The Capability of Localized Neural Network Approximation

Abstract

Abstract. In this paper, we investigate a localized approximation of a continuously differentiable function by neural networks. To do this, we first approximate a continuously differentiable function by B-spline functions and then approximate B-spline functions by neural networks. Our proofs are constructive and we give numerical results to support our theory.

Degree of Approximation by Periodic Neural Networks

“…where l j (f ) ∈ C. Now we have to show e ijx in (3.6) is approximated arbitrarily well by periodic neural networks. Using a Riemann sum, we obtained the following lemma in [3]. π −π σ(t)e −it dt = 0.…”

“…Approximation by neural networks has been investigated by many researchers [2,3,4,5,7,8] because it has been widely applied in engineering such as robotics, signal processing and etc.…”

“…The periodic neural network approximation was suggested in [2,3]. In fact, Guanzhen [2] suggested a periodic neural network approximation based on scattered nodes.…”

“…In fact, Guanzhen [2] suggested a periodic neural network approximation based on scattered nodes. On the other hand, Hahm and Hong [3] proved the approximation capability of continuous 2π-periodic functions by periodic neural networks.…”

Constructive Approximation by Neural Networks With Positive Integer Weights

“…Because of its applications in engineering such as robotics and signal processing, neural network approximation has been investigated by many mathematicians ( [3], [4], [6], [7], [8]). A general form of feedforward neural network with one hidden layer is…”