A reinforcement learning strategy to automate and accelerate h/p-multigrid solvers

A reinforcement learning strategy to automate and accelerate h / p-multigrid solvers David Huergo a,, Laura Alonso a, Saumitra Joshi a, Adrian Juanicotena a, Gonzalo Rubio a,b, Esteban Ferrer a,b a ETSIAE-UPM-School of Aeronautics, Universidad Polit ecnica de Madrid, Plaza Cardenal Cisneros 3, E-28040 Madrid, Spain b Center for Computational Simulation, Universidad Polit ecnica de Madrid, Campus de Montegancedo, Boadilla del Monte, 28660 Madrid, Spain Abstract We explore a reinforcement learning strategy to automate and accelerate h /p-multigrid methods in high-order solvers. Multigrid methods are very e fficient but require fine-tuning of numerical parameters, such as the number of smoothing sweeps per level and the correction fraction (i.e., proportion of the corrected solution that is transferred from a coarser grid to a finer grid). The objective of this paper is to use a proximal policy optimization algorithm to automatically tune the multigrid parameters and, by doing so, improve stability and e ffi ciency of the h / p-multigrid strategy. Our findings reveal that the proposed reinforcement learning h / p-multigrid approach significantly accelerates and improves the robustness of steady-state simulations for one dimensional advection-di ff usion and nonlinear Burgers' equations, when discretized using high-order h / p methods, on uniform and nonuniform grids. Keywords: Reinforcement Learning, Proximal Policy Optimization, PPO, Advection-di ff usion, Burgers' equation, High-order flux reconstruction, h / p-multigrid 1. Introduction Multigrid methods are widely recognized for minimizing time-to-convergence [36, 28] in numerical solvers and have become an essential tool also in the family of high-order (HO) methods [7]. These methods exploit the fact that errors represented on coarser discrete spaces have higher spatial frequencies than on the original discrete space, enabling faster damping. Traditional multigrid methods, proposed in the 1970s [3], rely on successively coarser meshes and are thus termed h-multigrid . In the context of HO solvers, we can represent high-order errors on lower orders, e ff ectively coarsening the polynomial order ( P). The resulting p-multigrid o ffers simplicity in transfer operations between levels, making it a natural choice to accelerate HO methods.