Neural network approximation of the composition of fractional operators and its application to the fractional Euler-Bernoulli beam equation

In this paper, we propose a new approach to approximation of the left and the right fractional Riemann - Liouville integrals as well as the compositions of these two operators, based on a shallow neural network with $ReLU$ as an activation function. We apply the proposed method to the fractional Euler - Bernoulli beam equation with fixed-supported and fixed-free ends, and we provide numerical simulations for constant, power and trigonometric functions. Finally, we compare the obtained results with the exact solutions of the considered problems.