Finite element and neural network methods for fractional stochastic differential equations

This manuscript develops a fractional stochastic model with a tempered mixed Hadamard-Prabhakar operator.The proposed operator combines logarithmic Hadamard memory, Prabhakar-type Mittag-Leffler modulation, and exponential tempering, and therefore permits finite-memory damping while retaining nonlocal fractional behavior.The resulting neutral stochastic equation is reformulated as a stochastic Volterra integral equation.Existence and uniqueness are proved in a mean-square adapted Banach space by a contraction argument based on explicit kernel bounds.Finite-time mean-square stability and Ulam-Hyers stability are then established with computable constants.A memory finite-element discretization is derived for the new kernel, including deterministic convolution weights and stochastic Brownian increments.Numerical tests, convergence tables, stability perturbation tests, and computational examples are provided.Finally, an artificial neural network and a residual-informed deep-learning surrogate are trained on finite-element data to approximate the stochastic fractional trajectory.The numerical and learning results support the theoretical stability estimates and demonstrate the computational applicability of the proposed operator.