Constructive approximate interpolation by neural networks

Constructive approximation to real function by wavelet neural networks

“…To verify the above theory and to show the wavelet neural networks have faster convergence rate than the neural networks in [12,13], we give five examples through f ðxÞ À X a ðx; AðnÞÞ j j ¼ f ðxÞ À f ðx i Þ þ X e ðx i ; AðnÞÞ À X a ðx; AðnÞÞ j j ¼ f ðxÞ À f ðx i Þ þ X e ðx i ; AðnÞÞ À X e ðx; AðnÞÞ þ X e ðx; AðnÞÞ À X a ðx; AðnÞÞ j j…”

“…In this section, we discuss the uniform approximation to continuous functions for the given exact interpolation (9) and approximate interpolation (12). We consider the uniform approximation with the following features:…”

“…In [12], Llanas and Sainz present a constructive method to get the outer weights c j . The given networks can approximately interpolate n + 1 distinct samples (x i , f i ), where i = 0,1,…,n, without training, and uniformly approximate any continuous function of one variable on [a,b].…”

Interpolation and rates of convergence for a class of neural networks

“…Furthermore, the fact ''If / is sigmoidal, continuous and there exists a point c such that / 0 ðcÞ-0, then an interpolation problem with 2n þ 1 samples can be approximated with arbitrary precision by a net with n þ 1 neurons" was given. Recently, Llanas and Sainz [22] studied the existence and construction of e-approximate interpolation networks. They first considered that the activation function / is a nondecreasing sigmoidal function satisfying the condition (2) and gave a new and quantitative proof of the fact that n þ 1 hidden neurons can learn n þ 1 distinct samples with zero error.…”

Multivariate numerical approximation using constructive $$ L^{2} (\mathbb{R}) $$ RBF neural network

“…The literature [5][6][7][8] has studied the approximation problems of one variable functions by constructive feedforward neural networks, but when the related conclusions are extended to multidimensional functions, the question becomes quite complex and too strict conditions of convergence are requested, the actual operation becomes difficulty relatively. But in the literature [9], the connection weight of RBF neural networks is obtained through the various learning algorithms; therefore, the weight has certain instability.…”