Quantitative Estimates for Neural Network Operators Implied by the Asymptotic Behaviour of the Sigmoidal Activation Functions

In this paper, for the neural network (NN) operators activated by sigmoidal functions, we will obtain quantitative estimates in direct connection with the asymptotic behavior of their activation functions. We will cover all cases of discrete and Kantorovich type versions of NN operators in both univariate and multivariate settings since the proofs of the latter ones can be reduced in a convenient way to the proofs in the univariate cases. The above mentioned quantitative estimates have been achieved for continuous functions and using the well-known modulus of continuity. Further, in the case of the Kantorovich version of the NN operators, we also provide quantitative estimates for Lp functions with respect to the Lp norm; this has been done using the well-known Peetre K-functionals. At the end of the paper, some examples of sigmoidal functions are presented; also the case of NN operators activated by the rectified linear unit (ReLU) functions are considered. Furthermore, numerical examples in both univariate and multivariate cases have been provided in order to show the approximation performances of all the above operators.