Deep Learning of Preconditioners for Conjugate Gradient Solvers in Urban Water Related Problems

TL;DR

This paper introduces a machine learning-based approach using CNNs to automatically design effective preconditioners for conjugate gradient solvers in large-scale water engineering problems, outperforming traditional handcrafted methods.

Contribution

It presents a novel method employing CNNs to generate preconditioners automatically, reducing reliance on expert knowledge and improving convergence rates in fluid simulation applications.

Findings

Learned preconditioners outperform traditional methods like incomplete Cholesky.

The approach significantly accelerates convergence in fluid simulation problems.

Machine learning can effectively tailor preconditioners to specific problem instances.

Abstract

Solving systems of linear equations is a problem occuring frequently in water engineering applications. Usually the size of the problem is too large to be solved via direct factorization. One can resort to iterative approaches, in particular the conjugate gradients method if the matrix is symmetric positive definite. Preconditioners further enhance the rate of convergence but hitherto only handcrafted ones requiring expert knowledge have been used. We propose an innovative approach employing Machine Learning, in particular a Convolutional Neural Network, to unassistedly design preconditioning matrices specifically for the problem at hand. Based on an in-depth case study in fluid simulation we are able to show that our learned preconditioner is able to improve the convergence rate even beyond well established methods like incomplete Cholesky factorization or Algebraic MultiGrid.