Learning optimal multigrid smoothers via neural networks

2021/02/24 by Ru Huang, Huang, Ru, Ruipeng Li +3 · 6 citations
Computer Science · Engineering · Physics and Astronomy · #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (cs.LG) #Model Reduction and Neural Networks #Numerical Analysis (math.NA)

paper · pdf · doi:10.48550/arxiv.2102.12071

openalex publication_date 2021/02/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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.

Citations

• Learning to Optimize Multigrid PDE Solvers
• Universality of deep convolutional neural networks
• Optimizing Geometric Multigrid Methods with Evolutionary Computation
• Learning to Control PDEs with Differentiable Physics

Cited by

• Nodal Hybrid Neural Solvers for Parametric PDE Systems
• A robust and stable hybrid neural network/finite element method for 2D flows that generalizes to different geometries
• Data-driven linear solver selection and performance tuning for multiphysics simulations in porous media
• A Greedy PDE Router for Blending Neural Operators and Classical Methods
• Leveraging Operator Learning to Accelerate Convergence of the Preconditioned Conjugate Gradient Method
• Graph Lineages and Skeletal Graph Products

Related

• Learning Optimal Multigrid Smoothers via Neural Networks
• Learning Relaxation for Multigrid
• Learning to Optimize Multigrid PDE Solvers
• MGCNN: a learnable multigrid solver for sparse linear systems from PDEs on structured grids
• UGrid: An Efficient-And-Rigorous Neural Multigrid Solver for Linear PDEs