A comparative analysis of dynamics in the generalized Lorenz model with feedforward neural network validation

This paper presents a comprehensive analysis of the Lorenz model under the generalized $\psi$-Hilfer fractional derivative, which offers remarkable flexibility through its fractional order $\theta$, type parameter $\kappa$, and auxiliary function $\psi (t)$. The primary contributions of this work are threefold: First, we provided a detailed qualitative study establishing the existence, uniqueness, and Ulam-Hyers stability using fixed point theory. Second, we developed a novel hybrid numerical scheme specifically designed for the $\psi$-Hilfer fractional Lorenz system. Third, we validated the numerical solutions using a feedforward neural network trained on numerically generated data, providing independent verification of our results. Quantitative performance metrics confirmed excellent agreement between numerical solutions and neural network approximations, with mean squared error values ranging from 1.085 to 5.186 and $R^2$ values between 0.937 and 0.983 across all state variables. MATLAB simulations comprise two-dimensional and three-dimensional visualizations, which demonstrated how variations in the fractional order $\theta$ and type parameter $\kappa$ significantly influence solution profiles, with decreasing fractional order enhancing memory effects and stabilizing system dynamics. Our work demonstrates how the additional flexibility of the $\psi$-Hilfer operator enables more accurate modeling of memory effects in chaotic systems, while neural network validation provides a robust framework for solution verification.