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Learning to Optimize Multigrid PDE Solvers
Daniel Greenfeld, Meirav Galun, Ron Kimmel, Irad Yavneh, Ronen Basri
TL;DR
Multigrid solvers for new PDEs can require problem-specific prolongation operators, making them challenging to devise. The paper trains a neural network to map a family of discretized diffusion problems to prolongation operators, and reports improved convergence over Black Box multigrid. It also reports generalization across problem sizes and boundary conditions, and to some extent across coefficient distributions.
Problem
Applying off-the-shelf multigrid algorithms to new problems is often non-optimal because suitable prolongation operators can require expert design.
Source: §1 · p. 1Method
Results
Experiments on 2D diffusion equations show improved convergence rates compared with the classical Black Box multigrid scheme.
Source: §5 · p. 8Takeaways & Limitations
The experiments show generalization across problem size and boundary conditions, and to some extent across the underlying coefficient distribution.
Source: §4 · p. 6, §5 · p. 8The multigrid discussion assumes that A is symmetric positive definite and P has full rank.
Source: §2.1 · p. 4
Abstract
from arXiv · showhide
Constructing fast numerical solvers for partial differential equations (PDEs) is crucial for many scientific disciplines. A leading technique for solving large-scale PDEs is using multigrid methods. At the core of a multigrid solver is the prolongation matrix, which relates between different scales of the problem. This matrix is strongly problem-dependent, and its optimal construction is critical to the efficiency of the solver. In practice, however, devising multigrid algorithms for new problems often poses formidable challenges. In this paper we propose a framework for learning multigrid solvers. Our method learns a (single) mapping from a family of parameterized PDEs to prolongation operators. We train a neural network once for the entire class of PDEs, using an efficient and unsupervised loss function. Experiments on a broad class of 2D diffusion problems demonstrate improved convergence rates compared to the widely used Black-Box multigrid scheme, suggesting that our method successfully learned rules for constructing prolongation matrices.
1. Introduction
Off-the-shelf multigrid methods may need expertly designed prolongation operators for new problems. This paper learns operators for families of diffusion equations and reports improved convergence over Black Box multigrid; earlier neural PDE approaches often require separate training for each equation.
1. Introduction
- Prolongation operators are critical to efficient multigrid solvers, yet applying off-the-shelf algorithms to new problems is often non-optimal.The paper identifies expert design of these operators as a challenge for new problems.Source: §1 · p. 1
- A single neural network learns an unsupervised mapping from discretized diffusion equations to prolongation operators for a family of problems.Training uses algebraic properties rather than ground-truth operators or numerical solutions.Source: §1 · p. 1
- The method reports superior convergence rates to Black Box multigrid across varied scenarios, including changes in grid size, boundary conditions, and instance type.Training uses block-periodic instances, while testing includes general instances and different sizes and boundary conditions.Source: §1 · p. 2
1.1. Previous efforts
- Earlier neural PDE approaches often target specific equations and require separate training for each; another line uses multilevel strategies to accelerate image-classification training.The cited PDE approaches include methods for specific problems and multigrid operators that do not generalize without retraining.Source: §1.1 · p. 2
2. Multigrid background and problem setting
The section presents a two-dimensional diffusion problem and explains how multigrid combines relaxation with coarse-grid correction. Its efficiency depends on a prolongation operator that captures smooth error while remaining sparse.
2. Multigrid background and problem setting
- The problem setting discretizes two-dimensional diffusion on a square grid, yielding a sparse matrix whose stencil couples each vertex unknown to its neighbors.The discrete diffusion coefficients are at cell centers, while the solution and right-hand side are at grid vertices; the matrix is positive definite for Dirichlet boundary conditions.Source: §2 · p. 2
- Gauss-Seidel is inexpensive per iteration but converges slowly as a stand-alone solver on large ill-conditioned systems; a few sweeps smooth the error for coarse-grid treatment.The passage gives O(n^2) arithmetic operations per iteration and explains that the remaining error varies slowly relative to mesh size.Source: §2 · p. 3
2.1. Multigrid Cycle
- Relaxation smooths the error left on the fine grid, allowing coarse-grid correction to approximate it; the correction is transferred back through prolongation.A multigrid cycle repeats this process on successively coarser grids, with linear computational complexity; W cycles cost more than V cycles but may help on challenging problems.Source: §2.1 · p. 3
- The two-grid error propagation matrix governs convergence, and relaxation and coarse-grid correction must reduce complementary components of the error.The coarse-grid correction subtracts the error component in the range of the prolongation matrix.Source: §2.1 · pp. 3–4
- A good prolongation operator must represent algebraically smooth error in its range while remaining sparse for computational efficiency.With the relaxation scheme and sweep count fixed, cycle efficiency is governed by the prolongation operator.Source: §2.1 · p. 4
3. Method
The method learns sparse prolongation operators from discretization matrices using a neural network and a convergence-oriented objective. Local stencil inputs and block Fourier analysis make the construction and training practical for diffusion problems.
3. Method
- A neural network maps discretization matrices A to sparse prolongations Pθ(A), optimizing expected error-propagation spectral radius under fixed Gauss-Seidel relaxation.The learning objective is defined over a distribution of discretization matrices.Source: §3 · p. 4
3.1. Inferring P from local information
- For each coarse point, the network uses five neighboring 3 × 3 stencils; it learns four weights, while four diagonal weights enforce Au = 0.Applying the same network across coarse points constructs the full prolongation matrix; local inference has constant-time computation per coarse point and transfers across grid sizes.Source: §3.1 · pp. 4–5, Figure 2 · p. 4
3.2. Fourier analysis for efficient training
- Block Fourier analysis decomposes the training loss into small blocks, while a squared Frobenius-norm surrogate avoids direct spectral-radius optimization.Block-periodic training permits small-matrix inversions; the authors report good large-grid performance after training on a 32×32 grid, and expect local dependence to transfer to general matrices.Source: §3.2 · pp. 5–6
4. Experiments
The experiments compare the learned multigrid solver with Black Box multigrid across diffusion problems and evaluation settings. The learned method generally has better convergence, including when test conditions differ from training.
4. Experiments
- The network is trained on relatively small periodic block-circulant problems and tested across grid sizes, general coefficients, Dirichlet boundaries, and another domain; cycle runtime matches Black Box, but setup is costlier.The experiments train and test on log-normal coefficients; the network's once-per-problem setup phase costs more because it computes the prolongation with the trained network rather than explicit formulas.Source: §4 · p. 6
4.1. Evaluation
- The network achieves a lower average spectral radius than Black Box multigrid on 64 × 64 Dirichlet problems, despite differences between training and test conditions.Training used block-periodic coefficients, Frobenius-norm minimization, and smaller grids; testing used general coefficients, Dirichlet boundaries, spectral radius, and larger grids.Source: §4.1 · p. 7, Table 1 · p. 7
- For 1024 × 1024 W-cycle tests, asymptotic error reduction factors are about 0.16 for the network-based method and 0.2 for Black Box multigrid.The experiment averaged 100 problems with Dirichlet boundary conditions.Source: §4.1 · p. 7, Figure 3 · p. 7
- Across 32 × 32 to 1024 × 1024 grids, the learned method has superior asymptotic convergence factors without significant deterioration on larger grids; Table 2 reports V- and W-cycle success rates.Success rate is the percentage of instances in which the network outperformed Black Box multigrid.Source: §4.1 · p. 7, Figure 4 · p. 7, Table 2 · p. 8
- When trained on log-normal coefficients and tested on uniform coefficients over [0, 1], the network retains superior convergence factors, but its advantage narrows.Figure 5 presents V- and W-cycle results for this distribution shift.Source: §4.1 · p. 8, Figure 5 · p. 8
- With a positive diagonal term, the network retains its advantage across tested εh2 values, including after training with εh2 = 10−8.The passage identifies this setting as relevant to time-dependent parabolic PDEs.Source: §4.1 · p. 8, Figure 6 · p. 8
5. Conclusion
The paper frames multigrid solver construction for parametric PDE families as learning a mapping from discretization matrices to prolongation operators. Experiments report improved convergence and generalization on 2D diffusion problems.
5. Conclusion
- Experiments on 2D diffusion equations show improved convergence rates over the classical Black Box scheme.The reported generalization extends to problem size and boundary conditions, and to some extent to the underlying distribution.Source: §5 · p. 8
- The framework learns a single mapping from discretization matrices to prolongation operators using an efficient, unsupervised procedure.The target is a family of parametric PDEs.Source: §5 · p. 8
- Experiments show generalization with respect to problem size, boundary conditions, and, to some extent, the underlying distribution.Source: §5 · p. 8
- The authors identify triangulated and unstructured grids, and simpler regression models for faster inference, as future directions.Source: §5 · p. 8
A. Appendix: Block Fourier analysis
The appendix uses block Fourier analysis to transform a block-circulant matrix into smaller blocks. It describes the indexing and Fourier construction behind that transformation, then notes its extension to higher dimensions.
A. Appendix: Block Fourier analysis
- The analysis starts with an n × n block-circulant matrix, where n = kb, and decomposes row and column indices into within-block and block indices.The indices are numbered from zero; the decomposition is l = l0 + tk and j = j0 + sk.Source: A. Appendix: Block Fourier analysis · p. 10
- The matrix W is built from the first b Fourier components; the appendix states that bW is unitary and that the transformed matrix has b blocks of size k × k.Figure 7 illustrates W for k = 3 and b = 4.Source: A. Appendix: Block Fourier analysis · p. 10, Figure 7 · p. 11
- The proof uses the index decomposition to show that AWj vanishes unless the block indices match, yielding the stated block-periodic form.For matching indices, the proof says the terms in the final sum equal 1, so the sum is b.Source: A. Appendix: Block Fourier analysis · p. 11
- The appendix states that block Fourier analysis extends recursively to any PDE dimension, and that stencil diameter at most k lets each B element be computed from one A element.Source: A. Appendix: Block Fourier analysis · p. 11