This work tackles the issue of enhancing heat transmission and minimizing entropy formation in tiny enclosures pertinent to thermal energy storage. It looks at how magnetohydrodynamic (MHD), non-Darcian porous media, a ternary hybrid nanofluid composition (Fe3O4–hBN–CuO/water), and triangle corrugation work together in a corrugated rectangular cavity. The goal is to figure out how these things affect convection, entropy formation, and the overall efficiency of the thermodynamic system. Utilizing the Galerkin finite element technique (GFEM), we found numerical solutions to the mathematical models for momentum, energy, and entropy generation. The effects of the porosity parameter, ternary nanoparticle concentration, Hartmann number, Darcy number, and Rayleigh number were carefully studied for the cavities' flat and triangular corrugated walls. Artificial Neural Network (ANN) model was developed and trained to predict the average Nusselt number and total entropy generation with high precision, using fewer computational resources compared to conventional CFD approaches. It is observed that the ANN model is used mostly as an ancillary prediction instrument derived from FEM-generated data, rather than as the principal computational framework. The results show that corrugated shapes improve local heat transfer by increasing the surface area and causing flow disruptions. However, too many corrugations lower the average Nusselt numbers because they cause recirculation. Higher Rayleigh numbers make buoyancy-driven convection stronger, whereas larger magnetic fields make circulation weaker, which makes conduction-dominated transport more likely and lowers entropy generation. The porosity and Darcy number have a big effect on convective intensity and entropy formation. On the other hand, the right number of nanoparticles may boost thermal conductivity without making irreversibility too high. The ANN model showed great prediction ability (MSE≈1.12 × 10⁻⁷), which proved that it works well for quickly testing Multiphysics systems. These results show that integrating ternary nanofluids, controlling porous media, and changing the magnetic field may improve thermal performance in advanced applications, including solar collectors, cooling electronics, and thermal energy storage devices. Combining ANN prediction gives us a solid base for designing and improving next-generation heat management solutions in a way that works well.