Learnable stability-aware computational grid coarsening for accelerating physics simulations

Abstract Efficient simulations of complex physical systems described by partial differential equations (PDEs) require computational methods that can reduce the resource demands without sacrificing the accuracy. Here, we introduce a framework based on graph neural networks for learnable self-supervised differentiable coarsening of unstructured computational grids. We leverage graph-based representation of the physical system and offer a computational grid coarsening method which preserves the underlying physical properties together with the stability of the chosen numerical scheme with the help of the designed loss terms. The coarsening model is trained in a self-supervised way by minimizing the error between the output of the simulations on the coarsened and original grids. We evaluate the approach on four PDE systems of different types, covering both linear and nonlinear regimes, including subsurface flow and wave propagation. We demonstrate that the proposed framework exhibits the ability to maintain high fidelity in simulation outputs even after 95% reduction in the number of nodes, significantly reducing computational overhead. We show that the model exhibits generalizability to unseen scenarios, thereby outperforming the baselines. The generality of the developed framework is also proven by its possibility to adapt to implicit numerical schemes used to model stiff systems of PDEs. Thus, the developed approach demonstrates the ability to accelerate physical simulations without compromising accuracy.