modeling of continuous time dynamical systems with input by recurrent neural networks

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1、IEEETRANSACTIONSONCIRCUITSANDSYSTEMS—I:FUNDAMENTALTHEORYANDAPPLICATIONS,VOL.47,NO.4,APRIL2000575ModelingofContinuousTimeDynamicalSystemswithtemscanbeapproximatelyrealizedbyRNN.ThispaperisorganizedInputbyRecurrentNeuralNetworksasfollows.Themodelofacontinuous-timeRNNisgiveninSectionII.S

2、ectionIIIpresentstheapproximationrealizationtheorem.SectionTommyW.S.ChowandXiao-DongLiIVpresentsthepreliminariesofourproof.SectionVgivestheproofofthetheorem.SectionVIdescribestheextensionofthetheorem.Fi-nally,SectionVIIconcludesthepaper.Abstract—Thispaperprovesthatanyfinitetimetraject

3、oryofagiven-dimensionaldynamicalcontinuoussystemwithinputcanbeapproxi-matedbytheinternalstateoftheoutputunitsofacontinuous-timerecur-II.DYNAMICALRECURRENTNEURALNETWORKSrentneuralnetwork(RNN).TheproofisbasedontheideaofembeddingAdynamicalrecurrentnetworks(RNN)isacomplexnonlineardy-the-d

4、imensionaldynamicalsystemintoahigherdimensionalone.Asaresult,weareabletoconfirmthatanycontinuousdynamicalsystemcanbenamicsystemdescribedbyasetofnonlineardifferentialordifferencemodeledbyanRNN.equationswithextensiveconnectionweights.Inthispaper,onlycon-tinuous-timeversionofananalogRNNw

5、ithtime-varyinginputsisdis-IndexTerms—Approximation,continuous-timerecurrentneuralnetworks,dynamicalsystem.cussed.AGeneralexpressionofthistypeofRNNwithLneuralunitsisgivenbythefollowingcontinuousnonlinearsystem:I.INTRODUCTIONz_=0z+f(W1;z;W2;u)(1)Feedforwardneuralnetworks(FNN’s)andrecu

6、rrentneuralnet-Lmworks(RNN’s)arethetwomajorclassesofneuralnetworks(NN’s)wherez2Randu2Raretheneuralstateandtheinputvec-L2LL2mwidelyused.Intheareaofdynamicalsystemsithasbeenshowntors,respectively,andW12R;W22RaretheconnectionthatFNNiscapableofapproximatingnotonlyacontinuousfunctionweight

7、matricesassociatedwiththeneuralstateandtheinputvectors,butalsoitsderivativestoanarbitrarydegreeofaccuracy.Thereisrespectively.isafixedconstantcontrollingthedecayingstateandisLmLalsoincreasinginterestinstudyingtheapproximationcapabilityofchosenas0<<1andf:R2R!Risanappropriatelydynamica

8、lRNN,