Legendre based neural networks integrated with heuristic algorithms for the analysis of Lorenz chaotic model: an intelligent and comparative study

The optimal control methods for managing Lorenz model are achieved using an innovative intelligent computing framework that integrates artificial neural networks with stochastic unsupervised learning-based optimizers, specifically Firefly (FA) and Archimedes algorithm (AOA). The hyper parameters (weights & biases) of unsupervised neural networks are refined through Legendre polynomials Artificial Neural Networks (LENN) optimized with global search techniques, including FA-AOA collectively referred to as LENN-FA-AOA. This design approach is employed to Lorenz model across three (3) different scenarios using various step sizes and input intervals. The study's findings reveal that to minimize the computational cost to find the solution of nonlinear chaotic systems by intelligence strategies. The absolute errors values from LENN-FA-AOA with reference solution are being ranged from 3.22 × 10^-5 to 3.06 × 10^-7, 4.56 × 10^-5 to 7.27 × 10^-8 and 5.17 × 10^-5 to 2.11 v 10^-7. Data validation through extensive graphical simulations confirms the effectiveness and robustness of the proposed intelligent solver. The LENN-FA-AOA solver is tested under different initial conditions of Lorenz model to assess its reliability, safety, and tolerance. Through this advanced LENN intelligent design framework, an objective/fitness optimization function is developed within a feedforward neural network. The hybrid FA-AOA optimization is also investigated to verify the LENN model accuracy and reliability. Mean square error (MSE) and TIC graphs are constructed to evaluate the proposed method integrity and efficiency.