{"id":"https://openalex.org/W144883304","doi":null,"title":"Analysing the behaviour of neural networks","display_name":"Analysing the behaviour of neural networks","publication_year":2004,"publication_date":"2004-01-01","ids":{"openalex":"https://openalex.org/W144883304","mag":"144883304"},"language":"en","primary_location":{"id":"mag:144883304","doi":null,"is_oa":false,"landing_page_url":"https://eprints.qut.edu.au/15943/","pdf_url":null,"source":null,"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":null},"volume":null,"type":"dissertation","is_traditional":true,"indexed_in":[],"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/A5063489665","display_name":"Stephan Breutel","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Stephan Breutel","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5063489665"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":false,"cited_by_count":4,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10142","display_name":"Formal Methods in Verification","score":0.8345000147819519,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"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/T10142","display_name":"Formal Methods in Verification","score":0.8345000147819519,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"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/T10743","display_name":"Software Testing and Debugging Techniques","score":0.0812000036239624,"subfield":{"id":"https://openalex.org/subfields/1712","display_name":"Software"},"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/T10126","display_name":"Logic, programming, and type systems","score":0.017100000753998756,"subfield":{"id":"https://openalex.org/subfields/1712","display_name":"Software"},"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/feedforward-neural-networks","display_name":"feedforward neural networks","score":0.8960000276565552},{"id":"https://openalex.org/keywords/preconditions-and-postconditions","display_name":"preconditions and postconditions","score":0.7900000214576721},{"id":"https://openalex.org/keywords/polyhedra","display_name":"polyhedra","score":0.8980000019073486},{"id":"https://openalex.org/keywords/safety-critical-software","display_name":"safety-critical software","score":0.718999981880188},{"id":"https://openalex.org/keywords/affine-transformation","display_name":"affine transformation","score":0.7099999785423279},{"id":"https://openalex.org/keywords/nonlinear-programming","display_name":"nonlinear programming","score":0.7239999771118164}],"concepts":[{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.7239567637443542},{"id":"https://openalex.org/C54829058","wikidata":"https://www.wikidata.org/wiki/Q172937","display_name":"Polyhedron","level":2,"score":0.6490074396133423},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.5896422863006592},{"id":"https://openalex.org/C177148314","wikidata":"https://www.wikidata.org/wiki/Q170084","display_name":"Generalization","level":2,"score":0.5415397882461548},{"id":"https://openalex.org/C113843644","wikidata":"https://www.wikidata.org/wiki/Q901882","display_name":"Interface (matter)","level":4,"score":0.5335296392440796},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.4872854948043823},{"id":"https://openalex.org/C12362212","wikidata":"https://www.wikidata.org/wiki/Q728435","display_name":"Linear subspace","level":2,"score":0.45508697628974915},{"id":"https://openalex.org/C84135550","wikidata":"https://www.wikidata.org/wiki/Q3154022","display_name":"Linear inequality","level":3,"score":0.4410724639892578},{"id":"https://openalex.org/C2777904410","wikidata":"https://www.wikidata.org/wiki/Q7397","display_name":"Software","level":2,"score":0.4408216178417206},{"id":"https://openalex.org/C80444323","wikidata":"https://www.wikidata.org/wiki/Q2878974","display_name":"Theoretical computer science","level":1,"score":0.426467627286911},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.41757136583328247},{"id":"https://openalex.org/C45374587","wikidata":"https://www.wikidata.org/wiki/Q12525525","display_name":"Computation","level":2,"score":0.4107869863510132},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.29806041717529297},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.24491319060325623},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.15440723299980164},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.13849708437919617},{"id":"https://openalex.org/C157915830","wikidata":"https://www.wikidata.org/wiki/Q2928001","display_name":"Bubble","level":2,"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/C129307140","wikidata":"https://www.wikidata.org/wiki/Q6795880","display_name":"Maximum bubble pressure method","level":3,"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/C173608175","wikidata":"https://www.wikidata.org/wiki/Q232661","display_name":"Parallel computing","level":1,"score":0.0},{"id":"https://openalex.org/C45555294","wikidata":"https://www.wikidata.org/wiki/Q28113351","display_name":"Inequality","level":2,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"mag:144883304","doi":null,"is_oa":false,"landing_page_url":"https://eprints.qut.edu.au/15943/","pdf_url":null,"source":null,"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":null}],"best_oa_location":null,"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":55,"referenced_works":["https://openalex.org/W54409344","https://openalex.org/W181457951","https://openalex.org/W368812709","https://openalex.org/W1495591997","https://openalex.org/W1497207529","https://openalex.org/W1515448847","https://openalex.org/W1529128749","https://openalex.org/W1542013646","https://openalex.org/W1551186145","https://openalex.org/W1554663460","https://openalex.org/W1566822127","https://openalex.org/W1575869149","https://openalex.org/W1578474339","https://openalex.org/W1587493372","https://openalex.org/W1589168849","https://openalex.org/W1597196612","https://openalex.org/W1614345895","https://openalex.org/W1622025140","https://openalex.org/W1678380195","https://openalex.org/W1851430230","https://openalex.org/W1939375757","https://openalex.org/W1963547452","https://openalex.org/W1965229818","https://openalex.org/W1973822995","https://openalex.org/W1981249599","https://openalex.org/W1986698569","https://openalex.org/W1991848143","https://openalex.org/W2040870580","https://openalex.org/W2061415549","https://openalex.org/W2063046703","https://openalex.org/W2065833635","https://openalex.org/W2067619114","https://openalex.org/W2074367725","https://openalex.org/W2088203270","https://openalex.org/W2097442255","https://openalex.org/W2100885571","https://openalex.org/W2112036221","https://openalex.org/W2117929680","https://openalex.org/W2119475352","https://openalex.org/W2120619312","https://openalex.org/W2124776405","https://openalex.org/W2133450829","https://openalex.org/W2134788617","https://openalex.org/W2137983211","https://openalex.org/W2145742290","https://openalex.org/W2149416133","https://openalex.org/W2149706766","https://openalex.org/W2157041604","https://openalex.org/W2159219354","https://openalex.org/W2161278885","https://openalex.org/W2163532598","https://openalex.org/W2166322089","https://openalex.org/W2766736793","https://openalex.org/W2798909945","https://openalex.org/W3119651796"],"related_works":["https://openalex.org/W1542013646","https://openalex.org/W3009821482","https://openalex.org/W2791251367","https://openalex.org/W2905431656","https://openalex.org/W3127075497","https://openalex.org/W2967175067","https://openalex.org/W3187871678","https://openalex.org/W4415035383","https://openalex.org/W2903571520","https://openalex.org/W2137989406"],"abstract_inverted_index":{"A":[0],"new":[1],"method":[2,192],"is":[3,36,95,200,218],"developed":[4],"to":[5,38,219],"determine":[6],"a":[7,59,97,150,187,203,223,231],"set":[8,204],"of":[9,34,84,92,101,109,121,144,161,175,205,215,230],"informative":[10],"and":[11,49,54,71,126,148,227],"refined":[12,134],"interface":[13,51,56,78,135],"assertions\r\n\r\nsatisfied":[14],"by":[15,20],"functions":[16],"that":[17],"are":[18,65,81,83,130,158,168,182],"represented":[19],"feed-forward":[21,151,188],"neural":[22,60,111,152,189,196],"networks.":[23],"Neural":[24],"networks":[25],"have":[26,39,46],"often":[27],"been":[28],"criticized":[29],"for":[30,58,67,72,171,212],"their":[31],"low":[32],"degree":[33],"comprehensibility.It":[35],"difficult":[37],"confidence":[40],"in":[41,96,115,132,164],"software":[42,63,76],"components":[43],"if":[44,88],"they":[45],"no":[47],"clear":[48],"valid":[50,206],"description.":[52],"Precise":[53],"understandable":[55],"assertions":[57,79,217],"network":[61,112,153],"based":[62],"component":[64],"required":[66],"safety":[68],"critical":[69],"applications":[70],"theintegration":[73],"into":[74],"larger":[75],"systems.\r\n\r\n\r\n\r\nThe":[77],"we":[80],"considering":[82],"the":[85,89,93,102,106,110,116,122,142,145,159,213,221,228],"form":[86],"&quote":[87,125],"input":[90,103],"x":[91],"neural\r\n\r\nnetwork":[94],"region":[98,117],"(alpha":[99],"symbol)":[100,119],"space":[104,124],"then":[105],"output":[107,123],"f(x)":[108],"will":[113],"be":[114,139],"(beta":[118],"vice":[127],"versa.":[128],"We":[129],"interested":[131],"computing":[133],"assertions,":[136],"which":[137],"can":[138],"viewed":[140],"as":[141],"computation":[143,214],"strongest":[146],"pre-":[147],"postconditions":[149],"fulfills.":[154],"Unions":[155],"ofpolyhedra":[156],"(polyhedra":[157],"generalization":[160],"convex":[162],"polygons":[163],"higher":[165,176],"dimensional":[166,177],"spaces)":[167],"well":[169],"suited":[170],"describing":[172],"arbitrary":[173],"regions":[174],"vector":[178],"spaces.":[179],"Additionally,":[180],"polyhedra":[181],"closed":[183],"under":[184],"affine":[185],"transformations.\r\n\r\n\r\n\r\nGiven":[186],"network,":[190],"our":[191],"produces":[193],"an":[194],"annotated":[195,201],"network,\r\n\r\nwhere":[197],"each":[198],"layer":[199],"with":[202],"linear":[207],"inequality":[208],"predicates.\r\n\r\nThe":[209],"main":[210],"challenges":[211],"these":[216],"compute":[220],"solution\r\n\r\nof":[222],"non-linear":[224],"optimization":[225],"problem":[226],"projection":[229],"polyhedron":[232],"onto":[233],"a\r\n\r\nlower-dimensional":[234],"subspace.":[235]},"counts_by_year":[{"year":2016,"cited_by_count":1},{"year":2013,"cited_by_count":1}],"updated_date":"2026-11-04T10:54:05.112534","created_date":"2016-06-24T00:00:00"}
