Convergence Estimates of Neural Network Operators

In this article, the aim is to investigate approximation error that is produced by the approximation operators activated by sigmoidal function of logarithmic type. For this we study a family of feed-forward neural network operators and approximate a continuous function that is specified on a close and bounded set to any desired level of precision by increasing amount of hidden neurons, also construct a class of quasi-interpolation operators. We have to obtain the approximation order for continuous derivative functions of first order by using neural network operator activated by logistic sigmoid and hyperbolic tangent functions and also establish some preliminary results. We have to investigate the case of multivariate neural networks’ approximation associated with sigmoidal activation function, and estimated quasi-interpolation operator for multivariate NN operators, constructed using sigmoidal functions and calculate the convergence and approximation order for linear-positive multivariate neural network operators. We study the estimation of the error for continuous and differentiable functions in uniform norm and constructed the corresponding approximation neural network operators for the bivariate functions and calculate the error for a function specified on compact interval by applying the approach of approximate approximation. The aim is to investigate univariate-quantitative approximation for the functions with real and complex values on a closed and bounded interval by employing quasi-interpolation hyperbolic tangent neural network operators and elaborate feed-forward neural network with single hidden layer activated by hyperbolic tangent function. We have to investigate the approximation and convergence of continuous function by using Hadamard-type fractional exponential sampling neural network Kantorovich operators for one and two dimensional case and establish the practical application in image processing. We have to employ Reimann-Liouville neural network Kantorovich operators to improve the function’s approximation ability by demonstrating it numerically and practically in image processing and also apply in real life case by taking COVID cases and deaths. We have to obtain Voronovskaja-type theorem and quantitative estimates for the Kantorovich type exponential sampling operators, and improve the order of approximation, using linear combination of convex type for the proposed operator. Convergence like pointwise and uniform must be studied for the class of Kantorovich type multivariate NN operators and have to establish neural network operators associated with ramp functions and proved its approximation properties.