Accelerating Algebraic Multigrid Methods via Artificial Neural Networks

Abstract

We present a novel deep learning-based algorithm to accelerate—through the use of Artificial Neural Networks (ANNs)—the convergence of Algebraic Multigrid (AMG) methods for the iterative solution of the linear systems of equations stemming from finite element discretizations of Partial Differential Equations (PDE). We show that ANNs can be successfully used to predict the strong connection parameter that enters in the construction of the sequence of increasingly smaller matrix problems standing at the basis of the AMG algorithm, so as to maximize the corresponding convergence factor of the AMG scheme. To demonstrate the practical capabilities of the proposed algorithm, which we call AMG-ANN, we consider the iterative solution of the algebraic system of equations stemming from finite element discretizations of two-dimensional model problems. First, we consider an elliptic equation with a highly heterogeneous diffusion coefficient and then a stationary Stokes problem. We train (off-line) our ANN with a rich dataset and present an in-depth analysis of the effects of tuning the strong threshold parameter on the convergence factor of the resulting AMG iterative scheme.