By George A. Anastassiou (auth.)
This short monograph is the 1st one to deal solely with the quantitative approximation through man made neural networks to the identity-unit operator. the following we learn with premiums the approximation houses of the "right" sigmoidal and hyperbolic tangent synthetic neural community confident linear operators. particularly we examine the measure of approximation of those operators to the unit operator within the univariate and multivariate instances over bounded or unbounded domain names. this is often given through inequalities and with using modulus of continuity of the concerned functionality or its greater order by-product. We learn the genuine and complicated cases.
For the ease of the reader, the chapters of this e-book are written in a self-contained style.
This treatise is determined by author's final years of comparable examine work.
Advanced classes and seminars may be taught out of this short publication. All priceless history and motivations are given consistent with bankruptcy. A comparable checklist of references is given additionally consistent with bankruptcy. The uncovered effects are anticipated to discover purposes in lots of parts of computing device technological know-how and utilized arithmetic, akin to neural networks, clever platforms, complexity thought, studying conception, imaginative and prescient and approximation thought, and so forth. As such this monograph is acceptable for researchers, graduate scholars, and seminars of the above matters, additionally for all technological know-how libraries.
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Additional info for Intelligent Systems: Approximation by Artificial Neural Networks
21) β So for 0 < α < 1 we get (1−α) e−2n < n−α . 22) (1−α) Thus be given ﬁxed A, B > 0, for the linear combination An−α +Be−2n the (dominant) rate of convergence to zero is n−α . The closer α is to 1 we get faster and better rate of convergence to zero. 3 Real Neural Network Quantitative Approximations Here we give a series of neural network approximation to a function given with rates. 14. Let f ∈ C ([a, b]) , 0 < α < 1, n ∈ N, x ∈ [a, b]. 23) and ii) Fn (f ) − f where · ∞ is the supremum norm.
F (N ) (w) − F (N) e−x dw ≤ Case of nk − x ≤ n1α . k i) Subcase of x ≥ nk . e. e− n ≥ e−x . k 1 |μ| ≤ (N − 1)! e− n e−x −k n e 1 (N − 1)! k e− n − w e−x k e− n − w N −1 N−1 ω1 F (N) , w − e−x k k 1 ω1 F (N) , e− n − e−x (N − 1)! k e− n − e−x e x − nk N! e−x k n k n ≤ ω1 F (N ) , e−a x − k n ≤ N . nαN nα Hence when x ≥ k e− n − w N N! n n . dw ≤ dw ≤ 52 2 Univariate Hyperbolic Tangent Neural Network ii) Subcase of |μ| ≤ k n 1 (N − 1)! 1 (N − 1)! k ≥ x. Then e− n ≤ e−x and e−x N −1 k w − e− n −k e n e−x N −1 k w − e− n k e− n F (N ) (w) − F (N ) e−x ω1 F (N ) , w − e−x e−x 1 k ω1 F (N ) , e−x − e− n (N − 1)!
Nb k= na Therefore again lim 1 − n→∞ Therefore we ﬁnd that Ψ (na − k) > 0. nb Ψ (nx − k) = 1, lim n→∞ k= na for at least some x ∈ [a, b]. 7. Let f ∈ C ([a, b]) and n ∈ N such that na ≤ nb . b] . For large enough n we always obtain na ≤ nb . Also a ≤ na ≤ k ≤ nb . 1) k n ≤ b, iﬀ We study here the pointwise and uniform convergence of Fn (f, x) to f (x) with rates. 3) . 4) . Consequently we derive |Fn (f, x) − f (x)| ≤ 1 F ∗ (f, x) − f (x) Ψ (1) n nb Ψ (nx − k) . 1488766) f k= na k n − f (x) Ψ (nx − k) .
Intelligent Systems: Approximation by Artificial Neural Networks by George A. Anastassiou (auth.)