{"id":"https://openalex.org/W4396987364","doi":"https://doi.org/10.1007/s10444-025-10273-5","title":"Analysis of the geometric structure of neural networks and neural ODEs via morse functions","display_name":"Analysis of the geometric structure of neural networks and neural ODEs via morse functions","publication_year":2026,"publication_date":"2026-02-01","ids":{"openalex":"https://openalex.org/W4396987364","doi":"https://doi.org/10.1007/s10444-025-10273-5"},"language":"en","primary_location":{"id":"doi:10.1007/s10444-025-10273-5","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10444-025-10273-5","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10444-025-10273-5.pdf","source":{"id":"https://openalex.org/S57050787","display_name":"Advances in Computational Mathematics","issn_l":"1019-7168","issn":["1019-7168","1572-9044"],"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":"Advances in Computational Mathematics","raw_type":"journal-article"},"type":"preprint","indexed_in":["arxiv","crossref","datacite"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007/s10444-025-10273-5.pdf","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":[{"id":"https://openalex.org/I4210132734","display_name":"Max Planck Computing and Data Facility","ror":"https://ror.org/03e21z229","country_code":"DE","type":"facility","lineage":["https://openalex.org/I149899117","https://openalex.org/I4210132734","https://openalex.org/I4210139716"]},{"id":"https://openalex.org/I4403386549","display_name":"Munich Center for Machine Learning","ror":"https://ror.org/02nfy3535","country_code":null,"type":"education","lineage":["https://openalex.org/I4403386549","https://openalex.org/I62916508","https://openalex.org/I8204097"]},{"id":"https://openalex.org/I62916508","display_name":"Technical University of Munich","ror":"https://ror.org/02kkvpp62","country_code":"DE","type":"education","lineage":["https://openalex.org/I62916508"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Christian Kuehn","raw_affiliation_strings":["Department of Mathematics, Technical University of Munich, School of Computation, Information and Technology, Boltzmannstra\u00dfe 3, Garching, 85748, Germany","Munich Center for Machine Learning (MCML), M\u00fcnchen, Germany","Munich Data Science Institute (MDSI), Garching, Germany"],"raw_orcid":"https://orcid.org/0000-0002-7063-6173","affiliations":[{"raw_affiliation_string":"Department of Mathematics, Technical University of Munich, School of Computation, Information and Technology, Boltzmannstra\u00dfe 3, Garching, 85748, Germany","institution_ids":["https://openalex.org/I62916508"]},{"raw_affiliation_string":"Munich Center for Machine Learning (MCML), M\u00fcnchen, Germany","institution_ids":["https://openalex.org/I4403386549"]},{"raw_affiliation_string":"Munich Data Science Institute (MDSI), Garching, Germany","institution_ids":["https://openalex.org/I4210132734"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5075355265","display_name":"Sara-Viola Kuntz","orcid":"https://orcid.org/0009-0000-4611-9742"},"institutions":[{"id":"https://openalex.org/I4210132734","display_name":"Max Planck Computing and Data Facility","ror":"https://ror.org/03e21z229","country_code":"DE","type":"facility","lineage":["https://openalex.org/I149899117","https://openalex.org/I4210132734","https://openalex.org/I4210139716"]},{"id":"https://openalex.org/I4403386549","display_name":"Munich Center for Machine Learning","ror":"https://ror.org/02nfy3535","country_code":null,"type":"education","lineage":["https://openalex.org/I4403386549","https://openalex.org/I62916508","https://openalex.org/I8204097"]},{"id":"https://openalex.org/I62916508","display_name":"Technical University of Munich","ror":"https://ror.org/02kkvpp62","country_code":"DE","type":"education","lineage":["https://openalex.org/I62916508"]}],"countries":["DE"],"is_corresponding":true,"raw_author_name":"Sara-Viola Kuntz","raw_affiliation_strings":["Department of Mathematics, Technical University of Munich, School of Computation, Information and Technology, Boltzmannstra\u00dfe 3, Garching, 85748, Germany","Munich Center for Machine Learning (MCML), M\u00fcnchen, Germany","Munich Data Science Institute (MDSI), Garching, Germany"],"raw_orcid":"https://orcid.org/0009-0000-4611-9742","affiliations":[{"raw_affiliation_string":"Department of Mathematics, Technical University of Munich, School of Computation, Information and Technology, Boltzmannstra\u00dfe 3, Garching, 85748, Germany","institution_ids":["https://openalex.org/I62916508"]},{"raw_affiliation_string":"Munich Center for Machine Learning (MCML), M\u00fcnchen, Germany","institution_ids":["https://openalex.org/I4403386549"]},{"raw_affiliation_string":"Munich Data Science Institute (MDSI), Garching, Germany","institution_ids":["https://openalex.org/I4210132734"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5075355265"],"corresponding_institution_ids":["https://openalex.org/I4210132734","https://openalex.org/I4403386549","https://openalex.org/I62916508"],"apc_list":{"value":2390,"currency":"EUR","value_usd":2990},"apc_paid":{"value":2390,"currency":"EUR","value_usd":2990},"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.00192901,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"52","issue":"1","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11245","display_name":"Advanced Numerical Analysis Techniques","score":0.9505000114440918,"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"}},"topics":[{"id":"https://openalex.org/T11245","display_name":"Advanced Numerical Analysis Techniques","score":0.9505000114440918,"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/morse-code","display_name":"Morse code","score":0.8015844821929932},{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.7344658374786377},{"id":"https://openalex.org/keywords/ode","display_name":"Ode","score":0.6138495802879333},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.43362587690353394},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.4241548180580139},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.36139625310897827},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.3338615894317627},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.2684881091117859}],"concepts":[{"id":"https://openalex.org/C140031139","wikidata":"https://www.wikidata.org/wiki/Q79897","display_name":"Morse code","level":2,"score":0.8015844821929932},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.7344658374786377},{"id":"https://openalex.org/C34862557","wikidata":"https://www.wikidata.org/wiki/Q178985","display_name":"Ode","level":2,"score":0.6138495802879333},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.43362587690353394},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.4241548180580139},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.36139625310897827},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.3338615894317627},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.2684881091117859},{"id":"https://openalex.org/C76155785","wikidata":"https://www.wikidata.org/wiki/Q418","display_name":"Telecommunications","level":1,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1007/s10444-025-10273-5","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10444-025-10273-5","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10444-025-10273-5.pdf","source":{"id":"https://openalex.org/S57050787","display_name":"Advances in Computational Mathematics","issn_l":"1019-7168","issn":["1019-7168","1572-9044"],"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":"Advances in Computational Mathematics","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:2405.09351","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2405.09351","pdf_url":"https://arxiv.org/pdf/2405.09351","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":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"pmh:doi:10.48550/arxiv.2405.09351","is_oa":true,"landing_page_url":null,"pdf_url":null,"source":{"id":"https://openalex.org/S4406922384","display_name":"Open MIND","issn_l":null,"issn":null,"is_oa":false,"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":"Article"},{"id":"doi:10.48550/arxiv.2405.09351","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2405.09351","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":"doi:10.1007/s10444-025-10273-5","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10444-025-10273-5","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10444-025-10273-5.pdf","source":{"id":"https://openalex.org/S57050787","display_name":"Advances in Computational Mathematics","issn_l":"1019-7168","issn":["1019-7168","1572-9044"],"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":"Advances in Computational Mathematics","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G1854101341","display_name":null,"funder_award_id":"SPP 2298","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"},{"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":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4396987364.pdf","grobid_xml":"https://content.openalex.org/works/W4396987364.grobid-xml"},"referenced_works_count":26,"referenced_works":["https://openalex.org/W2896556344","https://openalex.org/W4297826903","https://openalex.org/W1506426323","https://openalex.org/W4221038245","https://openalex.org/W4297778933","https://openalex.org/W4250903974","https://openalex.org/W2089876099","https://openalex.org/W2194775991","https://openalex.org/W4235840289","https://openalex.org/W2137983211","https://openalex.org/W4289468615","https://openalex.org/W4226047355","https://openalex.org/W3108320661","https://openalex.org/W4385967988","https://openalex.org/W3195349637","https://openalex.org/W2810898455","https://openalex.org/W2946265133","https://openalex.org/W1486721501","https://openalex.org/W1525848129","https://openalex.org/W2158581396","https://openalex.org/W596095475","https://openalex.org/W1604264128","https://openalex.org/W1995238279","https://openalex.org/W2276891868","https://openalex.org/W2600297185","https://openalex.org/W4288276633"],"related_works":["https://openalex.org/W4391375266","https://openalex.org/W2371448224","https://openalex.org/W3134495997","https://openalex.org/W2363680170","https://openalex.org/W2376428685","https://openalex.org/W4224264613","https://openalex.org/W4256082577","https://openalex.org/W625783435","https://openalex.org/W652105763","https://openalex.org/W2048742619"],"abstract_inverted_index":{"Abstract":[0],"Besides":[1],"classical":[2],"feed-forward":[3,35],"neural":[4,11,38,50,87,224,235,286,289],"networks":[5,51,287],"such":[6],"as":[7,29],"multilayer":[8],"perceptrons,":[9],"also":[10],"ordinary":[12],"differential":[13],"equations":[14],"(neural":[15],"ODEs)":[16],"have":[17,242],"gained":[18],"particular":[19],"interest":[20],"in":[21,212,255],"recent":[22],"years.":[23],"Neural":[24],"ODEs":[25],"can":[26,134],"be":[27,135],"interpreted":[28],"an":[30,90,98],"infinite":[31,48,259],"depth":[32,49,260],"limit":[33],"of":[34,45,69,80,94,102,115,129,160,170,199,248,283,299,307],"or":[36,167,177],"residual":[37],"networks.":[39],"We":[40,107,150,202],"study":[41],"the":[42,56,59,78,95,112,116,118,125,158,161,168,171,180,184,196,200,213,226,237,250,256,278,296,300],"input\u2013output":[43,119,301],"dynamics":[44],"finite":[46,67,257],"and":[47,127,232,258,273,280,288],"with":[52,65],"scalar":[53,142],"output.":[54],"In":[55,84,183],"finite-depth":[57,223],"case,":[58],"input":[60,96,181],"is":[61,148,164,174,219],"a":[62,66,86,207,304],"state":[63,79],"associated":[64],"number":[68],"nodes,":[70],"which":[71,140],"maps":[72,89,284],"under":[73],"multiple":[74],"non-linear":[75],"transformations":[76,240],"to":[77,97,179,267],"one":[81],"output":[82],"node.":[83],"analogy,":[85],"ODE":[88],"affine":[91,99,238],"linear":[92,100,239],"transformation":[93,101],"its":[103],"time-":[104],"T":[105],"map.":[106],"show":[108,203],"that,":[109,204],"depending":[110,194],"on":[111,195,270,277,295],"specific":[113,197],"structure":[114,298],"network,":[117],"map":[120,302],"has":[121,229],"different":[122],"properties":[123,133,252],"regarding":[124],"existence":[126],"regularity":[128,193],"critical":[130,146,153,187,217],"points.":[131],"These":[132],"characterized":[136],"via":[137],"Morse":[138],"functions,":[139],"are":[141,253],"functions":[143],"where":[144],"every":[145],"point":[147,218],"non-degenerate.":[149],"prove":[151],"that":[152,186],"points":[154,188],"cannot":[155],"exist":[156],"if":[157,221,233],"dimension":[159,169],"hidden":[162],"layer":[163],"monotonically":[165],"decreasing":[166],"phase":[172],"space":[173],"smaller":[175],"than":[176,313],"equal":[178],"dimension.":[182],"case":[185],"exist,":[189],"we":[190],"classify":[191],"their":[192],"architecture":[198],"network.":[201],"except":[205],"for":[206,222,234],"Lebesgue":[208],"measure":[209],"zero":[210],"set":[211],"weight":[214],"space,":[215],"each":[216,246],"non-degenerate":[220],"networks,":[225],"underlying":[227],"graph":[228],"no":[230],"bottleneck,":[231],"ODEs,":[236],"used":[241],"full":[243],"rank.":[244],"For":[245],"type":[247],"architecture,":[249],"proven":[251],"comparable":[254],"cases.":[261],"The":[262],"established":[263],"theorems":[264],"allow":[265],"us":[266],"formulate":[268],"results":[269],"universal":[271,274],"embedding":[272],"approximation,":[275],"i.e.,":[276],"exact":[279],"approximate":[281],"representation":[282],"by":[285],"ODEs.":[290],"Our":[291],"dynamical":[292],"systems":[293],"viewpoint":[294],"geometric":[297],"provides":[303],"fundamental":[305],"understanding":[306],"why":[308],"certain":[309],"architectures":[310],"perform":[311],"better":[312],"others.":[314]},"counts_by_year":[],"updated_date":"2026-05-17T08:19:37.847499","created_date":"2024-05-17T00:00:00"}