Computational Intelligence of Numerical Dynamics of Nanofluidic Model

This study investigates the flow dynamics of a nanofluid by modeling a system of nonlinear ordinary differential equations (ODEs). The system is transformed into a real dataset and solved using artificial neural networks (ANNs) trained via the Levenberg–Marquardt backpropagation (neural networks with backpropagation and machine learning [NN-BPML]) method, incorporating the explicit Runge–Kutta (ERK) numerical approach. The ANNs are trained to approximate the solutions of the nonlinear system, with particular attention given to the physical relevance of parameters, notably the “⅄†governing nanofluid movement. A comprehensive analysis involving training, testing, validation, performance evaluation, and regression analysis is conducted. Numerical experiments explore both rapid and slow steady-state behaviors, revealing characteristics rarely observed in the integer-order models. The accuracy and stability of the proposed model are assessed through mean-squared error, error histograms, and regression plots, confirming the reliability of the developed computational framework.