Multiple general sigmoids based Banach space valued neural network multivariate approximation
Here we present multivariate quantitative approximations of Banach space valued continuous multivariate functions on a box or \(\mathbb{R}^{N},\) \(N\in \mathbb{N}\), by the multivariate normalized, quasi-interpolation, Kantorovich type and quadrature type neural network operators. We treat also the...
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Universidad de La Frontera
2023-12-01
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| Series: | Cubo |
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| Online Access: | https://cubo.ufro.cl/index.php/cubo/article/view/3579/2328 |
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| author | George A. Anastassiou |
| author_facet | George A. Anastassiou |
| author_sort | George A. Anastassiou |
| collection | DOAJ |
| description | Here we present multivariate quantitative approximations of Banach space valued continuous multivariate functions on a box or \(\mathbb{R}^{N},\) \(N\in \mathbb{N}\), 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 several different among themselves general sigmoid functions. This is done on the purpose to activate as many as possible neurons. The approximations are pointwise and uniform. The related feed-forward neural network is with one hidden layer. We finish with related \(L_{p}\) approximations. |
| format | Article |
| id | doaj-art-10cb9bfbcb234efead033e849d0396d0 |
| institution | OA Journals |
| issn | 0719-0646 |
| language | English |
| publishDate | 2023-12-01 |
| publisher | Universidad de La Frontera |
| record_format | Article |
| series | Cubo |
| spelling | doaj-art-10cb9bfbcb234efead033e849d0396d02025-08-20T02:10:32ZengUniversidad de La FronteraCubo0719-06462023-12-01253411439https://doi.org/10.56754/0719-0646.2503.411Multiple general sigmoids based Banach space valued neural network multivariate approximationGeorge A. Anastassiou0https://orcid.org/0000-0002-3781-9824Department of Mathematical Sciences, University of Memphis Memphis, TN 38152, U.S.A.Here we present multivariate quantitative approximations of Banach space valued continuous multivariate functions on a box or \(\mathbb{R}^{N},\) \(N\in \mathbb{N}\), 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 several different among themselves general sigmoid functions. This is done on the purpose to activate as many as possible neurons. The approximations are pointwise and uniform. The related feed-forward neural network is with one hidden layer. We finish with related \(L_{p}\) approximations.https://cubo.ufro.cl/index.php/cubo/article/view/3579/2328general sigmoid functionsmultivariate neural network approximationquasi-interpolation operatorkantorovich type operatorquadrature type operatormultivariate modulus of continuityabstract approximationiterated approximationlp approximation |
| spellingShingle | George A. Anastassiou Multiple general sigmoids based Banach space valued neural network multivariate approximation Cubo general sigmoid functions multivariate neural network approximation quasi-interpolation operator kantorovich type operator quadrature type operator multivariate modulus of continuity abstract approximation iterated approximation lp approximation |
| title | Multiple general sigmoids based Banach space valued neural network multivariate approximation |
| title_full | Multiple general sigmoids based Banach space valued neural network multivariate approximation |
| title_fullStr | Multiple general sigmoids based Banach space valued neural network multivariate approximation |
| title_full_unstemmed | Multiple general sigmoids based Banach space valued neural network multivariate approximation |
| title_short | Multiple general sigmoids based Banach space valued neural network multivariate approximation |
| title_sort | multiple general sigmoids based banach space valued neural network multivariate approximation |
| topic | general sigmoid functions multivariate neural network approximation quasi-interpolation operator kantorovich type operator quadrature type operator multivariate modulus of continuity abstract approximation iterated approximation lp approximation |
| url | https://cubo.ufro.cl/index.php/cubo/article/view/3579/2328 |
| work_keys_str_mv | AT georgeaanastassiou multiplegeneralsigmoidsbasedbanachspacevaluedneuralnetworkmultivariateapproximation |