2,"score":0.4059000015258789},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.40389999747276306},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.388700008392334},{"id":"https://openalex.org/C114275822","wikidata":"https://www.wikidata.org/wiki/Q621512","display_name":"Linear dynamical system","level":3,"score":0.35179999470710754},{"id":"https://openalex.org/C48044578","wikidata":"https://www.wikidata.org/wiki/Q727490","display_name":"Scalability","level":2,"score":0.3287000060081482},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.32409998774528503},{"id":"https://openalex.org/C204323151","wikidata":"https://www.wikidata.org/wiki/Q905424","display_name":"Range (aeronautics)","level":2,"score":0.31700000166893005},{"id":"https://openalex.org/C147168706","wikidata":"https://www.wikidata.org/wiki/Q1457734","display_name":"Recurrent neural network","level":3,"score":0.31220000982284546},{"id":"https://openalex.org/C117251300","wikidata":"https://www.wikidata.org/wiki/Q1849855","display_name":"Parametric statistics","level":2,"score":0.3021000027656555},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.30140000581741333},{"id":"https://openalex.org/C137836250","wikidata":"https://www.wikidata.org/wiki/Q984063","display_name":"Optimization problem","level":2,"score":0.2935999929904938},{"id":"https://openalex.org/C33962884","wikidata":"https://www.wikidata.org/wiki/Q378637","display_name":"Dynamical system (definition)","level":3,"score":0.2856999933719635},{"id":"https://openalex.org/C2776359362","wikidata":"https://www.wikidata.org/wiki/Q2145286","display_name":"Representation (politics)","level":3,"score":0.27639999985694885},{"id":"https://openalex.org/C41949839","wikidata":"https://www.wikidata.org/wiki/Q1756677","display_name":"Stability theory","level":3,"score":0.2685999870300293},{"id":"https://openalex.org/C25854792","wikidata":"https://www.wikidata.org/wiki/Q1028945","display_name":"Lyapunov equation","level":4,"score":0.263700008392334},{"id":"https://openalex.org/C3031470","wikidata":"https://www.wikidata.org/wiki/Q818544","display_name":"State observer","level":3,"score":0.2630999982357025},{"id":"https://openalex.org/C167964875","wikidata":"https://www.wikidata.org/wiki/Q17011487","display_name":"Exponential stability","level":3,"score":0.26089999079704285},{"id":"https://openalex.org/C47702885","wikidata":"https://www.wikidata.org/wiki/Q5441227","display_name":"Feedforward neural network","level":3,"score":0.25760000944137573}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.30819","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.30819","pdf_url":"https://arxiv.org/pdf/2609.30819","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":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2609.30819","doi":"https://doi.org/10.48550/arxiv.2609.30819","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.30819","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:2609.30819","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.30819","pdf_url":"https://arxiv.org/pdf/2609.30819","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":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320322927","display_name":"Zhejiang University","ror":"https://ror.org/00a2xv884"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7214578138.pdf","grobid_xml":"https://content.openalex.org/works/W7214578138.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W4364322883","https://openalex.org/W1982925297","https://openalex.org/W2044378619","https://openalex.org/W2135836059","https://openalex.org/W4310743495","https://openalex.org/W7133228086","https://openalex.org/W7164777317","https://openalex.org/W4402260331","https://openalex.org/W4394782266","https://openalex.org/W1993829530"],"abstract_inverted_index":{"In":[0],"many":[1],"safety-critical":[2],"applications,":[3],"control":[4,133],"of":[5,166],"uncertain":[6],"dynamical":[7],"systems":[8],"relies":[9],"on":[10,131],"observers":[11,20,145],"that":[12,42,97,140],"estimate":[13],"states":[14],"and":[15,85,115,127,135,153],"external":[16],"disturbances.":[17],"Neural":[18],"network":[19,66,144],"can":[21],"improve":[22],"estimation":[23,83],"accuracy,":[24],"but":[25],"certifying":[26],"their":[27],"Lyapunov":[28,76,103],"stability":[29,87,104,113],"via":[30],"Linear":[31],"Matrix":[32],"Inequality":[33],"(LMI)":[34],"constraints":[35],"leads":[36],"to":[37,45],"large-scale":[38],"semidefinite":[39],"programs":[40],"(SDPs)":[41],"are":[43],"difficult":[44],"solve":[46],"for":[47,62,111],"large":[48],"networks.":[49],"To":[50],"overcome":[51],"this":[52],"scalability":[53],"bottleneck,":[54],"we":[55],"propose":[56],"a":[57,74,100,119,164],"novel":[58],"two-stage":[59],"training":[60],"framework":[61],"provably":[63],"stable":[64],"neural":[65,143],"observers.":[67],"Our":[68],"approach":[69],"decouples":[70],"the":[71],"optimization":[72],"into":[73],"point-guided":[75],"pre-training":[77],"phase,":[78],"which":[79],"rapidly":[80],"achieves":[81],"high":[82],"accuracy":[84,162],"local":[86,112],"over":[88,118,163],"sampled":[89],"states,":[90],"followed":[91],"by":[92],"an":[93],"LMI":[94],"fine-tuning":[95],"phase":[96],"efficiently":[98],"satisfies":[99],"strict":[101],"global":[102],"certificate.":[105],"We":[106],"provide":[107],"formal":[108],"theoretical":[109],"guarantees":[110],"radii":[114],"probabilistic":[116],"coverage":[117],"prescribed":[120],"compact":[121],"error-state":[122],"domain":[123],"under":[124],"specified":[125],"regularity":[126],"sampling":[128],"assumptions.":[129],"Experiments":[130],"nonlinear":[132],"benchmarks":[134],"X-29":[136],"aircraft":[137],"ablations":[138],"show":[139],"our":[141],"LMI-certified":[142],"train":[146],"significantly":[147],"faster":[148],"than":[149],"direct":[150],"LMI-based":[151],"methods":[152],"generalize":[154],"robustly":[155],"across":[156],"diverse":[157],"systems,":[158],"achieving":[159],"improved":[160],"tracking":[161],"range":[165],"observer":[167],"baselines.":[168],"The":[169],"code":[170],"is":[171],"available":[172],"at":[173],"https://github.com/Berry-Myon/LearningNeuralNetworkObserver.":[174]},"counts_by_year":[],"updated_date":"2026-11-06T12:05:46.655746","created_date":"2026-09-29T00:00:00"}