Artificial Neural Network Simulation of Radiative Hybrid Nanofluid Flow with Forchheimer Resistance on a Permeable Sheet

This study presents a numerical simulation utilizing the Levenberg–Marquardt backpropagation (LMBP) algorithm to examine the radiative flow of hybridized cobalt ferrite \(\left( {CoFe_{2} O_{4} } \right)\) and magnetite \(\left( {Fe_{3} O_{4} } \right)\) nanofluids over a permeable stretching surface, with a focus on the Forchheimer effect. Hybrid nanofluids, which consist of nanoparticles with distinct thermophysical properties, are known to exhibit enhanced heat transfer performance compared to conventional single-phase fluids. The incorporation of radiation heat transfer and the Forchheimer effect into the fluid model offers a more comprehensive representation of complex heat transport processes relevant to industrial applications. A novel hybrid nanofluid composed of cobalt ferrite and magnetite nanoparticles is introduced, demonstrating a significant improvement in the thermal characteristics of the base fluid. A similarity transformation was applied to convert the governing partial differential equations into ordinary differential equations (ODEs). The impacts of various parameters on the flow field were investigated and visualized through graphical plots. Numerical solutions obtained via the BVP4C (boundary value problem 4C) method was compared with existing literature, showing excellent agreement. The findings contribute to the optimization of thermal management systems in advanced engineering and industrial processes. Additionally, the LMBP algorithm was validated through error analysis, regression metrics, adaptive control parameter evaluations, and iterative learning curve studies, demonstrating convergence, accuracy, and efficiency. Performance verification was conducted using regression plots, mean squared error (MSE), and error histograms, ensuring the robustness of the LMBP approach in solving the problem. In addition, the numerical solution is obtained by training the network with the LMPB algorithm by developing an accurate surrogate using 70%/10%/10% split for training/testing/validation. The LMPB algorithm obtained a best MSE of \(4 \cdot 04419 \times 10^{ - 9}\) at epoch \(890\), with errors closely grouped around zero and an overall regression coefficient \(R \cong 1\) across the process demonstrating excellent agreement with the numerical targets.