{"id":"https://openalex.org/W2105712817","doi":"https://doi.org/10.1109/tnn.2008.2011267","title":"Neural Network Output Optimization Using Interval Analysis","display_name":"Neural Network Output Optimization Using Interval Analysis","publication_year":2009,"publication_date":"2009-02-26","ids":{"openalex":"https://openalex.org/W2105712817","doi":"https://doi.org/10.1109/tnn.2008.2011267","mag":"2105712817","pmid":"https://pubmed.ncbi.nlm.nih.gov/19244020"},"language":"en","primary_location":{"id":"doi:10.1109/tnn.2008.2011267","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tnn.2008.2011267","pdf_url":null,"source":{"id":"https://openalex.org/S42080949","display_name":"IEEE Transactions on Neural Networks","issn_l":"1045-9227","issn":["1045-9227","1941-0093"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Neural Networks","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","pubmed"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5016458559","display_name":"Elwin de Weerdt","orcid":"https://orcid.org/0000-0001-8199-4737"},"institutions":[{"id":"https://openalex.org/I98358874","display_name":"Delft University of Technology","ror":"https://ror.org/02e2c7k09","country_code":"NL","type":"education","lineage":["https://openalex.org/I98358874"]}],"countries":["NL"],"is_corresponding":true,"raw_author_name":"E. de Weerdt","raw_affiliation_strings":["Control and Simulation Division, Faculty of Aerospace Engineering, Delft University of Technology, Delft, The Netherlands. E.deWeerdt@TUDelft.nl","Control & Simulation Div., Delft Univ. of Technol., Delft"],"affiliations":[{"raw_affiliation_string":"Control and Simulation Division, Faculty of Aerospace Engineering, Delft University of Technology, Delft, The Netherlands. E.deWeerdt@TUDelft.nl","institution_ids":["https://openalex.org/I98358874"]},{"raw_affiliation_string":"Control & Simulation Div., Delft Univ. of Technol., Delft","institution_ids":["https://openalex.org/I98358874"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5041758470","display_name":"Q. P. Chu","orcid":"https://orcid.org/0000-0002-1260-3672"},"institutions":[{"id":"https://openalex.org/I98358874","display_name":"Delft University of Technology","ror":"https://ror.org/02e2c7k09","country_code":"NL","type":"education","lineage":["https://openalex.org/I98358874"]}],"countries":["NL"],"is_corresponding":false,"raw_author_name":"Q.P. Chu","raw_affiliation_strings":["Control and Simulation Division, Faculty of AeroSpace Engineering, Delft University of Technnology, Delft, Netherlands","Control & Simulation Div., Delft Univ. of Technol., Delft"],"affiliations":[{"raw_affiliation_string":"Control and Simulation Division, Faculty of AeroSpace Engineering, Delft University of Technnology, Delft, Netherlands","institution_ids":["https://openalex.org/I98358874"]},{"raw_affiliation_string":"Control & Simulation Div., Delft Univ. of Technol., Delft","institution_ids":["https://openalex.org/I98358874"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5112104825","display_name":"J.A. Mulder","orcid":null},"institutions":[{"id":"https://openalex.org/I98358874","display_name":"Delft University of Technology","ror":"https://ror.org/02e2c7k09","country_code":"NL","type":"education","lineage":["https://openalex.org/I98358874"]}],"countries":["NL"],"is_corresponding":false,"raw_author_name":"J.A. Mulder","raw_affiliation_strings":["Control and Simulation Division, Faculty of AeroSpace Engineering, Delft University of Technnology, Delft, Netherlands","Control & Simulation Div., Delft Univ. of Technol., Delft"],"affiliations":[{"raw_affiliation_string":"Control and Simulation Division, Faculty of AeroSpace Engineering, Delft University of Technnology, Delft, Netherlands","institution_ids":["https://openalex.org/I98358874"]},{"raw_affiliation_string":"Control & Simulation Div., Delft Univ. of Technol., Delft","institution_ids":["https://openalex.org/I98358874"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5016458559"],"corresponding_institution_ids":["https://openalex.org/I98358874"],"apc_list":null,"apc_paid":null,"fwci":3.1614,"has_fulltext":false,"cited_by_count":75,"citation_normalized_percentile":{"value":0.92257278,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":90,"max":100},"biblio":{"volume":"20","issue":"4","first_page":"638","last_page":"653"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10320","display_name":"Neural Networks and Applications","score":0.9994999766349792,"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"}},"topics":[{"id":"https://openalex.org/T10320","display_name":"Neural Networks and Applications","score":0.9994999766349792,"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"}},{"id":"https://openalex.org/T11206","display_name":"Model Reduction and Neural Networks","score":0.995199978351593,"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/T11236","display_name":"Control Systems and Identification","score":0.9922000169754028,"subfield":{"id":"https://openalex.org/subfields/2207","display_name":"Control and Systems Engineering"},"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/interval-arithmetic","display_name":"Interval arithmetic","score":0.6676185131072998},{"id":"https://openalex.org/keywords/interval","display_name":"Interval (graph theory)","score":0.6023144721984863},{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.5908262133598328},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.5685954689979553},{"id":"https://openalex.org/keywords/optimization-problem","display_name":"Optimization problem","score":0.5553315877914429},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.5503654479980469},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.5425177216529846},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.493068665266037},{"id":"https://openalex.org/keywords/dependency","display_name":"Dependency (UML)","score":0.4763070344924927},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.4719494879245758},{"id":"https://openalex.org/keywords/reinforcement-learning","display_name":"Reinforcement learning","score":0.4416697323322296},{"id":"https://openalex.org/keywords/taylor-series","display_name":"Taylor series","score":0.41400203108787537},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.38101351261138916},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.3600582480430603},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.1464666724205017}],"concepts":[{"id":"https://openalex.org/C191252586","wikidata":"https://www.wikidata.org/wiki/Q1671453","display_name":"Interval arithmetic","level":3,"score":0.6676185131072998},{"id":"https://openalex.org/C2778067643","wikidata":"https://www.wikidata.org/wiki/Q166507","display_name":"Interval (graph theory)","level":2,"score":0.6023144721984863},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.5908262133598328},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.5685954689979553},{"id":"https://openalex.org/C137836250","wikidata":"https://www.wikidata.org/wiki/Q984063","display_name":"Optimization problem","level":2,"score":0.5553315877914429},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.5503654479980469},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.5425177216529846},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.493068665266037},{"id":"https://openalex.org/C19768560","wikidata":"https://www.wikidata.org/wiki/Q320727","display_name":"Dependency (UML)","level":2,"score":0.4763070344924927},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.4719494879245758},{"id":"https://openalex.org/C97541855","wikidata":"https://www.wikidata.org/wiki/Q830687","display_name":"Reinforcement learning","level":2,"score":0.4416697323322296},{"id":"https://openalex.org/C158946198","wikidata":"https://www.wikidata.org/wiki/Q131187","display_name":"Taylor series","level":2,"score":0.41400203108787537},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.38101351261138916},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.3600582480430603},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.1464666724205017},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.0},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"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/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.0},{"id":"https://openalex.org/C50522688","wikidata":"https://www.wikidata.org/wiki/Q189833","display_name":"Economic growth","level":1,"score":0.0},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1109/tnn.2008.2011267","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tnn.2008.2011267","pdf_url":null,"source":{"id":"https://openalex.org/S42080949","display_name":"IEEE Transactions on Neural Networks","issn_l":"1045-9227","issn":["1045-9227","1941-0093"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Neural Networks","raw_type":"journal-article"},{"id":"pmid:19244020","is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/19244020","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":"IEEE transactions on neural networks","raw_type":null},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.332.8668","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.332.8668","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://repository.tudelft.nl/assets/uuid:b0d2cc9f-c6b9-49b0-91e5-d5ef37e32ecb/deWeerdt_2009.pdf","raw_type":"text"},{"id":"pmh:oai:tudelft.nl:uuid:b0d2cc9f-c6b9-49b0-91e5-d5ef37e32ecb","is_oa":false,"landing_page_url":"http://resolver.tudelft.nl/uuid:b0d2cc9f-c6b9-49b0-91e5-d5ef37e32ecb","pdf_url":null,"source":{"id":"https://openalex.org/S4306400906","display_name":"Research Repository (Delft University of Technology)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I98358874","host_organization_name":"Delft University of Technology","host_organization_lineage":["https://openalex.org/I98358874"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"journal article"}],"best_oa_location":null,"sustainable_development_goals":[{"score":0.6899999976158142,"id":"https://metadata.un.org/sdg/10","display_name":"Reduced inequalities"}],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":49,"referenced_works":["https://openalex.org/W138885771","https://openalex.org/W143062967","https://openalex.org/W173078835","https://openalex.org/W285242605","https://openalex.org/W560268932","https://openalex.org/W597975726","https://openalex.org/W1485608573","https://openalex.org/W1499226581","https://openalex.org/W1506281249","https://openalex.org/W1515851193","https://openalex.org/W1520048352","https://openalex.org/W1526272043","https://openalex.org/W1545877870","https://openalex.org/W1554663460","https://openalex.org/W1576452626","https://openalex.org/W1580809199","https://openalex.org/W1593960608","https://openalex.org/W1966295103","https://openalex.org/W1983014207","https://openalex.org/W1995341919","https://openalex.org/W2000115948","https://openalex.org/W2031335922","https://openalex.org/W2035331087","https://openalex.org/W2038128936","https://openalex.org/W2052222946","https://openalex.org/W2060330798","https://openalex.org/W2085683223","https://openalex.org/W2099209964","https://openalex.org/W2103496339","https://openalex.org/W2115886818","https://openalex.org/W2117812871","https://openalex.org/W2124776405","https://openalex.org/W2125664521","https://openalex.org/W2184579898","https://openalex.org/W2322762024","https://openalex.org/W2334118273","https://openalex.org/W2400458653","https://openalex.org/W2483775066","https://openalex.org/W2601626682","https://openalex.org/W2799061466","https://openalex.org/W2911283634","https://openalex.org/W3146520007","https://openalex.org/W4206262190","https://openalex.org/W4214717370","https://openalex.org/W4244494905","https://openalex.org/W4285719527","https://openalex.org/W4298255709","https://openalex.org/W4388297464","https://openalex.org/W6631515168"],"related_works":["https://openalex.org/W4306904969","https://openalex.org/W2138720691","https://openalex.org/W4362501864","https://openalex.org/W4380318855","https://openalex.org/W3084456289","https://openalex.org/W2024136090","https://openalex.org/W4391331176","https://openalex.org/W2067317451","https://openalex.org/W2031695474","https://openalex.org/W2586732548"],"abstract_inverted_index":{"The":[0,22,64],"problem":[1,23,36,81],"of":[2,10,59,67,82,127],"output":[3,84,151,179],"optimization":[4,35,180],"within":[5],"a":[6,124],"specified":[7],"input":[8,117],"space":[9],"neural":[11],"networks":[12],"(NNs)":[13],"with":[14,61],"fixed":[15],"weights":[16],"is":[17,24,37,47,110,137],"discussed":[18],"in":[19,39],"this":[20],"paper.":[21],"(highly)":[25],"nonlinear":[26,28],"when":[27,114],"activation":[29],"functions":[30],"are":[31,54,76,89,152],"used.":[32],"This":[33,139],"global":[34],"encountered":[38],"the":[40,80,104,132,143,146,149,169,173,177],"reinforcement":[41],"learning":[42],"(RL)":[43],"community.":[44],"Interval":[45],"analysis":[46],"applied":[48],"to":[49,56,91,122,158],"guarantee":[50],"that":[51,145,168],"all":[52],"solutions":[53],"found":[55],"any":[57],"degree":[58],"accuracy":[60],"guaranteed":[62],"bounds.":[63],"major":[65],"drawbacks":[66],"interval":[68,163],"analysis,":[69],"i.e.,":[70],"dependency":[71,105],"effect":[72,106],"and":[73,109],"high-computational":[74],"load,":[75],"both":[77],"present":[78],"for":[79,100,176],"NN":[83,178],"optimization.":[85],"Taylor":[86],"models":[87],"(TMs)":[88],"introduced":[90],"reduce":[92],"these":[93],"drawbacks.":[94],"They":[95],"have":[96],"excellent":[97],"convergence":[98],"properties":[99],"small":[101],"intervals.":[102],"However,":[103],"still":[107],"remains":[108],"even":[111],"made":[112],"worse":[113],"evaluating":[115],"large":[116],"domains.":[118],"As":[119],"an":[120],"alternative":[121],"TMs,":[123],"different":[125],"form":[126],"polynomial":[128,133],"inclusion":[129],"functions,":[130],"called":[131],"set":[134],"(PS)":[135],"method,":[136],"introduced.":[138],"new":[140],"method":[141,171],"has":[142],"property":[144],"bounds":[147],"on":[148],"network":[150],"tighter":[153],"or":[154],"at":[155],"least":[156],"equal":[157],"those":[159],"obtained":[160],"through":[161],"standard":[162],"arithmetic":[164],"(IA).":[165],"Experiments":[166],"show":[167],"PS":[170],"outperforms":[172],"other":[174],"methods":[175],"problem.":[181]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":5},{"year":2023,"cited_by_count":12},{"year":2022,"cited_by_count":19},{"year":2021,"cited_by_count":13},{"year":2020,"cited_by_count":5},{"year":2019,"cited_by_count":2},{"year":2018,"cited_by_count":3},{"year":2016,"cited_by_count":2},{"year":2015,"cited_by_count":1},{"year":2014,"cited_by_count":1},{"year":2013,"cited_by_count":2},{"year":2012,"cited_by_count":4}],"updated_date":"2026-04-05T17:49:38.594831","created_date":"2025-10-10T00:00:00"}
