{"id":"https://openalex.org/W4413332308","doi":"https://doi.org/10.3390/e27080878","title":"Geometric Neural Ordinary Differential Equations: From Manifolds to Lie Groups","display_name":"Geometric Neural Ordinary Differential Equations: From Manifolds to Lie Groups","publication_year":2025,"publication_date":"2025-08-19","ids":{"openalex":"https://openalex.org/W4413332308","doi":"https://doi.org/10.3390/e27080878","pmid":"https://pubmed.ncbi.nlm.nih.gov/40870350"},"language":"en","primary_location":{"id":"doi:10.3390/e27080878","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e27080878","pdf_url":"https://www.mdpi.com/1099-4300/27/8/878/pdf?version=1755616773","source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Entropy","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj","pubmed"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/1099-4300/27/8/878/pdf?version=1755616773","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5032431599","display_name":"Yannik P. Wotte","orcid":"https://orcid.org/0000-0003-3002-4479"},"institutions":[{"id":"https://openalex.org/I94624287","display_name":"University of Twente","ror":"https://ror.org/006hf6230","country_code":"NL","type":"education","lineage":["https://openalex.org/I94624287"]}],"countries":["NL"],"is_corresponding":true,"raw_author_name":"Yannik P. Wotte","raw_affiliation_strings":["Robotics and Mechatronics, EEMCS, University of Twente (UT), Drienerlolaan 5, 7522 NB Enschede, The Netherlands"],"raw_orcid":"https://orcid.org/0000-0003-3002-4479","affiliations":[{"raw_affiliation_string":"Robotics and Mechatronics, EEMCS, University of Twente (UT), Drienerlolaan 5, 7522 NB Enschede, The Netherlands","institution_ids":["https://openalex.org/I94624287"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5075056195","display_name":"Federico Califano","orcid":"https://orcid.org/0000-0001-8693-0900"},"institutions":[{"id":"https://openalex.org/I94624287","display_name":"University of Twente","ror":"https://ror.org/006hf6230","country_code":"NL","type":"education","lineage":["https://openalex.org/I94624287"]}],"countries":["NL"],"is_corresponding":false,"raw_author_name":"Federico Califano","raw_affiliation_strings":["Robotics and Mechatronics, EEMCS, University of Twente (UT), Drienerlolaan 5, 7522 NB Enschede, The Netherlands"],"raw_orcid":"https://orcid.org/0000-0001-8693-0900","affiliations":[{"raw_affiliation_string":"Robotics and Mechatronics, EEMCS, University of Twente (UT), Drienerlolaan 5, 7522 NB Enschede, The Netherlands","institution_ids":["https://openalex.org/I94624287"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5047812798","display_name":"Stefano Stramigioli","orcid":"https://orcid.org/0000-0001-8212-7387"},"institutions":[{"id":"https://openalex.org/I94624287","display_name":"University of Twente","ror":"https://ror.org/006hf6230","country_code":"NL","type":"education","lineage":["https://openalex.org/I94624287"]}],"countries":["NL"],"is_corresponding":false,"raw_author_name":"Stefano Stramigioli","raw_affiliation_strings":["Robotics and Mechatronics, EEMCS, University of Twente (UT), Drienerlolaan 5, 7522 NB Enschede, The Netherlands"],"raw_orcid":"https://orcid.org/0000-0001-8212-7387","affiliations":[{"raw_affiliation_string":"Robotics and Mechatronics, EEMCS, University of Twente (UT), Drienerlolaan 5, 7522 NB Enschede, The Netherlands","institution_ids":["https://openalex.org/I94624287"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5032431599"],"corresponding_institution_ids":["https://openalex.org/I94624287"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.15113818,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"27","issue":"8","first_page":"878","last_page":"878"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11206","display_name":"Model Reduction and Neural Networks","score":0.9995999932289124,"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":0.9995999932289124,"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/T10320","display_name":"Neural Networks and Applications","score":0.9229999780654907,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"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/ode","display_name":"Ode","score":0.8480321764945984},{"id":"https://openalex.org/keywords/lie-group","display_name":"Lie group","score":0.7104945182800293},{"id":"https://openalex.org/keywords/ordinary-differential-equation","display_name":"Ordinary differential equation","score":0.6382779479026794},{"id":"https://openalex.org/keywords/manifold","display_name":"Manifold (fluid mechanics)","score":0.5655379891395569},{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.5578432083129883},{"id":"https://openalex.org/keywords/differentiable-function","display_name":"Differentiable function","score":0.5359913110733032},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.521787166595459},{"id":"https://openalex.org/keywords/dynamical-systems-theory","display_name":"Dynamical systems theory","score":0.47493842244148254},{"id":"https://openalex.org/keywords/differential-topology","display_name":"Differential topology","score":0.4190019369125366},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.4084862172603607},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.3716500401496887},{"id":"https://openalex.org/keywords/differential-equation","display_name":"Differential equation","score":0.34515270590782166},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.3415898084640503},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.3259120285511017},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.22301265597343445},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.20868921279907227},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.10789120197296143},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.08715963363647461},{"id":"https://openalex.org/keywords/ricci-flat-manifold","display_name":"Ricci-flat manifold","score":0.06337961554527283}],"concepts":[{"id":"https://openalex.org/C34862557","wikidata":"https://www.wikidata.org/wiki/Q178985","display_name":"Ode","level":2,"score":0.8480321764945984},{"id":"https://openalex.org/C187915474","wikidata":"https://www.wikidata.org/wiki/Q622679","display_name":"Lie group","level":2,"score":0.7104945182800293},{"id":"https://openalex.org/C51544822","wikidata":"https://www.wikidata.org/wiki/Q465274","display_name":"Ordinary differential equation","level":3,"score":0.6382779479026794},{"id":"https://openalex.org/C529865628","wikidata":"https://www.wikidata.org/wiki/Q1790740","display_name":"Manifold (fluid mechanics)","level":2,"score":0.5655379891395569},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.5578432083129883},{"id":"https://openalex.org/C202615002","wikidata":"https://www.wikidata.org/wiki/Q783507","display_name":"Differentiable function","level":2,"score":0.5359913110733032},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.521787166595459},{"id":"https://openalex.org/C79379906","wikidata":"https://www.wikidata.org/wiki/Q3174497","display_name":"Dynamical systems theory","level":2,"score":0.47493842244148254},{"id":"https://openalex.org/C516171914","wikidata":"https://www.wikidata.org/wiki/Q1224402","display_name":"Differential topology","level":5,"score":0.4190019369125366},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.4084862172603607},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.3716500401496887},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.34515270590782166},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.3415898084640503},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.3259120285511017},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.22301265597343445},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.20868921279907227},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.10789120197296143},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.08715963363647461},{"id":"https://openalex.org/C203934109","wikidata":"https://www.wikidata.org/wiki/Q7322952","display_name":"Ricci-flat manifold","level":4,"score":0.06337961554527283},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","level":0,"score":0.0},{"id":"https://openalex.org/C78519656","wikidata":"https://www.wikidata.org/wiki/Q101333","display_name":"Mechanical engineering","level":1,"score":0.0},{"id":"https://openalex.org/C195065555","wikidata":"https://www.wikidata.org/wiki/Q214881","display_name":"Curvature","level":2,"score":0.0},{"id":"https://openalex.org/C12520029","wikidata":"https://www.wikidata.org/wiki/Q1147161","display_name":"Scalar curvature","level":3,"score":0.0}],"mesh":[],"locations_count":5,"locations":[{"id":"doi:10.3390/e27080878","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e27080878","pdf_url":"https://www.mdpi.com/1099-4300/27/8/878/pdf?version=1755616773","source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Entropy","raw_type":"journal-article"},{"id":"pmid:40870350","is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/40870350","pdf_url":null,"source":{"id":"https://openalex.org/S4306525036","display_name":"PubMed","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https://openalex.org/I1299303238"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Entropy (Basel, Switzerland)","raw_type":null},{"id":"pmh:oai:ris.utwente.nl:publications/989c3e86-86b4-46cf-be52-067f87563a65","is_oa":true,"landing_page_url":"https://research.utwente.nl/en/publications/989c3e86-86b4-46cf-be52-067f87563a65","pdf_url":"https://ris.utwente.nl/ws/files/506592648/entropy-27-00878-v2.pdf","source":{"id":"https://openalex.org/S4406922991","display_name":"University of Twente Research Information","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":"Wotte, Y P, Califano, F & Stramigioli, S 2025, 'Geometric Neural Ordinary Differential Equations : From Manifolds to Lie Groups', Entropy, vol. 27, no. 8, 878. https://doi.org/10.3390/e27080878","raw_type":"info:eu-repo/semantics/publishedVersion"},{"id":"pmh:oai:doaj.org/article:240c135d171f4b628ff35634eba3f2eb","is_oa":true,"landing_page_url":"https://doaj.org/article/240c135d171f4b628ff35634eba3f2eb","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","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-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Entropy, Vol 27, Iss 8, p 878 (2025)","raw_type":"article"},{"id":"pmh:oai:europepmc.org:11193404","is_oa":true,"landing_page_url":"https://www.ncbi.nlm.nih.gov/pmc/articles/12385718","pdf_url":null,"source":{"id":"https://openalex.org/S4306400806","display_name":"Europe PMC (PubMed Central)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1303153112","host_organization_name":"European Bioinformatics Institute","host_organization_lineage":["https://openalex.org/I1303153112"],"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"}],"best_oa_location":{"id":"doi:10.3390/e27080878","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e27080878","pdf_url":"https://www.mdpi.com/1099-4300/27/8/878/pdf?version=1755616773","source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Entropy","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4413332308.pdf","grobid_xml":"https://content.openalex.org/works/W4413332308.grobid-xml"},"referenced_works_count":50,"referenced_works":["https://openalex.org/W1524590822","https://openalex.org/W1557324374","https://openalex.org/W1969242270","https://openalex.org/W1980330256","https://openalex.org/W1994616650","https://openalex.org/W2014613771","https://openalex.org/W2134385637","https://openalex.org/W2151249889","https://openalex.org/W2902809050","https://openalex.org/W2930930439","https://openalex.org/W2996897098","https://openalex.org/W3008051283","https://openalex.org/W3014592032","https://openalex.org/W3025713978","https://openalex.org/W3093884544","https://openalex.org/W3102864525","https://openalex.org/W3118584630","https://openalex.org/W3169337152","https://openalex.org/W3175104376","https://openalex.org/W3204466930","https://openalex.org/W4243055393","https://openalex.org/W4310634128","https://openalex.org/W4313214554","https://openalex.org/W4313592055","https://openalex.org/W4381248243","https://openalex.org/W4381686075","https://openalex.org/W4381851224","https://openalex.org/W4385222493","https://openalex.org/W4386972987","https://openalex.org/W4388917619","https://openalex.org/W4399658447","https://openalex.org/W4400644035","https://openalex.org/W4401024547","https://openalex.org/W4401396180","https://openalex.org/W4405354153","https://openalex.org/W4408673128","https://openalex.org/W4408815123","https://openalex.org/W4408918775","https://openalex.org/W4409364195","https://openalex.org/W6679244297","https://openalex.org/W6752307458","https://openalex.org/W6760904531","https://openalex.org/W6760979436","https://openalex.org/W6763340073","https://openalex.org/W6774419874","https://openalex.org/W6775127026","https://openalex.org/W6779435032","https://openalex.org/W6810971026","https://openalex.org/W6838988421","https://openalex.org/W6996403240"],"related_works":["https://openalex.org/W2050640136","https://openalex.org/W4237019278","https://openalex.org/W4232354475","https://openalex.org/W4238738468","https://openalex.org/W611253958","https://openalex.org/W3111356014","https://openalex.org/W2026220633","https://openalex.org/W939179477","https://openalex.org/W4388195638","https://openalex.org/W3181613066"],"abstract_inverted_index":{"Neural":[0],"ordinary":[1],"differential":[2],"equations":[3],"(neural":[4],"ODEs)":[5],"are":[6,47],"a":[7,63,72,97],"well-established":[8],"tool":[9],"for":[10,44,56],"optimizing":[11],"the":[12,84,87,106],"parameters":[13],"of":[14,30,66,90,100],"dynamical":[15,28],"systems,":[16],"with":[17],"applications":[18],"in":[19],"image":[20],"classification,":[21],"optimal":[22],"control,":[23],"and":[24,37,61],"physics":[25],"learning.":[26],"Although":[27],"systems":[29],"interest":[31],"often":[32],"evolve":[33],"on":[34,50,59,93],"Lie":[35,94,107],"groups":[36],"more":[38],"general":[39],"differentiable":[40,79],"manifolds,":[41],"theoretical":[42],"results":[43,55,68,85],"neural":[45,57,91,102],"ODEs":[46,58,92,103],"frequently":[48],"phrased":[49],"Rn.":[51],"We":[52,81],"collect":[53],"recent":[54,88],"manifolds":[60],"present":[62],"unifying":[64],"derivation":[65],"various":[67],"that":[69,104],"serves":[70],"as":[71],"tutorial":[73],"to":[74,78,86],"extend":[75,83],"existing":[76],"methods":[77],"manifolds.":[80],"also":[82],"class":[89],"groups,":[95],"highlighting":[96],"non-trivial":[98],"extension":[99],"manifold":[101],"exploits":[105],"group":[108],"structure.":[109]},"counts_by_year":[],"updated_date":"2026-05-21T09:19:25.381259","created_date":"2025-10-10T00:00:00"}