Neural Multigrid Architectures

Abstract

We propose a convenient matrix-free neural architecture for the multigrid method. The architecture is simple enough to be implemented in less than fifty lines of code, yet it encompasses a large number of distinct multigrid solvers. We argue that a fixed neural network without dense layers can not realize an efficient iterative method. Because of that, standard training protocols do not lead to competitive solvers. To overcome this difficulty, we use parameter sharing and serialization of layers. The resulting network can be trained on linear problems with thousands of unknowns and retains its efficiency on problems with millions of unknowns. From the point of view of numerical linear algebra network's training corresponds to finding optimal smoothers for the geometric multigrid method. We demonstrate our approach on a few second-order elliptic equations. For tested linear systems, we obtain from two to five times smaller spectral radius of the error propagation matrix compare to a basic linear multigrid with Jacobi smoother.

References in corpus (6)

• Spectral Networks and Locally Connected Networks on Graphs
• Learning Neural PDE Solvers with Convergence Guarantees
• On accuracy of approximation of the spectral radius by the Gelfand formula
• Learning to Optimize Multigrid PDE Solvers
• Learning Algebraic Multigrid Using Graph Neural Networks
• Optimizing Geometric Multigrid Methods with Evolutionary Computation

Cited by in corpus (1)

• Evolving Generalizable Multigrid-Based Helmholtz Preconditioners with Grammar-Guided Genetic Programming