General multi-composite sigmoid based Banach space valued multivariate neural network approximations

Here we expose multi-composite multivariate quantitative approximations of Banach space valued continuous multivariate functions on a box or \(\mathbb {R} ^{N},\) ${\mathbb{R}}^N,$ \(N\in \mathbb {N}\) $N\in \mathbb{N}$ , by the multi-composite multivariate normalized, quasi-interpolation, Kantorovich type and quadrature type neural network operators. We treat also the case of approximation by multi-composite iterated multi-layer operators of the last four types. These approximations are derived by establishing multi-composite 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 multi-composite multidimensional density function induced by a multi-composite general sigmoid activation function. The approximations are pointwise and uniform. The related feed-forward neural network is with one hidden layer.