Learning to Optimize Multigrid PDE Solvers

Daniel Greenfeld, Meirav Galun, Ronen Basri, Irad Yavneh, Ron Kimmel
Proceedings of the 36th International Conference on Machine Learning, PMLR 97:2415-2423, 2019.

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 discretized PDEs to prolongation operators for a broad class of 2D diffusion problems. We train a neural network once for the entire class of PDEs, using an efficient and unsupervised loss function. Our tests demonstrate improved convergence rates compared to the widely used Black-Box multigrid scheme, suggesting that our method successfully learned rules for constructing prolongation matrices.

Cite this Paper

BibTeX
@InProceedings{pmlr-v97-greenfeld19a, title = {Learning to Optimize Multigrid {PDE} Solvers}, author = {Greenfeld, Daniel and Galun, Meirav and Basri, Ronen and Yavneh, Irad and Kimmel, Ron}, booktitle = {Proceedings of the 36th International Conference on Machine Learning}, pages = {2415--2423}, year = {2019}, editor = {Chaudhuri, Kamalika and Salakhutdinov, Ruslan}, volume = {97}, series = {Proceedings of Machine Learning Research}, month = {09--15 Jun}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v97/greenfeld19a/greenfeld19a.pdf}, url = {https://proceedings.mlr.press/v97/greenfeld19a.html}, 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 discretized PDEs to prolongation operators for a broad class of 2D diffusion problems. We train a neural network once for the entire class of PDEs, using an efficient and unsupervised loss function. Our tests demonstrate improved convergence rates compared to the widely used Black-Box multigrid scheme, suggesting that our method successfully learned rules for constructing prolongation matrices.} }
Endnote
%0 Conference Paper %T Learning to Optimize Multigrid PDE Solvers %A Daniel Greenfeld %A Meirav Galun %A Ronen Basri %A Irad Yavneh %A Ron Kimmel %B Proceedings of the 36th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2019 %E Kamalika Chaudhuri %E Ruslan Salakhutdinov %F pmlr-v97-greenfeld19a %I PMLR %P 2415--2423 %U https://proceedings.mlr.press/v97/greenfeld19a.html %V 97 %X 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 discretized PDEs to prolongation operators for a broad class of 2D diffusion problems. We train a neural network once for the entire class of PDEs, using an efficient and unsupervised loss function. Our tests demonstrate improved convergence rates compared to the widely used Black-Box multigrid scheme, suggesting that our method successfully learned rules for constructing prolongation matrices.
APA
Greenfeld, D., Galun, M., Basri, R., Yavneh, I. & Kimmel, R.. (2019). Learning to Optimize Multigrid PDE Solvers. Proceedings of the 36th International Conference on Machine Learning, in Proceedings of Machine Learning Research 97:2415-2423 Available from https://proceedings.mlr.press/v97/greenfeld19a.html.

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