Hybrid finite volume physics – informed neural network framework for stationary bifurcation prediction in magnetohydrodynamic convection

Full abstract

Read the full abstract

Purpose In this study, two finite – volume physics – informed neural network (FV–PINN) strategies are investigated for the analysis of stationary bifurcations in magnetohydrodynamic (MHD) convection. This study aims to reduce the computational cost of classical bifurcation analysis while using the predictive capability of PINNs to determine the critical Reynolds number (Rec) over a range of Hartmann (Ha) and Grashof (Gr) numbers. Design/methodology/approach A benchmark problem of rectangular lid-driven cavity with asymmetric driving is considered. In the first approach, finite – volume (FV) simulations provide data to train the neural network, and a relaxed bifurcation constraint is used to construct the Rec – Ha stability boundary. In the second approach, FV discretization is explicitly embedded within the PINN training through residual evaluation with the reconstruction of a Rec – (Ha Gr) stability surface. The training data are generated using a FV-based algorithm coupled with a classical bifurcation detection method. Findings The training dynamics are analyzed through the evolution of the total loss function, revealing bounded convergence in both approaches. The comparison with input data has shown that the first method gives comparatively better results than the second method for one-parameter predictions (Rec – Ha) but fails when the constraints are upgraded for a two-parameter space (Rec – (Ha, Gr)). For the first method, oscillatory behavior is observed with relaxed constraints due to competition between bifurcation and monotonicity losses, while for the second method, the total loss monotonically approaches a stable minimum. Originality/value To the best of the authors’ knowledge, this work represents the first application of hybrid FV – PINNs to bifurcation analysis in MHD convection.