Artificial neural network approach for solving fuzzy differential equations求解模糊微分方程的人工神经网络方法

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1、InformationSciences180(2010)1434–1457ContentslistsavailableatScienceDirectInformationSciencesjournalhomepage:www.elsevier.com/locate/insArtificialneuralnetworkapproachforsolvingfuzzydifferentialequationsa,*bSohrabEffati,MortezaPakdamanaDepartmentofAppliedMathematics,FerdowsiUnive

2、rsityofMashhad,Mashhad,IranbSamaOrganization(affiliatedwithIslamicAzadUniversity)-MashhadBranch,Mashhad,IranarticleinfoabstractArticlehistory:Thecurrentresearchattemptstoofferanovelmethodforsolvingfuzzydifferentialequa-Received23May2007tionswithinitialconditionsbasedontheuseoffee

3、d-forwardneuralnetworks.First,theReceivedinrevisedform19December2009fuzzydifferentialequationisreplacedbyasystemofordinarydifferentialequations.AtrialAccepted21December2009solutionofthissystemiswrittenasasumoftwoparts.Thefirstpartsatisfiestheinitialconditionandcontainsnoadjustable

4、parameters.Thesecondpartinvolvesafeed-forwardneuralnetworkcontainingadjustableparameters(theweights).Hencebyconstruction,theKeywords:initialconditionissatisfiedandthenetworkistrainedtosatisfythedifferentialequations.FuzzydifferentialequationsThismethod,incomparisonwithexistingnum

5、ericalmethods,showsthattheuseofneuralFuzzyCauchyproblemArtificialneuralnetworksnetworksprovidessolutionswithgoodgeneralizationandhighaccuracy.Theproposedmethodisillustratedbyseveralexamples.Ó2009ElsevierInc.Allrightsreserved.1.IntroductionUncertaintyisanattributeofinformation,[28

6、]andtheuseoffuzzydifferentialequations(FDEs)isanaturalwaytomodeldynamicsystemswithembeddeduncertainty.MostpracticalproblemscanbemodeledasFDEs(e.g.[5,8]andSection3.2).ThemethodoffuzzymappingwasinitiallyintroducedbyChangandZadeh[10].Later,DuboisandPrade[11]presentedaformofelementa

7、ryfuzzycalculusbasedontheextensionprinciple[27].PuriandRalescue[23]suggestedtwodefinitionsforthefuzzyderivativeoffuzzyfunctions.ThefirstmethodwasbasedonH-differencenotationandwasfurtherinvestigatedbyKaleva[16].SeveralapproacheswerelaterproposedforFDEsandtheexistenceoftheirsolution

8、s(e.g.[15,19,21,24,26]).TheapproachbasedonH-der