This research investigates heat and mass transfer in hydro-nanofluid (Fe 3 O 4 -H 2 O) flow over an inclined, rotating disk embedded in a porous medium, accounting for viscous dissipation and Joule heating. The analysis is conducted by means of a stochastic solver leveraging Levenberg–Marquardt backpropagation neural networks (LMNN). Inclined rotating disks portray a significant role in gas turbines and aerospace engines, where they are used to regulate thermal loads under extreme operating conditions. These parts are especially essential to industrial gearboxes and wind turbines, where high rotational speeds and temperature gradients subject them to severe mechanical and thermal strains. The Darcy–Forchheimer model (DFM) is used to account for inertial and porous media effects, providing a more precise and physically realistic depiction of the system. The governing nonlinear PDEs are reduced to a set of nonlinear ODEs using scaling group transformations. The differential transform approach is used to obtain analytical solutions for ODEs. The generated data are used as the neural network's training set. The neural network's learning procedure involves validation, training, and testing phases to accurately map and predict results across various scenarios. These scenarios are produced by amending key physical parameters, including the porosity parameter, the Prandtl number, and the strength of the magnetic field. The effect of parameters on all profiles has been analyzed thoroughly and plotted. The Levenberg–Marquardt backpropagation neural network algorithm has been strictly validated by convergence assessment, stability confirmation, and computational performance evaluation. The model's accurateness is systematically measured using performance metrics, regression plots, error histogram, and goodness of fit analyses for the study of hydro-nanofluid system. The exactness of the results evaluated by the differential transform method has been ratified completely through comparison with established results from the literature and numerical simulations, demonstrating excellent agreement. A graphical abstract summarizing the key findings is provided in Fig. 1 . Description of problem
Reshu Gupta (Sat,) studied this question.