<abstract xmlns="http://www.w3.org/1999/xhtml">

<p>Here we examine the multivariate quantitative approximations of Banach space valued continuous multivariate functions on a box or ℝ<sup><italic>N</italic></sup> , <italic>N</italic> ∈ ℕ, by the multivariate normalized, quasi-interpolation, Kantorovich type and quadrature type neural network operators. We treat also the case of approximation by iterated operators of the last four types. These approximations are derived 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 by using a multidimensional density function induced by a hyperbolic tangent like sigmoid function. The approximations are pointwise and uniform. The related feed-forward neural network is with one hidden layer.</p>
</abstract>

Here we examine the multivariate quantitative approximations of Banach space valued continuous multivariate functions on a box or ℝN , N ∈ ℕ, by the multivariate normalized, quasi-interpolation, Kantorovich type and quadrature type neural network operators. We treat also the case of approximation by iterated operators of the last four types. These approximations are derived 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 by using a multidimensional density function induced by a hyperbolic tangent like sigmoid function. The approximations are pointwise and uniform. The related feed-forward neural network is with one hidden layer.

Hyperbolic Tangent Like Relied Banach Space Valued Neural Network Multivariate Approximations

Annals of West University of Timisoara - Mathematics and Computer Science

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Here we examine the multivariate quantitative approximations of Banach space valued continuous multivariate functions on a box or ℝN , N ∈ ℕ, by the...

Hyperbolic Tangent Like Relied Banach Space Valued Neural Network Multivariate Approximations

Published Online: May 03, 2023

Page range: 45 - 68

DOI: https://doi.org/10.2478/awutm-2023-0005

Keywords
Hyperbolic tangent like sigmoid function, multivariate neural network approximation, quasi-interpolation operator, Kantorovich type operator, quadrature type operator, multivariate modulus of continuity, abstract approximation, iterated approximation

© 2023 George A. Anastassiou, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Hyperbolic Tangent Like Relied Banach Space Valued Neural Network Multivariate Approximations

Published Online: May 03, 2023

Page range: 45 - 68

DOI: https://doi.org/10.2478/awutm-2023-0005

KeywordsHyperbolic tangent like sigmoid function, multivariate neural network approximation, quasi-interpolation operator, Kantorovich type operator, quadrature type operator, multivariate modulus of continuity, abstract approximation, iterated approximation

© 2023 George A. Anastassiou, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
Hyperbolic tangent like sigmoid function, multivariate neural network approximation, quasi-interpolation operator, Kantorovich type operator, quadrature type operator, multivariate modulus of continuity, abstract approximation, iterated approximation