Deep Learning of Preconditioners for Conjugate Gradient Solvers in Urban Water Related Problems
This work addresses the need for efficient iterative solvers in urban water engineering, offering a novel automated alternative to expert-dependent handcrafted preconditioners, though it is incremental as it builds on existing preconditioning concepts with a new method.
The paper tackles the problem of designing effective preconditioners for conjugate gradient solvers in large-scale linear systems from water engineering, proposing a machine learning approach that uses a convolutional neural network to automatically generate preconditioners, which improved convergence rates beyond established methods like incomplete Cholesky factorization and Algebraic MultiGrid in a fluid simulation case study.
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.