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本文讨论神经网络的能力问题及其在系统识别中的一些逼近问题。文中证明了:(1)函数g∈L_(Loc)~P(R~1)∩S′(R′)为—L~P-Tauber-Wiener函数的充要条件为g不是一个多项式;(2)当g∈(L~PTW)时,sum from i=1 to N c_ig(y_i·x+θ_i)全体在L~P(K)中稠密;(3)证明了用一元函数的复合可以逼近定义在L~P(K)上的连续(线性或非线性)泛函及L~(P1)(K_1)到L~P2(K_2)中的连续(线性或非张性)算子。上述结果表明任一非多项式的L_(Loc)~P∩S′(R′)中的函数可以作为神经网络隐层中的非线性元,以及神经网络算法可以以任意精度识别一个系统。
This article discusses the problem of neural network capabilities and some of its approximations in system identification. The paper proves that: (1) The necessary and sufficient condition for the function L ~ P-Tauber-Wiener to be a function g∈L_ (Loc) ~ P (R ~ 1) ∩S ’(R’) is that g is not a polynomial; ) From g = (L ~ PTW), all from i = 1 to N c_ig (y_i · x + θ_i) is dense in L ~ P (K); (3) CONTINUOUS (LINEAR OR NONLINEAR) FUNCTIONS AT L ~ P (K) AND CONTINUOUS (LINEAR OR NON-TENSOR) OPERATORS IN L ~ (P1) (K_1) TO L ~ P2 (K_2) The above results show that the functions in L_ (Loc) ~P∩S ’(R’) of any non-polynomial can be used as nonlinear elements in the hidden layer of neural network, and the neural network algorithm can identify a system with arbitrary precision.