{"id":"https://openalex.org/W7133318221","doi":"https://doi.org/10.1007/s10444-026-10296-6","title":"Multi-fidelity learning of reduced order models for parabolic PDE constrained optimization","display_name":"Multi-fidelity learning of reduced order models for parabolic PDE constrained optimization","publication_year":2026,"publication_date":"2026-03-03","ids":{"openalex":"https://openalex.org/W7133318221","doi":"https://doi.org/10.1007/s10444-026-10296-6"},"language":"en","primary_location":{"id":"doi:10.1007/s10444-026-10296-6","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10444-026-10296-6","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10444-026-10296-6.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":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007/s10444-026-10296-6.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5127925903","display_name":"Benedikt Klein","orcid":null},"institutions":[{"id":"https://openalex.org/I22465464","display_name":"University of M\u00fcnster","ror":"https://ror.org/00pd74e08","country_code":"DE","type":"education","lineage":["https://openalex.org/I22465464"]},{"id":"https://openalex.org/I63274643","display_name":"FH M\u00fcnster","ror":"https://ror.org/00pv45a02","country_code":"DE","type":"education","lineage":["https://openalex.org/I63274643"]}],"countries":["DE"],"is_corresponding":true,"raw_author_name":"Benedikt Klein","raw_affiliation_strings":["Mathematics M\u00fcnster, University of M\u00fcnster, Einsteinstrasse 62, M\u00fcnster, 48149, Germany"],"raw_orcid":"https://orcid.org/0009-0004-1909-3574","affiliations":[{"raw_affiliation_string":"Mathematics M\u00fcnster, University of M\u00fcnster, Einsteinstrasse 62, M\u00fcnster, 48149, Germany","institution_ids":["https://openalex.org/I22465464","https://openalex.org/I63274643"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5039193717","display_name":"Mario Ohlberger","orcid":"https://orcid.org/0000-0002-6260-3574"},"institutions":[{"id":"https://openalex.org/I22465464","display_name":"University of M\u00fcnster","ror":"https://ror.org/00pd74e08","country_code":"DE","type":"education","lineage":["https://openalex.org/I22465464"]},{"id":"https://openalex.org/I63274643","display_name":"FH M\u00fcnster","ror":"https://ror.org/00pv45a02","country_code":"DE","type":"education","lineage":["https://openalex.org/I63274643"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Mario Ohlberger","raw_affiliation_strings":["Mathematics M\u00fcnster, University of M\u00fcnster, Einsteinstrasse 62, M\u00fcnster, 48149, Germany"],"raw_orcid":"https://orcid.org/0000-0002-6260-3574","affiliations":[{"raw_affiliation_string":"Mathematics M\u00fcnster, University of M\u00fcnster, Einsteinstrasse 62, M\u00fcnster, 48149, Germany","institution_ids":["https://openalex.org/I22465464","https://openalex.org/I63274643"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5127925903"],"corresponding_institution_ids":["https://openalex.org/I22465464","https://openalex.org/I63274643"],"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":0.43993124,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"52","issue":"2","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":0.97079998254776,"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":0.97079998254776,"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/T10339","display_name":"Advanced Numerical Methods in Computational Mathematics","score":0.00419999985024333,"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/T11416","display_name":"Numerical methods for differential equations","score":0.003599999938160181,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/parameterized-complexity","display_name":"Parameterized complexity","score":0.7099999785423279},{"id":"https://openalex.org/keywords/hierarchy","display_name":"Hierarchy","score":0.6668999791145325},{"id":"https://openalex.org/keywords/a-priori-and-a-posteriori","display_name":"A priori and a posteriori","score":0.614300012588501},{"id":"https://openalex.org/keywords/estimator","display_name":"Estimator","score":0.5759000182151794},{"id":"https://openalex.org/keywords/constrained-optimization","display_name":"Constrained optimization","score":0.4797999858856201},{"id":"https://openalex.org/keywords/order","display_name":"Order (exchange)","score":0.4607999920845032},{"id":"https://openalex.org/keywords/optimization-problem","display_name":"Optimization problem","score":0.4077000021934509},{"id":"https://openalex.org/keywords/computational-science-and-engineering","display_name":"Computational Science and Engineering","score":0.4049000144004822}],"concepts":[{"id":"https://openalex.org/C165464430","wikidata":"https://www.wikidata.org/wiki/Q1570441","display_name":"Parameterized complexity","level":2,"score":0.7099999785423279},{"id":"https://openalex.org/C31170391","wikidata":"https://www.wikidata.org/wiki/Q188619","display_name":"Hierarchy","level":2,"score":0.6668999791145325},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.652999997138977},{"id":"https://openalex.org/C75553542","wikidata":"https://www.wikidata.org/wiki/Q178161","display_name":"A priori and a posteriori","level":2,"score":0.614300012588501},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6068000197410583},{"id":"https://openalex.org/C185429906","wikidata":"https://www.wikidata.org/wiki/Q1130160","display_name":"Estimator","level":2,"score":0.5759000182151794},{"id":"https://openalex.org/C55660270","wikidata":"https://www.wikidata.org/wiki/Q5164377","display_name":"Constrained optimization","level":2,"score":0.4797999858856201},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.4607999920845032},{"id":"https://openalex.org/C137836250","wikidata":"https://www.wikidata.org/wiki/Q984063","display_name":"Optimization problem","level":2,"score":0.4077000021934509},{"id":"https://openalex.org/C68597687","wikidata":"https://www.wikidata.org/wiki/Q362601","display_name":"Computational Science and Engineering","level":2,"score":0.4049000144004822},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.4043999910354614},{"id":"https://openalex.org/C186867907","wikidata":"https://www.wikidata.org/wiki/Q1575416","display_name":"Parabolic partial differential equation","level":3,"score":0.4002000093460083},{"id":"https://openalex.org/C2780801425","wikidata":"https://www.wikidata.org/wiki/Q5164392","display_name":"Construct (python library)","level":2,"score":0.37070000171661377},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.3587000072002411},{"id":"https://openalex.org/C2989514635","wikidata":"https://www.wikidata.org/wiki/Q5164377","display_name":"Constrained optimization problem","level":3,"score":0.3508000075817108},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.34610000252723694},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.31279999017715454},{"id":"https://openalex.org/C2776036281","wikidata":"https://www.wikidata.org/wiki/Q48769818","display_name":"Constraint (computer-aided design)","level":2,"score":0.30880001187324524},{"id":"https://openalex.org/C3309286","wikidata":"https://www.wikidata.org/wiki/Q4907693","display_name":"Bilevel optimization","level":3,"score":0.29350000619888306},{"id":"https://openalex.org/C49937458","wikidata":"https://www.wikidata.org/wiki/Q2599292","display_name":"Probabilistic logic","level":2,"score":0.2838999927043915},{"id":"https://openalex.org/C32230216","wikidata":"https://www.wikidata.org/wiki/Q7882499","display_name":"Uncertainty quantification","level":2,"score":0.2815999984741211},{"id":"https://openalex.org/C48753275","wikidata":"https://www.wikidata.org/wiki/Q11216","display_name":"Numerical analysis","level":2,"score":0.25060001015663147}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1007/s10444-026-10296-6","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10444-026-10296-6","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10444-026-10296-6.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"}],"best_oa_location":{"id":"doi:10.1007/s10444-026-10296-6","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10444-026-10296-6","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10444-026-10296-6.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/G1177974802","display_name":null,"funder_award_id":"EXC 2044-390685587","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"},{"id":"https://openalex.org/G1652504919","display_name":null,"funder_award_id":"EXC 2044","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"},{"id":"https://openalex.org/G2110285270","display_name":null,"funder_award_id":"2044-390685587","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"},{"id":"https://openalex.org/G2817606592","display_name":null,"funder_award_id":"390685587","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"},{"id":"https://openalex.org/G8646566273","display_name":null,"funder_award_id":"EXC 2044390685587","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/F4320322957","display_name":"Westf\u00e4lische Wilhelms-Universit\u00e4t M\u00fcnster","ror":"https://ror.org/00pd74e08"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7133318221.pdf","grobid_xml":"https://content.openalex.org/works/W7133318221.grobid-xml"},"referenced_works_count":43,"referenced_works":["https://openalex.org/W93748562","https://openalex.org/W2005799932","https://openalex.org/W2010160985","https://openalex.org/W2053197085","https://openalex.org/W2085335060","https://openalex.org/W2089245375","https://openalex.org/W2104225554","https://openalex.org/W2110917150","https://openalex.org/W2123224804","https://openalex.org/W2128702576","https://openalex.org/W2226462370","https://openalex.org/W2230918517","https://openalex.org/W2277401480","https://openalex.org/W2504586482","https://openalex.org/W2734721340","https://openalex.org/W2749106469","https://openalex.org/W2753365189","https://openalex.org/W2760972773","https://openalex.org/W2766298346","https://openalex.org/W2811395263","https://openalex.org/W2890578718","https://openalex.org/W2899283552","https://openalex.org/W2919115771","https://openalex.org/W2963473801","https://openalex.org/W3034891723","https://openalex.org/W3048421397","https://openalex.org/W3146713108","https://openalex.org/W3162220519","https://openalex.org/W3167031531","https://openalex.org/W3206057793","https://openalex.org/W4308658042","https://openalex.org/W4311780061","https://openalex.org/W4353056901","https://openalex.org/W4376128927","https://openalex.org/W4381415745","https://openalex.org/W4385198556","https://openalex.org/W4388692980","https://openalex.org/W4391435972","https://openalex.org/W4395081858","https://openalex.org/W4398238760","https://openalex.org/W4398757386","https://openalex.org/W4402976369","https://openalex.org/W7106270155"],"related_works":[],"abstract_inverted_index":{"Abstract":[0],"This":[1],"article":[2],"builds":[3],"on":[4],"the":[5,60,113,116],"recently":[6],"proposed":[7,84,117],"RB-ML-ROM":[8],"approach":[9,36],"for":[10,22,37],"parameterized":[11],"parabolic":[12,24],"PDEs":[13],"and":[14,78],"proposes":[15],"a":[16,32,51,70,74,79,95,97],"novel":[17],"hierarchical":[18,85],"trust":[19,104],"region":[20,105],"algorithm":[21],"solving":[23],"PDE":[25],"constrained":[26],"optimization":[27,62],"problems.":[28],"Instead":[29],"of":[30,54,69,115],"using":[31],"traditional":[33],"offline/online":[34],"splitting":[35],"model":[38,67,77],"order":[39,56,72,76],"reduction,":[40],"we":[41],"adopt":[42],"an":[43,102],"active":[44],"learning":[45,81],"or":[46],"enrichment":[47],"strategy":[48],"to":[49,111],"construct":[50],"multi-fidelity":[52,65],"hierarchy":[53,90],"reduced":[55,75],"models":[57],"on-the-fly":[58],"during":[59],"outer":[61],"loop.":[63],"The":[64,83],"surrogate":[66],"consists":[68],"full":[71],"model,":[73],"machine":[80],"model.":[82],"framework":[86],"adaptively":[87],"updates":[88],"its":[89],"when":[91],"querying":[92],"parameters,":[93],"utilizing":[94],"rigorous":[96],"posteriori":[98],"error":[99],"estimator":[100],"in":[101],"error-aware":[103],"framework.":[106],"Numerical":[107],"experiments":[108],"are":[109],"given":[110],"demonstrate":[112],"efficiency":[114],"approach.":[118]},"counts_by_year":[],"updated_date":"2026-05-05T08:41:31.759640","created_date":"2026-03-04T00:00:00"}
