Learning to Optimize Multigrid PDE Solvers

Daniel Greenfeld; Irad Yavneh; Meirav Galun; Ronen Basri; Ron Kimmel

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Learning to Optimize Multigrid PDE Solvers

Daniel Greenfeld, Meirav Galun, Ron Kimmel, Irad Yavneh, Ronen Basri

Numerical Analysis · Submitted 25 Feb 2019

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim 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.

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arXiv ID1902.10248 (version 3)

Pith IDVUWH5III, Pith's permanent record of this paper and its review

CitationDaniel Greenfeld, Irad Yavneh, Meirav Galun, Ronen Basri, Ron Kimmel. "Learning to Optimize Multigrid PDE Solvers." arXiv:1902.10248 (2019). https://pith.science/paper/VUWH5III

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What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.16 · human participants 0.01 · controlled comparison 0.88 · randomized assignment 0.01 · states quantitative result 0.09 · compares with prior results 0.97 · mentions code or data availability 0.02.

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arXiv listing abstract v3. Results and the main claim are underlined; show every role.

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.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Constructing fast numerical solvers for partial differential equations (PDEs) is crucial for many scientific disciplines.	background 1.00	0.04	0.02	asserted 1.00
s1	A leading technique for solving large-scale PDEs is using multigrid methods.	background 1.00	0.04	0.02	asserted 1.00
s2	At the core of a multigrid solver is the prolongation matrix, which relates between different scales of the problem.	background 1.00	0.07	0.02	asserted 1.00
s3	This matrix is strongly problem-dependent, and its optimal construction is critical to the efficiency of the solver.	background 0.97	0.06	0.02	asserted 1.00
s4	In practice, however, devising multigrid algorithms for new problems often poses formidable challenges.	gap 0.74	0.05	0.02	asserted 0.99
s5	In this paper we propose a framework for learning multigrid solvers.	contribution 0.99	0.87	0.09	asserted 0.99
s6	Our method learns a (single) mapping from a family of parameterized PDEs to prolongation operators.	method 0.89	0.91	0.07	asserted 1.00
s7	We train a neural network once for the entire class of PDEs, using an efficient and unsupervised loss function.	method 0.99	0.95	0.07	asserted 1.00
s8	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.	result 0.91	0.96	0.88	hedged possibility 0.65

What kind of paper the classifier reports this is: method or tool 1.00 · evidence from simulation or computation 0.74.

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Original abstract

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.

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References

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Learned smoothing in operator-adapted multigrid: controlled accuracy gains and computational limits
math.NA 2026-10 accept novelty 6.0 of 10

Learned smoothing in a fixed operator-adapted multigrid cycle achieves up to 6x energy error reduction on channels but is overtaken by classical cycles with more steps and is slower than converged solvers.
Learning to Optimize: A Primer and A Benchmark
math.OC 2021-03 conditional novelty 4.0 of 10

A survey plus open benchmark of learned optimizers finds that structure-aware learned methods beat classic solvers on in-distribution problems, while generic learned optimizers generalize poorly.

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7 arXiv papers cite this in 14 sentences

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Builds on it 1 paper

Generalizing to unstructured problems an approach used in [ref] , we reduce the training complexity by training on a limited class of $A$ matrices called block-circulant matrices.

Learning Algebraic Multigrid Using Graph Neural Networks · Efficient Training on Block-Circulant Matrices · 0.90

Cites it as background 3 papers

For instance, [ref] proposed a series of data-driven approaches to design prolongations and restrictions in MG.

Meta-MgNet: Meta Multigrid Networks for Solving Parameterized Partial Differential Equations · Related Work · 0.89
The idea of learning the elements of multi-grid preconditioners with neural networks has been pursued in a number of earlier publications, see, e.g., [ref] .

Gauge-equivariant neural networks as preconditioners in lattice QCD · Introduction · 0.80
The last group, very different from previous two, adopts NN to optimize traditional iterative methods [ref] .

Variational training of neural network approximations of solution maps for physical models · Related work · 0.93

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