多变元周期函数的神经网络逼近逼近阶估计.pdf

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1、第24卷第9期计算机学报Vol.24No.92001年9月CINESEJ.COMPUTERSSept.2001多变元周期函数的神经网络逼近:逼近阶估计曹飞龙徐宗本(西安交通大学理学院信息与系统科学研究所西安710049)摘要该文证明具有三角隐层单元的三层前向神经网络逼近多变元周期函数速度的上界估计~下界估计和饱和定理.揭示该类神经网络之隐层单元数与网络逼近速度~逼近函数结构之间的关系.特别指出二阶光滑模为该类神经网络的本质逼近阶并且当被逼近函数属于二阶Lipschitz函数类时该类神经网络的逼近能力完全取决于被逼近函数的光滑性.文中也证

2、明了该类神经网络的最大逼近能力以及达到最大逼近能力的一个充分必要条件.该文所获结果对于澄清该类神经网络的函数逼近能力与应用有重要指导意义.关键词三层人工神经网络函数逼近下界估计光滑模逆定理中图法分类号:TP18NeuralNetworkApproximationforMultivariatePeriodicfunctions:EstimatesonApproximationOrderCAOFei-LongXUZong-Ben(LnstzttefolLnfolmatzonandSystemSczenceScooofSczenceXz/an

3、JzaotongUnzUelsztyXz/an710049)AbstractFunctionapproximationisakeyissueineValuatingthecomputationalabilityofmulti-layerartificialforWardneuralnetWork.ThemainresultsofextensiVestudiesonthissubjectarethatthree-layerartificialforWardneuralnetWorkcanWithasufficientnumberofhid

4、den-layerunitsapproximatecontinuousfunctionsandLebesgue-integrablefunctionstoanydegreeofaccuracy.ButmostoftheseresultscontributealmostnothingtoansWersuchimportantguestionsashoWWecanconstructtheseapproximatingnetWorksandhoWmanyhidden-layerunitsWeneedtoapproximatespecificf

5、unctionsWithinsomespecifiederror.TosolVetheseproblemsShin[7]Sazukiconstructedathree-layerartificialforWardneuralnetWorkWithtrigonometrichidden-layerunitsandproVeditsconstructiVeapproximationtheorems.AupperboundonapproximationerrorforperiodicfunctionsWasestimatedbythefirs

6、tordermodulusofsmoothness.TheseresultshoWeVercannotpreciselycharacterizetheapproximationabilityofthenetWorkandtherelationshipbetWeentherateofapproximationandthetopologicalstructureofhidden-layer.InthispaperWefirstpointoutthatthesecondordermodulusofsmoothnessistheessentia

7、lorderofapproximationforthenetWork.ThenWeusethesecondordermodulusofsmoothnessandgiVeaestimateofupperboundsonapproximationrate.AsanimportantresultobtainedinthispaperanloWerboundedestimationonapproximationrateforthenetWorkisgiVenbymeansofthetheoriesofK-functionalandmodulus

8、ofsmoothnessinapproximationtheory.SincetheupperandloWerboundsofapproximationerrorobtainedbyushaVethesam