Brownian motion approximation by parametrized and deformed neural networks

The first author recently derived several approximation results by neural network operators see the new monograph (Anastassiou GA, Parametrized, deformed and general neural networks, accepted. Springer, Heidelberg, 2023). There, the approximation methods derived from the parametrized and deformed neural networks induced by the q-deformed and lambda-parametrized logistic and hyperbolic tangent activation functions. The results we apply here are univariate on a compact interval, regular and fractional. The outcome is the quantitative approximation of Brownian motion over the three dimensional sphere. We derive several Jackson type inequalities estimating the degree of convergence of our neural network operators to a general expectation function of Brownian motion. We give a detailed list of approximation applications regarding the expectation of well known functions of Brownian motion. Smoothness of our functions is taken into account producing higher speeds of approximation. At the end of the article we provide a rich list of interesting related references.