Richards’s curve induced Banach space valued multivariate neural network approximation

Here, we present multivariate quantitative approximations of Banach space valued continuous multivariate functions on a box or \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^{N},$$\end{document}RN, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\in {\mathbb {N}},$$\end{document}N∈N, by the multivariate normalized, quasi-interpolation, Kantorovich-type and quadrature-type neural network operators. We examine also the case of approximation by iterated operators of the last four types. These approximations are achieved by establishing multidimensional Jackson type inequalities involving the multivariate modulus of continuity of the engaged function or its high-order Fréchet derivatives. Our multivariate operators are defined using a multidimensional density function induced by the Richards’s curve, which is a generalized logistic function. The approximations are pointwise, uniform and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_{p}.$$\end{document}Lp. The related feed-forward neural network is with one hidden layer.

properties of the basic multivariate density function induced by the sigmoid function related to Richards's curve and defining our operators. Richards's curve among others has been used for modeling COVID-19 infection trajectory [26].
Feed-forward neural networks (FNNs) with one hidden layer, the only type of networks we deal with in this article, are mathematically expressed as where for 0 ≤ j ≤ n, b j ∈ R are the thresholds, a j ∈ R s are the connection weights, c j ∈ R are the coefficients, a j · x is the inner product of a j and x, and σ is the activation function of the network. In many fundamental network models, the activation function is based on the Richards's curve sigmoid function. About neural networks, see [25,27,28].
Theorem 2.2 [20] It holds so that G is a density function.
Theorem 2.3 [20] Let 0 < α < 1, μ > 0 and n ∈ N with n 1−α > 2. It holds Denote by · the integral part of the number and by · the ceiling of the number.
Theorem 2.4 [20] Let [a, b] ⊂ R and n ∈ N, so that na ≤ nb . It holds We make Remark 2.5 [20] (i) We have that We introduce It has the properties where k := (k 1 , . . . , k n ) ∈ Z N , ∀ x ∈ R N ; hence ∀ x ∈ R N ; n ∈ N, and (iv) that is, Z is a multivariate density function.
Here, denote where a := (a 1 , . . . , We obviously see that For 0 < β < 1 and n ∈ N, a fixed x ∈ R N , we have that In the last two sums, the counting is over disjoint vector sets of k's, because the condition k with n ∈ N : for at least some We introduce and define the following multivariate linear normalized neural network operator (x : For large enough n ∈ N, we always obtain Clearly, L n is a positive linear operator. We have that Therefore, we have that Since L n (1) = 1, we get that We call L n the companion operator of L n . For convenience, we call Consequently, we derive We will estimate the right-hand side of (31). For the last and others, we need the following. Definition 2.6 [15, p. 274] Let M be a convex and compact subset of R N , · p , p ∈ [1, ∞] , and X, · γ be a Banach space. Let f ∈ C (M, X ) . We define the first modulus of continuity of f as If δ > diam (M) , then Notice Clearly, we have also: f ∈ C U R N , X (uniformly continuous functions), iff ω 1 ( f, δ) → 0 as δ ↓ 0, where ω 1 is defined similarly to (32). The space C B R N , X denotes the continuous and bounded functions on R N .
n ∈ N, ∀ x ∈ R N , N ∈ N, the multivariate quasi-interpolation neural network operator.
Also, for f ∈ C B R N , X , we define the multivariate Kantorovich-type neural network operator Again, for f ∈ C B R N , X , N ∈ N, we define the multivariate neural network operator of quadrature-type D n ( f, x) , n ∈ N, as follows. Let In this article, we study the approximation properties of L n , B n , C n , D n neural network operators and as well of their iterates. That is, the quantitative pointwise and uniform convergence of these operators to the unit operator I.

Multivariate Richards's curve neural network approximations
Here, we present several vectorial neural network approximations to Banach space valued functions given with rates.
We give and We notice that lim n→∞ L n ( f ) · γ = f, pointwise and uniformly.
Above ω 1 is with respect to p = ∞ and the speed of convergence is max 1 Proof We observe that Thus ≤ ω 1 f, ≤ ω 1 f, So that Now, using (31), we finish the proof.
We make Let X, · γ be a general Banach space. Then, the space Let M be a non-empty convex and compact subset of be a continuous function, whose Fréchet derivatives (see [29] We will work with f | M . Then, by Taylor's formula [21], [29, p. 124], we get where the remainder is the Riemann integral here, we set is a (polynomial) spline function; see [1, p. 210-211]. Also, from there, we get with equality true only at t = 0. Therefore, it holds We have found that One can rewrite (52) as follows: a pointwise functional inequality on M.
Here, (· − x 0 ) j maps M into R N j and it is continuous, and also, f ( j) (x 0 ) maps R N j into X and it is continuous. Hence, their composition Let S N N ∈N be a sequence of positive linear operators' mapping C (M) into C (M) . Therefore, we obtain ⎛ Clearly, (54) is valid when M = N i=1 [a i , b i ] and S n = L n , see (23). All the above is preparation for the following theorem, where we assume Fr échet differentiability of functions.
We present the following high-order approximation results.
and let X, · γ be a general Banach space. Let m ∈ N and f ∈ C m (O, X ) , the space of m-times continuously Fréchet differentiable functions from O into X. We study the approximation of and (4) We need the following.
We make the following.
We have that When p = ∞, j = 1, . . . , m, we obtain We further have that That is Therefore, when p = ∞, for j = 1, . . . , m, we have proved and converges to zero, as n → ∞.
We conclude the following: In Theorem 3.3, the right-hand sides of (57) and (58) converge to zero as n → ∞, for any p ∈ [1, ∞] . Also in Corollary 3.6, the right-hand sides of (60) and (61) converge to zero as n → ∞, for any p ∈ [1, ∞] . Higher speed of convergence happens also to the left-hand side of (55).
We further give the following:
We give the following. (2) Thus, it holds (by (35)) We observe that proving the claim.
We also present the following. (2) Given that f ∈ C U R N , X ∩ C B R N , X , we obtain lim n→∞ D n ( f ) = f, uniformly.
Proof We have that [by (37)] proving the claim.
We make the following.

Definition 3.13
Let f ∈ C B R N , X , N ∈ N, where X, · γ is a Banach space. We define the general neural network operator We need the following.
Proof Lengthy and similar to the proof of Theorem 10 of [18], as such is omitted.

Remark 3.15 By (22), it is obvious that
Call K n any of the operators L n , B n , C n , D n . Clearly, then etc. Therefore, we get the contraction property. Also, we see that Here, K k n are bounded linear operators. Notation 3. 16 Here, N ∈ N, 0 < β < 1. Denote by and We give the condensed.
Clearly, we notice that the speed of convergence to the unit operator of the multiply iterated operator is not worse than the speed of K m 1 .
Proof As similar to [18] is omitted.
We continue with the following.

Theorem 3.22
Let all as in Corollary 3.9, and r ∈ N. Here, 3 (n) is as in (73). Then L r n f − f γ ∞ ≤ r L n f − f γ ∞ ≤ r 3 (n) .
Proof As similar to [18] is omitted.
Next, we present some L p 1 , p 1 ≥ 1, approximation related results.
Next come.
Theorem 3.25 All as in Theorem 3.24, but we use λ 3 (n) of (78). Then We have that lim n→∞ C n f − f γ p 1 ,P = 0 for f ∈ C U R N , X ∩ C B R N , X .

Theorem 3.26
All as in Theorem 3.24, but we use λ 4 (n) of (84). Then D n f − f γ p 1 ,P ≤ λ 4 (n) |P| We have that lim n→∞ D n f − f γ p 1 ,P = 0 for f ∈ C U R N , X ∩ C B R N , X .

Application 3.27
A typical application of all of our results is when X, · γ = (C, |·|) , where C is the set of the complex numbers.
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Declarations
Conflict of interest Not applicable.