{"id":"https://openalex.org/W2962692913","doi":"https://doi.org/10.1109/tnnls.2018.2808470","title":"Output Reachable Set Estimation and Verification for Multilayer Neural Networks","display_name":"Output Reachable Set Estimation and Verification for Multilayer Neural Networks","publication_year":2018,"publication_date":"2018-03-16","ids":{"openalex":"https://openalex.org/W2962692913","doi":"https://doi.org/10.1109/tnnls.2018.2808470","mag":"2962692913","pmid":"https://pubmed.ncbi.nlm.nih.gov/29993822"},"language":"en","primary_location":{"id":"doi:10.1109/tnnls.2018.2808470","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tnnls.2018.2808470","pdf_url":null,"source":{"id":"https://openalex.org/S4210175523","display_name":"IEEE Transactions on Neural Networks and Learning Systems","issn_l":"2162-237X","issn":["2162-237X","2162-2388"],"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 and Learning Systems","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/A5090573416","display_name":"Weiming Xiang","orcid":"https://orcid.org/0000-0001-9065-8428"},"institutions":[{"id":"https://openalex.org/I200719446","display_name":"Vanderbilt University","ror":"https://ror.org/02vm5rt34","country_code":"US","type":"education","lineage":["https://openalex.org/I200719446"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Weiming Xiang","raw_affiliation_strings":["Department of Electrical Engineering and Computer Science, Vanderbilt University, Nashville, TN, USA"],"raw_orcid":"https://orcid.org/0000-0001-9065-8428","affiliations":[{"raw_affiliation_string":"Department of Electrical Engineering and Computer Science, Vanderbilt University, Nashville, TN, USA","institution_ids":["https://openalex.org/I200719446"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5023289374","display_name":"Hoang-Dung Tran","orcid":"https://orcid.org/0000-0001-6946-9526"},"institutions":[{"id":"https://openalex.org/I200719446","display_name":"Vanderbilt University","ror":"https://ror.org/02vm5rt34","country_code":"US","type":"education","lineage":["https://openalex.org/I200719446"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Hoang-Dung Tran","raw_affiliation_strings":["Department of Electrical Engineering and Computer Science, Vanderbilt University, Nashville, TN, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Electrical Engineering and Computer Science, Vanderbilt University, Nashville, TN, USA","institution_ids":["https://openalex.org/I200719446"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5067901159","display_name":"Taylor T. Johnson","orcid":"https://orcid.org/0000-0001-8021-9923"},"institutions":[{"id":"https://openalex.org/I200719446","display_name":"Vanderbilt University","ror":"https://ror.org/02vm5rt34","country_code":"US","type":"education","lineage":["https://openalex.org/I200719446"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Taylor T. Johnson","raw_affiliation_strings":["Department of Electrical Engineering and Computer Science, Vanderbilt University, Nashville, TN, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Electrical Engineering and Computer Science, Vanderbilt University, Nashville, TN, USA","institution_ids":["https://openalex.org/I200719446"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5090573416"],"corresponding_institution_ids":["https://openalex.org/I200719446"],"apc_list":null,"apc_paid":null,"fwci":21.3417,"has_fulltext":false,"cited_by_count":270,"citation_normalized_percentile":{"value":0.99440908,"is_in_top_1_percent":true,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":90,"max":100},"biblio":{"volume":"29","issue":"11","first_page":"5777","last_page":"5783"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11689","display_name":"Adversarial Robustness in Machine Learning","score":0.9997000098228455,"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/T11689","display_name":"Adversarial Robustness in Machine Learning","score":0.9997000098228455,"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/T10876","display_name":"Fault Detection and Control Systems","score":0.998199999332428,"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"}},{"id":"https://openalex.org/T12072","display_name":"Machine Learning and Algorithms","score":0.9696999788284302,"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/artificial-neural-network","display_name":"Artificial neural network","score":0.6773077249526978},{"id":"https://openalex.org/keywords/set","display_name":"Set (abstract data type)","score":0.672393262386322},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.637403130531311},{"id":"https://openalex.org/keywords/sensitivity","display_name":"Sensitivity (control systems)","score":0.6362290382385254},{"id":"https://openalex.org/keywords/monotonic-function","display_name":"Monotonic function","score":0.6119358539581299},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.47584792971611023},{"id":"https://openalex.org/keywords/perceptron","display_name":"Perceptron","score":0.4606715440750122},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.4559992849826813},{"id":"https://openalex.org/keywords/convex-optimization","display_name":"Convex optimization","score":0.4150197207927704},{"id":"https://openalex.org/keywords/regular-polygon","display_name":"Regular polygon","score":0.396156907081604},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.3183220624923706},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.23850032687187195},{"id":"https://openalex.org/keywords/engineering","display_name":"Engineering","score":0.09629327058792114}],"concepts":[{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.6773077249526978},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.672393262386322},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.637403130531311},{"id":"https://openalex.org/C21200559","wikidata":"https://www.wikidata.org/wiki/Q7451068","display_name":"Sensitivity (control systems)","level":2,"score":0.6362290382385254},{"id":"https://openalex.org/C72169020","wikidata":"https://www.wikidata.org/wiki/Q194404","display_name":"Monotonic function","level":2,"score":0.6119358539581299},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.47584792971611023},{"id":"https://openalex.org/C60908668","wikidata":"https://www.wikidata.org/wiki/Q690207","display_name":"Perceptron","level":3,"score":0.4606715440750122},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.4559992849826813},{"id":"https://openalex.org/C157972887","wikidata":"https://www.wikidata.org/wiki/Q463359","display_name":"Convex optimization","level":3,"score":0.4150197207927704},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.396156907081604},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.3183220624923706},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.23850032687187195},{"id":"https://openalex.org/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","level":0,"score":0.09629327058792114},{"id":"https://openalex.org/C24326235","wikidata":"https://www.wikidata.org/wiki/Q126095","display_name":"Electronic engineering","level":1,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1109/tnnls.2018.2808470","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tnnls.2018.2808470","pdf_url":null,"source":{"id":"https://openalex.org/S4210175523","display_name":"IEEE Transactions on Neural Networks and Learning Systems","issn_l":"2162-237X","issn":["2162-237X","2162-2388"],"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 and Learning Systems","raw_type":"journal-article"},{"id":"pmid:29993822","is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/29993822","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 and learning systems","raw_type":null}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G129050612","display_name":null,"funder_award_id":"FA9550-16-1-0246","funder_id":"https://openalex.org/F4320338279","funder_display_name":"Air Force Office of Scientific Research"},{"id":"https://openalex.org/G1522347985","display_name":null,"funder_award_id":"FA9550-15-1-0258","funder_id":"https://openalex.org/F4320338279","funder_display_name":"Air Force Office of Scientific Research"},{"id":"https://openalex.org/G36914211","display_name":null,"funder_award_id":"FA9550-18-1-0122","funder_id":"https://openalex.org/F4320338279","funder_display_name":"Air Force Office of Scientific Research"},{"id":"https://openalex.org/G4261586539","display_name":null,"funder_award_id":"SHF 1527398","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G4308778893","display_name":null,"funder_award_id":"CNS 1464311","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G6560536672","display_name":null,"funder_award_id":"SHF 1736323","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G7819384135","display_name":null,"funder_award_id":"CNS 1713253","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320338279","display_name":"Air Force Office of Scientific Research","ror":"https://ror.org/011e9bt93"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":35,"referenced_works":["https://openalex.org/W618708471","https://openalex.org/W1486132939","https://openalex.org/W1969705022","https://openalex.org/W1977230515","https://openalex.org/W2026854879","https://openalex.org/W2070797207","https://openalex.org/W2076063813","https://openalex.org/W2088656430","https://openalex.org/W2091633396","https://openalex.org/W2107808739","https://openalex.org/W2110480233","https://openalex.org/W2137983211","https://openalex.org/W2144354855","https://openalex.org/W2165073069","https://openalex.org/W2167287735","https://openalex.org/W2167901303","https://openalex.org/W2257979135","https://openalex.org/W2342840547","https://openalex.org/W2344309565","https://openalex.org/W2492473950","https://openalex.org/W2523516295","https://openalex.org/W2543296129","https://openalex.org/W2551542663","https://openalex.org/W2594877703","https://openalex.org/W2605373881","https://openalex.org/W2605827175","https://openalex.org/W2606133973","https://openalex.org/W2748862026","https://openalex.org/W2752420316","https://openalex.org/W2766086755","https://openalex.org/W2777417342","https://openalex.org/W3146803896","https://openalex.org/W6727440262","https://openalex.org/W6729243212","https://openalex.org/W6734921443"],"related_works":["https://openalex.org/W2077314575","https://openalex.org/W4315701745","https://openalex.org/W1990290471","https://openalex.org/W4380682190","https://openalex.org/W2005710836","https://openalex.org/W2102386043","https://openalex.org/W2945307361","https://openalex.org/W2116636209","https://openalex.org/W2053698688","https://openalex.org/W2346559293"],"abstract_inverted_index":{"In":[0],"this":[1],"brief,":[2],"the":[3,39,55,81,90,105,108],"output":[4,56,82],"reachable":[5,57,83],"estimation":[6,59,85],"and":[7,27],"safety":[8,75,91],"verification":[9,76,92],"problems":[10],"for":[11,28,61,93],"multilayer":[12],"perceptron":[13],"(MLP)":[14],"neural":[15,62],"networks":[16,63],"are":[17,36],"addressed.":[18],"First,":[19],"a":[20,29,52,67,94],"conception":[21],"called":[22],"maximum":[23,40],"sensitivity":[24,41],"is":[25,64,77,101],"introduced,":[26],"class":[30],"of":[31,69,107],"MLPs":[32],"whose":[33],"activation":[34],"functions":[35],"monotonic":[37],"functions,":[38],"can":[42],"be":[43],"computed":[44],"via":[45],"solving":[46],"convex":[47],"optimization":[48,70],"problems.":[49,71],"Then,":[50],"using":[51],"simulation-based":[53],"method,":[54],"set":[58,84],"problem":[60],"formulated":[65],"into":[66],"chain":[68],"Finally,":[72],"an":[73],"automated":[74],"developed":[78],"based":[79],"on":[80],"result.":[86],"An":[87],"application":[88],"to":[89,103],"robotic":[95],"arm":[96],"model":[97],"with":[98],"two":[99],"joints":[100],"presented":[102],"show":[104],"effectiveness":[106],"proposed":[109],"approaches.":[110]},"counts_by_year":[{"year":2026,"cited_by_count":5},{"year":2025,"cited_by_count":17},{"year":2024,"cited_by_count":32},{"year":2023,"cited_by_count":46},{"year":2022,"cited_by_count":43},{"year":2021,"cited_by_count":52},{"year":2020,"cited_by_count":51},{"year":2019,"cited_by_count":17},{"year":2018,"cited_by_count":6},{"year":2012,"cited_by_count":1}],"updated_date":"2026-05-05T08:41:31.759640","created_date":"2025-10-10T00:00:00"}