Multivariate Neural Network Operators Research Articles

Fractional type multivariate neural network operators

In this paper, we introduce a novel family of multivariate neural network operators involving Riemann‐Liouville fractional integral operator of order α. Their pointwise and uniform approximation results are presented, and new results concerning the rate of convergence in terms of the modulus of continuity are estimated. Moreover, several graphical and numerical results are presented to demonstrate the accuracy, applicability, and efficiency of the operators through special activation functions. Finally, an illustrative real‐world example on the recent trend of novel corona virus Covid‐19 has been investigated in order to demonstrate the modeling capabilities of the proposed neural network operators.

Asymptotic expansions and Voronovskaja type theorems for the multivariate neural network operators

In this paper, an asymptotic formula for the so-called multivariate neural network (NN) operators has been established. As a direct consequence, a first and a second order pointwise Voronovskaja type theorem has been reached. At the end, the particular case of the NN operators activated by the logistic function has been treated in details.

Approximation theorems for a family of multivariate neural network operators in Orlicz-type spaces

In this paper, we study the theory of a Kantorovich version of the multivariate neural network operators. Such operators, are activated by suitable kernels generated by sigmoidal functions. In particular, the main result here proved is a modular convergence theorem in Orlicz spaces. As special cases, convergence theorem in $$L^p$$ -spaces, interpolation spaces, and exponential-type spaces can be deduced. In general, multivariate approximations by constructive neural network algorithms are useful for applications to neurocomputing processes involving high dimensional data. At the end of the paper, several examples of activation functions of sigmoidal-type for which the above theory holds have been described.

Convergence results for a family of Kantorovich max-product neural network operators in a multivariate setting

AbstractThe theory of multivariate neural network operators in a Kantorovich type version is here introduced and studied. The main results concerns the approximation of multivariate data, with respect to the uniform andLpnorms, for continuous andLpfunctions, respectively. The above family of operators, are based upon kernels generated by sigmoidal functions. Multivariate approximation by constructive neural network algorithms are useful for applications to neurocomputing processes involving high dimensional data. At the end of the paper, several examples of sigmoidal functions for which the above theory holds have been presented.

Pointwise and uniform approximation by multivariate neural network operators of the max-product type

In this article, the theory of multivariate max-product neural network (NN) and quasi-interpolation operators has been introduced. Pointwise and uniform approximation results have been proved, together with estimates concerning the rate of convergence. At the end, several examples of sigmoidal activation functions have been provided.

Multivariate neural network operators with sigmoidal activation functions

In this paper, we study pointwise and uniform convergence, as well as order of approximation, of a family of linear positive multivariate neural network (NN) operators with sigmoidal activation functions. The order of approximation is studied for functions belonging to suitable Lipschitz classes and using a moment-type approach. The special cases of NN operators, activated by logistic, hyperbolic tangent, and ramp sigmoidal functions are considered. Multivariate NNs approximation finds applications, typically, in neurocomputing processes. Our approach to NN operators allows us to extend previous convergence results and, in some cases, to improve the order of approximation. The case of multivariate quasi-interpolation operators constructed with sigmoidal functions is also considered.

Rate of convergence of some multivariate neural network operators to the unit

This paper deals with the determination of the rate of convergence to the unit of some multivariate neural network operators, namely, the Cardaliaguet-Euvrard and “squashing” operators. This is given through the multidimensional modulus of continuity of the involved multivariate function or its partial derivatives of specific order that appear in the right-hand side of the associated multivariate Jackson type inequality.