Learning to Optimize Multigrid PDE Solvers
This addresses the problem of devising fast multigrid algorithms for new PDEs in scientific computing, representing an incremental improvement with a novel learning-based approach.
The paper tackles the challenge of constructing efficient multigrid solvers for parameterized PDEs by learning a mapping from PDE families to prolongation operators, resulting in improved convergence rates compared to the Black-Box multigrid scheme on 2D diffusion problems.
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.