{"id":"https://openalex.org/W4396600857","doi":"https://doi.org/10.1090/mcom/4120","title":"Machine learning of continuous and discrete variational ODEs with convergence guarantee and uncertainty quantification","display_name":"Machine learning of continuous and discrete variational ODEs with convergence guarantee and uncertainty quantification","publication_year":2025,"publication_date":"2025-06-26","ids":{"openalex":"https://openalex.org/W4396600857","doi":"https://doi.org/10.1090/mcom/4120"},"language":"en","primary_location":{"id":"doi:10.1090/mcom/4120","is_oa":false,"landing_page_url":"https://doi.org/10.1090/mcom/4120","pdf_url":null,"source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2404.19626","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5015121881","display_name":"Christian Offen","orcid":"https://orcid.org/0000-0002-5940-8057"},"institutions":[{"id":"https://openalex.org/I206945453","display_name":"Paderborn University","ror":"https://ror.org/058kzsd48","country_code":"DE","type":"education","lineage":["https://openalex.org/I206945453"]}],"countries":["DE"],"is_corresponding":true,"raw_author_name":"Christian Offen","raw_affiliation_strings":["Department of Mathematics, Paderborn University, Warburger Str. 100, 33098 Paderborn, Germany"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Paderborn University, Warburger Str. 100, 33098 Paderborn, Germany","institution_ids":["https://openalex.org/I206945453"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5015121881"],"corresponding_institution_ids":["https://openalex.org/I206945453"],"apc_list":null,"apc_paid":null,"fwci":0.6481,"has_fulltext":true,"cited_by_count":2,"citation_normalized_percentile":{"value":0.57503431,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":97,"max":98},"biblio":{"volume":"95","issue":"361","first_page":"2339","last_page":"2386"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11583","display_name":"Advanced Measurement and Metrology Techniques","score":0.883899986743927,"subfield":{"id":"https://openalex.org/subfields/2210","display_name":"Mechanical Engineering"},"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/T11583","display_name":"Advanced Measurement and Metrology Techniques","score":0.883899986743927,"subfield":{"id":"https://openalex.org/subfields/2210","display_name":"Mechanical Engineering"},"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/T11115","display_name":"Topology Optimization in Engineering","score":0.8598999977111816,"subfield":{"id":"https://openalex.org/subfields/2205","display_name":"Civil and Structural Engineering"},"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/T11245","display_name":"Advanced Numerical Analysis Techniques","score":0.7745000123977661,"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/ode","display_name":"Ode","score":0.8776167035102844},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.707642674446106},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.49478429555892944},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.4482179880142212},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.43907949328422546},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.4283667504787445},{"id":"https://openalex.org/keywords/mathematical-economics","display_name":"Mathematical economics","score":0.33218812942504883},{"id":"https://openalex.org/keywords/economics","display_name":"Economics","score":0.25298628211021423}],"concepts":[{"id":"https://openalex.org/C34862557","wikidata":"https://www.wikidata.org/wiki/Q178985","display_name":"Ode","level":2,"score":0.8776167035102844},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.707642674446106},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.49478429555892944},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.4482179880142212},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.43907949328422546},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.4283667504787445},{"id":"https://openalex.org/C144237770","wikidata":"https://www.wikidata.org/wiki/Q747534","display_name":"Mathematical economics","level":1,"score":0.33218812942504883},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.25298628211021423},{"id":"https://openalex.org/C50522688","wikidata":"https://www.wikidata.org/wiki/Q189833","display_name":"Economic growth","level":1,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1090/mcom/4120","is_oa":false,"landing_page_url":"https://doi.org/10.1090/mcom/4120","pdf_url":null,"source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:2404.19626","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2404.19626","pdf_url":"https://arxiv.org/pdf/2404.19626","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"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:oai:pure.atira.dk:openaire_cris_publications/376e6f70-e377-445f-946e-f887a2c28686","is_oa":true,"landing_page_url":"https://research.birmingham.ac.uk/en/publications/376e6f70-e377-445f-946e-f887a2c28686","pdf_url":null,"source":{"id":"https://openalex.org/S4306402634","display_name":"University of Birmingham Research Portal (University of Birmingham)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I79619799","host_organization_name":"University of Birmingham","host_organization_lineage":["https://openalex.org/I79619799"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Offen, C 2025, 'Machine learning of continuous and discrete variational ODEs with convergence guarantee and uncertainty quantification', Mathematics of Computation. https://doi.org/10.1090/mcom/4120","raw_type":"article"},{"id":"doi:10.48550/arxiv.2404.19626","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2404.19626","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"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":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2404.19626","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2404.19626","pdf_url":"https://arxiv.org/pdf/2404.19626","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"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"},"sustainable_development_goals":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320328599","display_name":"Ministerium f\u00fcr Kultur und Wissenschaft des Landes Nordrhein-Westfalen","ror":null}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4396600857.pdf","grobid_xml":"https://content.openalex.org/works/W4396600857.grobid-xml"},"referenced_works_count":43,"referenced_works":["https://openalex.org/W2041099140","https://openalex.org/W2126479355","https://openalex.org/W2027506796","https://openalex.org/W4376122518","https://openalex.org/W2966473221","https://openalex.org/W2120575449","https://openalex.org/W2031392034","https://openalex.org/W2023149829","https://openalex.org/W3138278823","https://openalex.org/W4210889506","https://openalex.org/W3173623556","https://openalex.org/W4379378243","https://openalex.org/W1531605312","https://openalex.org/W4390535901","https://openalex.org/W4315488839","https://openalex.org/W3166279512","https://openalex.org/W2166318576","https://openalex.org/W2083364319","https://openalex.org/W4411338519","https://openalex.org/W3081814969","https://openalex.org/W4388918580","https://openalex.org/W2905558281","https://openalex.org/W2050061431","https://openalex.org/W2175634517","https://openalex.org/W1524590822","https://openalex.org/W2170039048","https://openalex.org/W1988473367","https://openalex.org/W4294946167","https://openalex.org/W4206724591","https://openalex.org/W4321276973","https://openalex.org/W4390661394","https://openalex.org/W6851142841","https://openalex.org/W2979712029","https://openalex.org/W3104709959","https://openalex.org/W1571870753","https://openalex.org/W2619381903","https://openalex.org/W3086666652","https://openalex.org/W4231103245","https://openalex.org/W2089245375","https://openalex.org/W3168833755","https://openalex.org/W3156927273","https://openalex.org/W2276138198","https://openalex.org/W2070330278"],"related_works":["https://openalex.org/W2371448224","https://openalex.org/W3134495997","https://openalex.org/W2363680170","https://openalex.org/W2376428685","https://openalex.org/W4256082577","https://openalex.org/W625783435","https://openalex.org/W2048742619","https://openalex.org/W2384641672","https://openalex.org/W2742179975","https://openalex.org/W2362647823"],"abstract_inverted_index":{"The":[0,17,114],"article":[1,115],"introduces":[2],"a":[3,76,134,146],"method":[4,18,66],"to":[5,51,62,123,148],"learn":[6],"dynamical":[7],"systems":[8],"that":[9,90],"are":[10,59],"governed":[11],"by":[12,36],"Euler\u2013Lagrange":[13],"equations":[14],"from":[15],"data.":[16],"is":[19,91,105],"based":[20],"on":[21],"Gaussian":[22],"process":[23],"regression":[24],"and":[25,31,53,101,119,141],"identifies":[26],"continuous":[27],"or":[28],"discrete":[29],"Lagrangians":[30,132],"is,":[32],"therefore,":[33],"structure":[34],"preserving":[35],"design.":[37],"A":[38],"rigorous":[39],"proof":[40],"of":[41,70,78,87,97,106,111,126,130,137,145],"convergence":[42,57,63],"as":[43],"the":[44,65,94,109,124,127],"distance":[45],"between":[46],"observation":[47],"data":[48],"points":[49],"converges":[50],"zero":[52],"lower":[54],"bounds":[55],"for":[56,68],"rates":[58],"provided.":[60],"Next":[61],"guarantees,":[64],"allows":[67],"quantification":[69,86],"model":[71],"uncertainty,":[72],"which":[73,104],"can":[74],"provide":[75,83],"basis":[77],"adaptive":[79],"sampling":[80],"techniques.":[81],"We":[82],"efficient":[84],"uncertainty":[85],"any":[88],"observable":[89],"linear":[92],"in":[93,108,152],"Lagrangian,":[95],"including":[96],"Hamiltonian":[98],"functions":[99],"(energy)":[100],"symplectic":[102],"structures,":[103],"interest":[107],"context":[110],"system":[112],"identification.":[113],"overcomes":[116],"major":[117],"practical":[118],"theoretical":[120],"difficulties":[121],"related":[122],"ill-posedness":[125],"identification":[128],"task":[129],"(discrete)":[131],"through":[133,142],"careful":[135],"design":[136],"geometric":[138],"regularisation":[139],"strategies":[140],"an":[143],"exploit":[144],"relation":[147],"convex":[149],"minimisation":[150],"problems":[151],"reproducing":[153],"kernel":[154],"Hilbert":[155],"spaces.":[156]},"counts_by_year":[{"year":2026,"cited_by_count":2}],"updated_date":"2026-08-26T07:47:46.906454","created_date":"2024-05-03T00:00:00"}