Numerical solution of nonlinear stochastic differential equations with fractional Brownian motion using fractional-order Genocchi deep neural networks

Abstract

In this work, a new computational scheme namely fractional-order Genocchi deep neural network (FGDNN) is introduced to solve a class of nonlinear stochastic differential equations (NSDEs) driven by fractional Brownian motion (FBM) with Hurst parameter H∈(0,1). For generating the fractional Brownian motion, derivative and fractional-order integral operational matrices based on the fractional-order Genocchi functions and the classic Brownian motion approximation with help of the Gauss–Legendre quadrature are obtained. The FGDNN method is utilized the fractional-order Genocchi functions and Tanh function as activation functions of the deep structure. By considering deep neural network’s ability in approximating a nonlinear function, we present a new approximate function to estimate unknown function. This approximate function contains the FGDNN with unknown weights. Using the classical optimization method and Newton’s iterative scheme, the weights are adjusted such that the approximate function satisfies the under study problem. The convergence analysis of the mentioned scheme is discussed. Finally, some illustrative examples are included to show the applicability, accuracy and efficiency of the new method. The FGDNN method is compared with the analytical solutions and the numerical results obtained through the Chebyshev cardinal wavelets and hat functions methods.