Deep Bernoulli Optimisation for Solving 2D/3D Ψ-Tempered Fractional Optimalcontrol Problems

Abstract

This study introduces two- and three-dimensional optimal control problems characterised by theψ-tempered fractional derivative and proposes a novel numerical methodology based on deepfractional-order Bernoulli optimisation to achieve their efficient solution. To this end, the problemsunder consideration are first reformulated as equivalent variational problems. Subsequently, a deepneural network employing the fractional-order Bernoulli functions and sinh as activation functions isutilised to approximate the state variable. To facilitate the effective implementation of the proposedmethod, we construct several integral operators of both integer and ψ-tempered fractional ordersbased on the basis functions derived from the deep neural network. These operators are discretisedvia the Fejér quadrature rule adapted to the ψ-tempered fractional calculus, ensuring stability andhigh-precision integration. The proposed approach converts the 2D/3D ψ-tempered fractional opti-mal control problem into an algebraic system using deep neural networks with integral operatorsand Gauss–Legendre integration, which is efficiently solved via Newton’s method. The proposedmethod combines simple implementation, low computational cost, and high accuracy. Its effective-ness and robustness are validated through several representative 2D and 3D examples, confirmingthe method’s applicability to complex fractional optimal control problems.