Learning optimal multigrid smoothers via neural networks
This addresses the challenge of problem-dependent smoothing in multigrid methods for PDEs and graph Laplacians, offering a domain-specific improvement that is incremental by building on existing neural network techniques.
The paper tackles the problem of finding optimal smoothing algorithms in multigrid methods for solving linear systems from PDEs by proposing an adaptive framework that learns optimized smoothers using convolutional neural networks trained on small-scale problems, resulting in improved convergence rates and solution time for anisotropic rotated Laplacian problems compared to classical methods.
Multigrid methods are one of the most efficient techniques for solving linear systems arising from Partial Differential Equations (PDEs) and graph Laplacians from machine learning applications. One of the key components of multigrid is smoothing, which aims at reducing high-frequency errors on each grid level. However, finding optimal smoothing algorithms is problem-dependent and can impose challenges for many problems. In this paper, we propose an efficient adaptive framework for learning optimized smoothers from operator stencils in the form of convolutional neural networks (CNNs). The CNNs are trained on small-scale problems from a given type of PDEs based on a supervised loss function derived from multigrid convergence theories, and can be applied to large-scale problems of the same class of PDEs. Numerical results on anisotropic rotated Laplacian problems demonstrate improved convergence rates and solution time compared with classical hand-crafted relaxation methods.