AI-enhanced iterative solvers for accelerating the solution of large scale parametrized linear systems of equations

Recent advances in the field of machine learning open a new era in high performance computing. Applications of machine learning algorithms for the development of accurate and cost-efficient surrogates of complex problems have already attracted major attention from scientists. Despite their powerful approximation capabilities, however, surrogates cannot produce the `exact' solution to the problem. To address this issue, this paper exploits up-to-date ML tools and delivers customized iterative solvers of linear equation systems, capable of solving large-scale parametrized problems at any desired level of accuracy. Specifically, the proposed approach consists of the following two steps. At first, a reduced set of model evaluations is performed and the corresponding solutions are used to establish an approximate mapping from the problem's parametric space to its solution space using deep feedforward neural networks and convolutional autoencoders. This mapping serves a means to obtain very accurate initial predictions of the system's response to new query points at negligible computational cost. Subsequently, an iterative solver inspired by the Algebraic Multigrid method in combination with Proper Orthogonal Decomposition, termed POD-2G, is developed that successively refines the initial predictions towards the exact system solutions. The application of POD-2G as a standalone solver or as preconditioner in the context of preconditioned conjugate gradient methods is demonstrated on several numerical examples of large scale systems, with the results indicating its superiority over conventional iterative solution schemes.