Artificial intelligence-enhanced algebraic multigrid for 3D finite element simulations

Abstract

The paper presents preliminary investigations into a strategy for solving linear systems resulting from 3D finite element simulations, based on the algebraic multigrid (AMG) method, enhanced using artificial intelligence techniques. In particular, we adapt to 3D problems the algorithm presented in Luz et al. (2020) for using a graph neural network to create the prolongation and restriction operators in a way that will improve convergence. The process of training the network proceeds on the basis of a set of system matrices obtainedfor tasks much smaller in scale than the target problem to be solved. Learning is aimed at decreasing the spectral radius of the error propagation matrix after applying modified prolongation and restriction. We describe some implementation details of the solver developed based on the presented strategy and show several numerical examples of its application for medium-sized problems.