A physics-informed neural network approach for natural convection boundary-layer flow in nanofluid-saturated non-darcy porous medium

This study presents a robust deep learning framework based on Physics-Informed Neural Networks (PINNs) for modeling natural convection boundary-layer flow over a vertical plate embedded in a non-Darcy porous medium saturated with nanofluid. The mathematical formulation is based on the extended Cheng–Minkowycz problem, incorporating the Buongiorno nanofluid model to account for Brownian motion and thermophoresis effects, along with the Forchheimer–Darcy model to describe nonlinear porous medium resistance. Through appropriate similarity transformations, the governing nonlinear partial differential equations are reduced to a system of coupled nonlinear ordinary differential equations. These equations are then solved using PINNs, enabling accurate and mesh-free approximation of the solution while preserving the underlying physical laws. A comprehensive analysis is conducted to investigate the influence of key deep learning hyperparameters, including the learning rate and neural network architecture, on convergence and solution accuracy. The reliability of the proposed PINN model is validated through comparison with numerical solutions obtained using the implicit finite difference method implemented via the bvp4c solver. Furthermore, a detailed parametric study is performed to examine the effects of governing physical parameters on velocity, temperature, and nanoparticle concentration profiles, as well as on engineering quantities of interest such as the local Nusselt number. The results reveal that both nanoparticle effects and porous medium resistance significantly influence heat and mass transfer characteristics, highlighting the effectiveness of the proposed PINN framework for complex transport phenomena.