{"id":"https://openalex.org/W4410112129","doi":"https://doi.org/10.1007/s00332-025-10159-2","title":"Heterogeneous Mixtures of Dictionary Functions to Approximate Subspace Invariance in Koopman Operators: Why Deep Koopman Operators Work","display_name":"Heterogeneous Mixtures of Dictionary Functions to Approximate Subspace Invariance in Koopman Operators: Why Deep Koopman Operators Work","publication_year":2025,"publication_date":"2025-05-06","ids":{"openalex":"https://openalex.org/W4410112129","doi":"https://doi.org/10.1007/s00332-025-10159-2"},"language":"en","primary_location":{"id":"doi:10.1007/s00332-025-10159-2","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00332-025-10159-2","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00332-025-10159-2.pdf","source":{"id":"https://openalex.org/S131276044","display_name":"Journal of Nonlinear Science","issn_l":"0938-8974","issn":["0938-8974","1432-1467"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Nonlinear Science","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007/s00332-025-10159-2.pdf","any_repository_has_fulltext":null},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5031137742","display_name":"Charles A. Johnson","orcid":"https://orcid.org/0000-0002-1975-2005"},"institutions":[{"id":"https://openalex.org/I154570441","display_name":"University of California, Santa Barbara","ror":"https://ror.org/02t274463","country_code":"US","type":"education","lineage":["https://openalex.org/I154570441"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Charles A. Johnson","raw_affiliation_strings":["Department of Mechanical Engineering, University of California, Santa Barbara, 93106, USA"],"affiliations":[{"raw_affiliation_string":"Department of Mechanical Engineering, University of California, Santa Barbara, 93106, USA","institution_ids":["https://openalex.org/I154570441"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5036831250","display_name":"Shara Balakrishnan","orcid":null},"institutions":[{"id":"https://openalex.org/I2799323099","display_name":"Montgomery County Public Schools","ror":"https://ror.org/05sc9ba13","country_code":"US","type":"education","lineage":["https://openalex.org/I2799323099"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Shara Balakrishnan","raw_affiliation_strings":["Mithrl Inc, 44 Montgomery St, San Francisco, CA, 94104, USA"],"affiliations":[{"raw_affiliation_string":"Mithrl Inc, 44 Montgomery St, San Francisco, CA, 94104, USA","institution_ids":["https://openalex.org/I2799323099"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5091530669","display_name":"Enoch Yeung","orcid":"https://orcid.org/0000-0001-7630-7429"},"institutions":[{"id":"https://openalex.org/I154570441","display_name":"University of California, Santa Barbara","ror":"https://ror.org/02t274463","country_code":"US","type":"education","lineage":["https://openalex.org/I154570441"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Enoch Yeung","raw_affiliation_strings":["Department of Mechanical Engineering, University of California, Santa Barbara, 93106, USA"],"affiliations":[{"raw_affiliation_string":"Department of Mechanical Engineering, University of California, Santa Barbara, 93106, USA","institution_ids":["https://openalex.org/I154570441"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5031137742"],"corresponding_institution_ids":["https://openalex.org/I154570441"],"apc_list":{"value":2290,"currency":"EUR","value_usd":2890},"apc_paid":{"value":2290,"currency":"EUR","value_usd":2890},"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.07424579,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"35","issue":"4","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/T11254","display_name":"Fluid Dynamics and Vibration Analysis","score":0.9783999919891357,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"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/T10360","display_name":"Fluid Dynamics and Turbulent Flows","score":0.9776999950408936,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"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/subspace-topology","display_name":"Subspace topology","score":0.6846709847450256},{"id":"https://openalex.org/keywords/operator-theory","display_name":"Operator theory","score":0.6436010003089905},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5594943165779114},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.5396106243133545},{"id":"https://openalex.org/keywords/linear-subspace","display_name":"Linear subspace","score":0.41767942905426025},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.4149673879146576},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.30307555198669434}],"concepts":[{"id":"https://openalex.org/C32834561","wikidata":"https://www.wikidata.org/wiki/Q660730","display_name":"Subspace topology","level":2,"score":0.6846709847450256},{"id":"https://openalex.org/C43929395","wikidata":"https://www.wikidata.org/wiki/Q1198874","display_name":"Operator theory","level":2,"score":0.6436010003089905},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5594943165779114},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.5396106243133545},{"id":"https://openalex.org/C12362212","wikidata":"https://www.wikidata.org/wiki/Q728435","display_name":"Linear subspace","level":2,"score":0.41767942905426025},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.4149673879146576},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.30307555198669434}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1007/s00332-025-10159-2","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00332-025-10159-2","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00332-025-10159-2.pdf","source":{"id":"https://openalex.org/S131276044","display_name":"Journal of Nonlinear Science","issn_l":"0938-8974","issn":["0938-8974","1432-1467"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Nonlinear Science","raw_type":"journal-article"},{"id":"pmh:oai:osti.gov:2564765","is_oa":true,"landing_page_url":"www.osti.gov/servlets/purl/2564765","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":null}],"best_oa_location":{"id":"doi:10.1007/s00332-025-10159-2","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00332-025-10159-2","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00332-025-10159-2.pdf","source":{"id":"https://openalex.org/S131276044","display_name":"Journal of Nonlinear Science","issn_l":"0938-8974","issn":["0938-8974","1432-1467"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Nonlinear Science","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G3833944892","display_name":null,"funder_award_id":"FA8750-19-2-0502","funder_id":"https://openalex.org/F4320332180","funder_display_name":"Defense Advanced Research Projects Agency"},{"id":"https://openalex.org/G5503258486","display_name":null,"funder_award_id":"528678","funder_id":"https://openalex.org/F4320338354","funder_display_name":"Pacific Northwest National Laboratory"},{"id":"https://openalex.org/G5524522455","display_name":null,"funder_award_id":"DARPA","funder_id":"https://openalex.org/F4320338281","funder_display_name":"Army Research Office"},{"id":"https://openalex.org/G6671297155","display_name":null,"funder_award_id":"CAREER","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G7174470830","display_name":"CAREER: A Novel Framework for Measuring and Engineering Twisting and Writhing in DNA","funder_award_id":"2240176","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G7452299184","display_name":null,"funder_award_id":"W911NF","funder_id":"https://openalex.org/F4320338281","funder_display_name":"Army Research Office"},{"id":"https://openalex.org/G8998121839","display_name":null,"funder_award_id":"911NF","funder_id":"https://openalex.org/F4320338281","funder_display_name":"Army Research Office"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320306078","display_name":"U.S. Department of Defense","ror":"https://ror.org/0447fe631"},{"id":"https://openalex.org/F4320306084","display_name":"U.S. Department of Energy","ror":"https://ror.org/01bj3aw27"},{"id":"https://openalex.org/F4320332180","display_name":"Defense Advanced Research Projects Agency","ror":"https://ror.org/02caytj08"},{"id":"https://openalex.org/F4320332815","display_name":"Advanced Research Projects Agency","ror":"https://ror.org/02caytj08"},{"id":"https://openalex.org/F4320337509","display_name":"Biological and Environmental Research","ror":"https://ror.org/0114b2m14"},{"id":"https://openalex.org/F4320338281","display_name":"Army Research Office","ror":"https://ror.org/05epdh915"},{"id":"https://openalex.org/F4320338354","display_name":"Pacific Northwest National Laboratory","ror":"https://ror.org/05h992307"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4410112129.pdf","grobid_xml":"https://content.openalex.org/works/W4410112129.grobid-xml"},"referenced_works_count":34,"referenced_works":["https://openalex.org/W234899517","https://openalex.org/W1591018827","https://openalex.org/W1988115241","https://openalex.org/W2014356541","https://openalex.org/W2067035505","https://openalex.org/W2101258119","https://openalex.org/W2126999002","https://openalex.org/W2151693816","https://openalex.org/W2164954534","https://openalex.org/W2166116275","https://openalex.org/W2172945660","https://openalex.org/W2239232218","https://openalex.org/W2551318894","https://openalex.org/W2564386551","https://openalex.org/W2594932112","https://openalex.org/W2739844829","https://openalex.org/W2777639606","https://openalex.org/W2886547831","https://openalex.org/W2963401716","https://openalex.org/W2963765797","https://openalex.org/W2963882206","https://openalex.org/W2964119648","https://openalex.org/W2973004389","https://openalex.org/W3038118776","https://openalex.org/W3046212064","https://openalex.org/W3100184721","https://openalex.org/W3102548437","https://openalex.org/W3113606885","https://openalex.org/W3193899040","https://openalex.org/W4239539940","https://openalex.org/W4251512373","https://openalex.org/W4287086219","https://openalex.org/W4378781518","https://openalex.org/W6852326679"],"related_works":["https://openalex.org/W3100286349","https://openalex.org/W2896134808","https://openalex.org/W4289378085","https://openalex.org/W4294291164","https://openalex.org/W3172436493","https://openalex.org/W1887135636","https://openalex.org/W4287164812","https://openalex.org/W2386063599","https://openalex.org/W1975884855","https://openalex.org/W3213150849"],"abstract_inverted_index":{"Abstract":[0],"Koopman":[1,26,134,160,257],"operators":[2],"model":[3],"nonlinear":[4,14,103,162,183,262],"dynamics":[5],"as":[6,16,153,192],"a":[7,13,25,34,40,44,59,145,159,214],"linear":[8],"dynamic":[9,49,82],"system":[10],"acting":[11],"on":[12],"function":[15],"the":[17,53,98,101,110,117,186,193,253],"state.":[18],"This":[19,227],"nonstandard":[20],"state":[21],"is":[22,29],"often":[23],"called":[24,80],"observable":[27],"and":[28,93,115,189,221,233],"usually":[30],"approximated":[31],"numerically":[32],"by":[33,212],"superposition":[35],"of":[36,62,100,138,147,174,182,218,241,255],"functions":[37,55,76,131,141,163,177,184,220,225],"drawn":[38,57,178],"from":[39,58,113,179],"dictionary":[41,54,75,130,140,176,216,229],".":[42],"In":[43,105,166],"widely":[45],"used":[46,71],"algorithm,":[47],"extended":[48],"mode":[50,83],"decomposition":[51,84],"(EDMD),":[52],"are":[56,164],"fixed":[60],"class":[61],"functions.":[63],"Deep":[64],"learning":[65,263],"combined":[66],"with":[67,97,238],"EDMD":[68],"has":[69],"been":[70],"to":[72,132,236,259],"learn":[73],"novel":[74],"in":[77,244],"an":[78,239],"algorithm":[79,196],"deep":[81],"(deepDMD).":[85],"The":[86],"learned":[87,111],"representation":[88],"both":[89],"(1)":[90],"accurately":[91],"models":[92,258],"(2)":[94],"scales":[95],"well":[96],"dimension":[99],"original":[102],"system.":[104],"this":[106,167,211],"paper,":[107,168],"we":[108,151,169],"analyze":[109],"dictionaries":[112],"deepDMD":[114,195,237],"explore":[116,125],"theoretical":[118],"basis":[119,224],"for":[120],"their":[121],"strong":[122],"performance.":[123],"We":[124,208],"State-Inclusive":[126],"Logistic":[127],"Lifting":[128],"(SILL)":[129],"approximate":[133,156],"observables.":[135],"Error":[136],"analysis":[137],"these":[139],"show":[142,210],"they":[143],"satisfy":[144],"property":[146],"subspace":[148],"approximation,":[149],"which":[150],"define":[152],"uniform":[154],"finite":[155],"closure.":[157],"Typically,":[158],"dictionary\u2019s":[161],"homogeneous.":[165],"discover":[170],"that":[171],"structured":[172],"mixing":[173],"heterogeneous":[175,215],"different":[180],"classes":[181],"achieve":[185],"same":[187],"accuracy":[188,232],"dimensional":[190,234],"scaling":[191,235],"deep-learning-based":[194],"Yeung":[197],"et":[198],"al.":[199],"(":[200],"In:":[201],"2019":[202],"American":[203],"Control":[204],"Conference":[205],"(ACC),":[206],"2019).":[207],"specifically":[209],"building":[213],"comprised":[217],"SILL":[219],"conjunctive":[222],"radial":[223],"(RBFs).":[226],"mixed":[228],"achieves":[230],"similar":[231],"order":[240],"magnitude":[242],"reduction":[243],"parameters,":[245],"while":[246],"maintaining":[247],"geometric":[248],"interpretability.":[249],"These":[250],"results":[251],"strengthen":[252],"viability":[254],"dictionary-based":[256],"solving":[260],"high-dimensional":[261],"problems.":[264]},"counts_by_year":[],"updated_date":"2026-04-21T08:09:41.155169","created_date":"2025-05-07T00:00:00"}