{"id":"https://openalex.org/W4417515787","doi":"https://doi.org/10.48550/arxiv.2505.07244","title":"The Influence of the Memory Capacity of Neural DDEs on the Universal Approximation Property","display_name":"The Influence of the Memory Capacity of Neural DDEs on the Universal Approximation Property","publication_year":2025,"publication_date":"2025-05-12","ids":{"openalex":"https://openalex.org/W4417515787","doi":"https://doi.org/10.48550/arxiv.2505.07244"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2505.07244","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2505.07244","pdf_url":"https://arxiv.org/pdf/2505.07244","source":{"id":"https://openalex.org/S4393918464","display_name":"ArXiv.org","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"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"},"type":"preprint","indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2505.07244","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5024817853","display_name":"Christian Kuehn","orcid":"https://orcid.org/0000-0002-7063-6173"},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Kuehn, Christian","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5075355265","display_name":"Sara-Viola Kuntz","orcid":"https://orcid.org/0009-0000-4611-9742"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Kuntz, Sara-Viola","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5024817853"],"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":0.5687999725341797,"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":0.5687999725341797,"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 methods for differential equations","score":0.04659999907016754,"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/T11347","display_name":"Neural Networks Stability and Synchronization","score":0.04129999876022339,"subfield":{"id":"https://openalex.org/subfields/1705","display_name":"Computer Networks and Communications"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.7875999808311462},{"id":"https://openalex.org/keywords/lipschitz-continuity","display_name":"Lipschitz continuity","score":0.7371000051498413},{"id":"https://openalex.org/keywords/residual","display_name":"Residual","score":0.58160001039505},{"id":"https://openalex.org/keywords/ode","display_name":"Ode","score":0.447299987077713},{"id":"https://openalex.org/keywords/limit","display_name":"Limit (mathematics)","score":0.4237000048160553},{"id":"https://openalex.org/keywords/control-theory","display_name":"Control theory (sociology)","score":0.41269999742507935},{"id":"https://openalex.org/keywords/property","display_name":"Property (philosophy)","score":0.40630000829696655},{"id":"https://openalex.org/keywords/delay-differential-equation","display_name":"Delay differential equation","score":0.4007999897003174},{"id":"https://openalex.org/keywords/ordinary-differential-equation","display_name":"Ordinary differential equation","score":0.3880000114440918},{"id":"https://openalex.org/keywords/recurrent-neural-network","display_name":"Recurrent neural network","score":0.3801000118255615}],"concepts":[{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.7875999808311462},{"id":"https://openalex.org/C22324862","wikidata":"https://www.wikidata.org/wiki/Q652707","display_name":"Lipschitz continuity","level":2,"score":0.7371000051498413},{"id":"https://openalex.org/C155512373","wikidata":"https://www.wikidata.org/wiki/Q287450","display_name":"Residual","level":2,"score":0.58160001039505},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.4788999855518341},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.4480000138282776},{"id":"https://openalex.org/C34862557","wikidata":"https://www.wikidata.org/wiki/Q178985","display_name":"Ode","level":2,"score":0.447299987077713},{"id":"https://openalex.org/C151201525","wikidata":"https://www.wikidata.org/wiki/Q177239","display_name":"Limit (mathematics)","level":2,"score":0.4237000048160553},{"id":"https://openalex.org/C47446073","wikidata":"https://www.wikidata.org/wiki/Q5165890","display_name":"Control theory (sociology)","level":3,"score":0.41269999742507935},{"id":"https://openalex.org/C189950617","wikidata":"https://www.wikidata.org/wiki/Q937228","display_name":"Property (philosophy)","level":2,"score":0.40630000829696655},{"id":"https://openalex.org/C147504518","wikidata":"https://www.wikidata.org/wiki/Q2146125","display_name":"Delay differential equation","level":3,"score":0.4007999897003174},{"id":"https://openalex.org/C51544822","wikidata":"https://www.wikidata.org/wiki/Q465274","display_name":"Ordinary differential equation","level":3,"score":0.3880000114440918},{"id":"https://openalex.org/C147168706","wikidata":"https://www.wikidata.org/wiki/Q1457734","display_name":"Recurrent neural network","level":3,"score":0.3801000118255615},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.3506999909877777},{"id":"https://openalex.org/C26517878","wikidata":"https://www.wikidata.org/wiki/Q228039","display_name":"Key (lock)","level":2,"score":0.3495999872684479},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.33730000257492065},{"id":"https://openalex.org/C177973122","wikidata":"https://www.wikidata.org/wiki/Q7860946","display_name":"Types of artificial neural networks","level":4,"score":0.33489999175071716},{"id":"https://openalex.org/C91873725","wikidata":"https://www.wikidata.org/wiki/Q3445816","display_name":"Function approximation","level":3,"score":0.3303999900817871},{"id":"https://openalex.org/C184720557","wikidata":"https://www.wikidata.org/wiki/Q7825049","display_name":"Topology (electrical circuits)","level":2,"score":0.31470000743865967},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.31380000710487366},{"id":"https://openalex.org/C86582703","wikidata":"https://www.wikidata.org/wiki/Q7617824","display_name":"Stochastic neural network","level":4,"score":0.3133000135421753},{"id":"https://openalex.org/C182365436","wikidata":"https://www.wikidata.org/wiki/Q50701","display_name":"Variable (mathematics)","level":2,"score":0.2903999984264374},{"id":"https://openalex.org/C38365724","wikidata":"https://www.wikidata.org/wiki/Q4677469","display_name":"Activation function","level":3,"score":0.2883000075817108},{"id":"https://openalex.org/C55689738","wikidata":"https://www.wikidata.org/wiki/Q15963867","display_name":"Discrete time and continuous time","level":2,"score":0.28630000352859497},{"id":"https://openalex.org/C112972136","wikidata":"https://www.wikidata.org/wiki/Q7595718","display_name":"Stability (learning theory)","level":2,"score":0.28600001335144043},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.2858999967575073},{"id":"https://openalex.org/C33962884","wikidata":"https://www.wikidata.org/wiki/Q378637","display_name":"Dynamical system (definition)","level":3,"score":0.2775999903678894},{"id":"https://openalex.org/C152306953","wikidata":"https://www.wikidata.org/wiki/Q621744","display_name":"Approximation property","level":3,"score":0.27489998936653137},{"id":"https://openalex.org/C122383733","wikidata":"https://www.wikidata.org/wiki/Q865920","display_name":"Approximation error","level":2,"score":0.2558000087738037},{"id":"https://openalex.org/C73586568","wikidata":"https://www.wikidata.org/wiki/Q2600211","display_name":"Parameter space","level":2,"score":0.25540000200271606},{"id":"https://openalex.org/C79379906","wikidata":"https://www.wikidata.org/wiki/Q3174497","display_name":"Dynamical systems theory","level":2,"score":0.25380000472068787}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2505.07244","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2505.07244","pdf_url":"https://arxiv.org/pdf/2505.07244","source":{"id":"https://openalex.org/S4393918464","display_name":"ArXiv.org","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"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.2505.07244","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2505.07244","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2505.07244","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2505.07244","pdf_url":"https://arxiv.org/pdf/2505.07244","source":{"id":"https://openalex.org/S4393918464","display_name":"ArXiv.org","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"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":[],"awards":[{"id":"https://openalex.org/G442688124","display_name":null,"funder_award_id":"SPP2298","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"}],"funders":[{"id":"https://openalex.org/F4320320879","display_name":"Deutsche Forschungsgemeinschaft","ror":"https://ror.org/018mejw64"},{"id":"https://openalex.org/F4320320882","display_name":"Volkswagen Foundation","ror":"https://ror.org/03bsmfz84"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4417515787.pdf","grobid_xml":"https://content.openalex.org/works/W4417515787.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"Neural":[0,13,24,42],"Ordinary":[1],"Differential":[2,26],"Equations":[3,27],"(Neural":[4,28],"ODEs),":[5],"which":[6,226],"are":[7,52],"the":[8,69,84,91,100,104,109,113,116,119,125,132,146,154,161,164,185,192,202,213,222,238,242],"continuous-time":[9],"analog":[10],"of":[11,38,108,115,121,163,206,246],"Residual":[12,41],"Networks":[14,43],"(ResNets),":[15],"have":[16,145],"gained":[17],"significant":[18],"attention":[19],"in":[20,68,74,87,225],"recent":[21],"years.":[22],"Similarly,":[23],"Delay":[25],"DDEs)":[29],"can":[30,167,200,220],"be":[31,168],"interpreted":[32],"as":[33,72],"an":[34],"infinite":[35],"depth":[36],"limit":[37],"Densely":[39],"Connected":[40],"(DenseResNets).":[44],"In":[45,78,118,189],"contrast":[46],"to":[47,255,261],"traditional":[48],"ResNet":[49],"architectures,":[50,123],"DenseResNets":[51],"feed-forward":[53,141],"networks":[54,143],"that":[55,152,235],"allow":[56],"for":[57,98,209],"shortcut":[58],"connections":[59,65],"across":[60],"all":[61],"layers.":[62],"These":[63],"additional":[64],"introduce":[66],"memory":[67,85,101,155,181,193,239,269],"network":[70,126],"architecture,":[71],"typical":[73],"many":[75],"modern":[76],"architectures.":[77],"this":[79],"work,":[80],"we":[81,199,219],"explore":[82],"how":[83],"capacity":[86,102,156,182,194,240],"neural":[88,137,142,165,172,176,207,214],"DDEs":[89,177,208,247],"influences":[90],"universal":[92,147,186,203,227,262,271],"approximation":[93,148,187,204,228,272],"property.":[94,149,188],"The":[95],"key":[96],"parameter":[97,223],"studying":[99],"is":[103,128,158,196,217,229,252],"product":[105],"$K":[106],"\u03c4$":[107],"Lipschitz":[110],"constant":[111],"and":[112,134,139],"delay":[114,250],"DDE.":[117],"case":[120],"non-augmented":[122,175],"where":[124],"width":[127],"not":[129,253],"larger":[130],"than":[131],"input":[133],"output":[135],"dimensions,":[136],"ODEs":[138],"classical":[140],"cannot":[144],"We":[150],"show":[151,234],"if":[153,191],"$K\u03c4$":[157,195],"sufficiently":[159,197],"small,":[160],"dynamics":[162],"DDE":[166,215],"approximated":[169],"by":[170,236],"a":[171,179,257,267],"ODE.":[173],"Consequently,":[174],"with":[178,248],"small":[180],"also":[183],"lack":[184],"contrast,":[190],"large,":[198],"establish":[201],"property":[205],"continuous":[210],"functions.":[211],"If":[212],"architecture":[216],"augmented,":[218],"expand":[221],"regions":[224],"possible.":[230],"Overall,":[231],"our":[232],"results":[233],"increasing":[237],"$K\u03c4$,":[241],"infinite-dimensional":[243],"phase":[244],"space":[245],"positive":[249],"$\u03c4>0$":[251],"sufficient":[254],"guarantee":[256],"direct":[258],"jump":[259],"transition":[260],"approximation,":[263],"but":[264],"only":[265],"after":[266],"certain":[268],"threshold,":[270],"holds.":[273]},"counts_by_year":[],"updated_date":"2026-04-21T08:09:41.155169","created_date":"2025-10-10T00:00:00"}