Machine learning techniques applied to preconditioning of linear systems

This work explores the use of machine learning techniques to improve existing methods for preconditioning and solving large, sparse linear systems. Standard feed-forward neural networks are utilized to predict optimal parameter choices for Incomplete $LU$ factorization, with the resulting models being tested on new, unseen systems from a variety of disciplines. Results show an improvement in convergence times over standard, fixed default values. Then, a graph neural network is constructed to select the entries for the prolongation operator in Algebraic Multigrid. The learned operator is tested on a difficult class of three-dimensional Poisson problems and shows an improvement over both standard operators and other machine learning operators from the literature.