{"id":"https://openalex.org/W7225560842","doi":"https://doi.org/10.5281/zenodo.24051388","title":"Numerical challenges in data driven learning of nonlinear dynamics","display_name":"Numerical challenges in data driven learning of nonlinear dynamics","publication_year":2026,"publication_date":"2026-11-08","ids":{"openalex":"https://openalex.org/W7225560842","doi":"https://doi.org/10.5281/zenodo.24051388"},"language":"en","primary_location":{"id":"doi:10.5281/zenodo.24051388","doi":"https://doi.org/10.5281/zenodo.24051388","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24051388","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"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":"article-journal"},"volume":null,"type":"article","is_traditional":true,"indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.5281/zenodo.24051388","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5068311458","display_name":"Zlatko Drma\u010d","orcid":"https://orcid.org/0000-0001-6845-332X","observed_orcids":["https://orcid.org/0000-0001-6845-332X"]},"institutions":[{"id":"https://openalex.org/I181343428","display_name":"University of Zagreb","ror":"https://ror.org/00mv6sv71","country_code":"HR","type":"education","lineage":["https://openalex.org/I181343428"]}],"countries":["HR"],"is_corresponding":true,"raw_author_name":"Drma\u010d, Zlatko","raw_affiliation_strings":["Department of Mathematics, Faculty of Science, University of Zagreb"],"raw_orcid":"https://orcid.org/0000-0001-6845-332X","affiliations":[{"raw_affiliation_string":"Department of Mathematics, Faculty of Science, University of Zagreb","institution_ids":["https://openalex.org/I181343428"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5068311458"],"corresponding_institution_ids":["https://openalex.org/I181343428"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.0,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"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":1.0,"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":1.0,"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/T11416","display_name":"Numerical Solvers for Ordinary Differential Equations","score":0.0,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"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/T11236","display_name":"Control Systems and Identification","score":0.0,"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/koopman-operator","display_name":"Koopman operator","score":0.8399999737739563},{"id":"https://openalex.org/keywords/dynamic-mode-decomposition","display_name":"dynamic mode decomposition","score":0.7730000019073486},{"id":"https://openalex.org/keywords/extended-dynamic-mode-decomposition","display_name":"extended dynamic mode decomposition","score":0.9020000100135803},{"id":"https://openalex.org/keywords/koopman-mode-decomposition","display_name":"Koopman mode decomposition","score":0.9829999804496765},{"id":"https://openalex.org/keywords/non-normal-operators","display_name":"non-normal operators","score":0.8320000171661377},{"id":"https://openalex.org/keywords/low-rank-approximation","display_name":"low-rank approximation","score":0.5730000138282776},{"id":"https://openalex.org/keywords/qr-decomposition","display_name":"QR decomposition","score":0.8899999856948853},{"id":"https://openalex.org/keywords/backward-error-analysis","display_name":"backward error analysis","score":0.7580000162124634},{"id":"https://openalex.org/keywords/condition-number","display_name":"condition number","score":0.6420000195503235},{"id":"https://openalex.org/keywords/model-reduction","display_name":"model reduction","score":0.6399999856948853},{"id":"https://openalex.org/keywords/numerical-stability","display_name":"numerical stability","score":0.5180000066757202},{"id":"https://openalex.org/keywords/invariant-subspaces","display_name":"invariant subspaces","score":0.5350000262260437}],"concepts":[{"id":"https://openalex.org/C2777032711","wikidata":"https://www.wikidata.org/wiki/Q5318993","display_name":"Dynamic mode decomposition","level":2,"score":0.9298999905586243},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.6247000098228455},{"id":"https://openalex.org/C188060507","wikidata":"https://www.wikidata.org/wiki/Q653242","display_name":"QR decomposition","level":3,"score":0.5412999987602234},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.5192000269889832},{"id":"https://openalex.org/C2780440489","wikidata":"https://www.wikidata.org/wiki/Q5227278","display_name":"Data-driven","level":2,"score":0.5116999745368958},{"id":"https://openalex.org/C17020691","wikidata":"https://www.wikidata.org/wiki/Q139677","display_name":"Operator (biology)","level":5,"score":0.4982999861240387},{"id":"https://openalex.org/C42355184","wikidata":"https://www.wikidata.org/wiki/Q1361088","display_name":"Matrix decomposition","level":3,"score":0.4693000018596649},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.45410001277923584},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.4487999975681305},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4471000134944916},{"id":"https://openalex.org/C2781089630","wikidata":"https://www.wikidata.org/wiki/Q21856745","display_name":"Realization (probability)","level":2,"score":0.43639999628067017},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.42590001225471497},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.3772999942302704},{"id":"https://openalex.org/C2777904410","wikidata":"https://www.wikidata.org/wiki/Q7397","display_name":"Software","level":2,"score":0.37560001015663147},{"id":"https://openalex.org/C164226766","wikidata":"https://www.wikidata.org/wiki/Q7293202","display_name":"Rank (graph theory)","level":2,"score":0.3723999857902527},{"id":"https://openalex.org/C48753275","wikidata":"https://www.wikidata.org/wiki/Q11216","display_name":"Numerical analysis","level":2,"score":0.3476000130176544},{"id":"https://openalex.org/C22789450","wikidata":"https://www.wikidata.org/wiki/Q420904","display_name":"Singular value decomposition","level":2,"score":0.3379000127315521},{"id":"https://openalex.org/C79379906","wikidata":"https://www.wikidata.org/wiki/Q3174497","display_name":"Dynamical systems theory","level":2,"score":0.3215000033378601},{"id":"https://openalex.org/C187834632","wikidata":"https://www.wikidata.org/wiki/Q188804","display_name":"Factorization","level":2,"score":0.31779998540878296},{"id":"https://openalex.org/C49766605","wikidata":"https://www.wikidata.org/wiki/Q207643","display_name":"Linear map","level":2,"score":0.29670000076293945},{"id":"https://openalex.org/C71139939","wikidata":"https://www.wikidata.org/wiki/Q910194","display_name":"Modal","level":2,"score":0.2921999990940094},{"id":"https://openalex.org/C169756996","wikidata":"https://www.wikidata.org/wiki/Q194919","display_name":"Eigendecomposition of a matrix","level":3,"score":0.27480000257492065},{"id":"https://openalex.org/C6802819","wikidata":"https://www.wikidata.org/wiki/Q1072174","display_name":"Linear system","level":2,"score":0.2621999979019165}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.5281/zenodo.24051388","doi":"https://doi.org/10.5281/zenodo.24051388","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24051388","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"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":"article-journal"},{"id":"doi:10.5281/zenodo.24051389","doi":"https://doi.org/10.5281/zenodo.24051389","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24051389","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"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":"article-journal"}],"best_oa_location":{"id":"doi:10.5281/zenodo.24051388","doi":"https://doi.org/10.5281/zenodo.24051388","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24051388","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"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":"article-journal"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320322674","display_name":"Hrvatska Zaklada za Znanost","ror":"https://ror.org/03n51vw80"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W3159420233","https://openalex.org/W4390306868","https://openalex.org/W7017554980","https://openalex.org/W1591018827","https://openalex.org/W3189932128","https://openalex.org/W7120351519","https://openalex.org/W7213961838","https://openalex.org/W3192787595","https://openalex.org/W6983517329","https://openalex.org/W4380887250"],"abstract_inverted_index":{"The":[0,20,61,87,135,193,305,323],"Dynamic":[1],"Mode":[2,82],"Decomposition":[3],"(DMD)":[4],"is":[5,32,34,40,111,153,310,317,336,388],"a":[6,51,76,94,276,313,333,339,348],"popular":[7],"and":[8,130,133,145,160,166,205,212,222,226,248,283,294,372],"versatile":[9],"numerical":[10,136,163,246],"method":[11],"for":[12,131,147],"data":[13,43,63],"driven":[14,44],"analysis":[15,149,324,387],"of":[16,47,50,71,84,107,118,138,150,157,162,171,184,196,199,224,244,254,264,307,327,332,344,350],"nonlinear":[17,62,102,120,151],"dynamical":[18,59],"systems.":[19],"theoretical":[21,359],"bedrock":[22],"upon":[23],"which":[24,110],"the":[25,35,57,69,72,85,101,108,119,139,158,172,177,182,185,200,203,209,213,242,265,278,284,292,296,328,379,384,391,397],"more":[26],"general":[27,267],"Extended":[28],"DMD":[29,144,201],"framework":[30,142],"(EDMD)":[31],"built":[33],"Koopman":[36,52,140,269,334],"composition":[37,53],"operator.":[38],"EDMD":[39],"based":[41,175,319,376],"on":[42,176,291,295,320,378],"Rayleigh-Ritz":[45],"extraction":[46],"spectral":[48],"information":[49],"operator":[54,109,141,270,293,335],"associated":[55],"with":[56,338,390],"underlying":[58],"system.":[60],"snapshots":[64],"are":[65,202,238,280],"then":[66],"represented":[67],"using":[68,104,312],"eigenvectors/eigenfunctions":[70],"operator,":[73],"resulting":[74],"in":[75,112,181,232,241,258,325,342,347],"modal":[77,340],"(spatio-temporal)":[78],"decomposition":[79,315,341],"KMD":[80,88,285],"(Koopman":[81],"Decomposition)":[83],"dynamics.":[86,121],"can":[89,123],"also":[90],"be":[91,124],"considered":[92],"as":[93,190,275,330],"model":[95],"order":[96],"reduction":[97],"tool":[98],"that":[99,316,353,367],"represents":[100],"dynamics":[103,152],"selected":[105,356],"eigenpairs":[106],"fact":[113],"an":[114,154,167],"infinite":[115],"dimensional":[116,303],"linearisation":[117],"It":[122],"used":[125,299],"to":[126,300,355,394],"reveal":[127],"coherent":[128],"states":[129],"forecasting":[132],"control.":[134],"realization":[137],"(i.e.":[143],"EDMD)":[146],"computational":[148],"excellent":[155],"illustration":[156],"importance":[159],"power":[161],"linear":[164],"algebra,":[165],"instructive":[168],"case":[169],"study":[170],"software":[173,187],"development":[174,243],"design":[178],"principles":[179],"introduced":[180],"state":[183],"art":[186],"packages":[188],"such":[189],"e.g.":[191],"LAPACK.":[192],"main":[194],"ingredients":[195],"all":[197],"variants":[198],"SVD":[204],"low":[206],"rank":[207],"approximations,":[208,221],"QR":[210],"factorization":[211],"orthogonal":[214],"projections,":[215],"structured":[216],"sparsity":[217],"constrained":[218],"least":[219],"squares":[220],"approximations":[223],"eigenvalues":[225],"eigenvectors":[227,279,309,329],"from":[228,364],"subspaces,":[229],"fast":[230],"updating":[231],"sreaming":[233],"real":[234],"time":[235],"applications.":[236],"There":[237],"many":[239],"challenges":[240],"robust":[245],"methods":[247],"this":[249],"talk":[250],"will":[251],"address":[252],"some":[253,259],"them.":[255],"For":[256],"example,":[257],"cases":[260],"numerically":[261,287],"computed":[262,398],"compression":[263],"(in":[266],"non-normal)":[268],"exhibits":[271],"high":[272],"non-normality,":[273],"and,":[274],"result,":[277],"highly":[281],"ill-conditioned":[282,308],"becomes":[286],"unstable.":[288],"This":[289],"depends":[290],"basis":[297],"(functions)":[298],"derive":[301],"finite":[302],"compression.":[304],"problem":[306],"solved":[311],"Koopman-Schur":[314],"entirely":[318],"unitary":[321],"transformations.":[322],"terms":[326,343],"modes":[331],"replaced":[337],"orthonormal":[345],"bases":[346],"flag":[349],"invariant":[351],"subspaces":[352],"correspond":[354],"eigenvalues.":[357],"Further,":[358],"guarantees":[360],"require":[361],"nontrivial":[362],"input":[363],"perturbation":[365],"theory":[366,393],"identifies":[368],"relevant":[369],"condition":[370],"numbers,":[371],"reliable":[373],"residual":[374],"estimates":[375],"solely":[377],"supplied":[380],"data.":[381],"In":[382],"particular,":[383],"backward":[385],"error":[386],"connected":[389],"shadowing":[392],"properly":[395],"interpret":[396],"results.":[399]},"counts_by_year":[],"updated_date":"2026-11-10T11:39:43.021295","created_date":"2026-11-09T00:00:00"}