Momentum-Accelerated Richardson(m) and Their Multilevel Neural Solvers

TL;DR

This paper introduces momentum-accelerated Richardson neural solvers and multilevel neural methods that significantly improve convergence speed and robustness for large-scale linear systems, especially PDEs, outperforming traditional iterative methods.

Contribution

The paper proposes novel neural solvers based on Richardson(m) with momentum and preconditioning, and develops multilevel neural solvers to enhance robustness and efficiency in PDE applications.

Findings

Faster convergence than Chebyshev methods.

Lower computational complexity.

Enhanced robustness with multilevel neural solvers.

Abstract

Recently, designing neural solvers for large-scale linear systems of equations has emerged as a promising approach in scientific and engineering computing. This paper first introduce the Richardson(m) neural solver by employing a meta network to predict the weights of the long-step Richardson iterative method. Next, by incorporating momentum and preconditioning techniques, we further enhance convergence. Numerical experiments on anisotropic second-order elliptic equations demonstrate that these new solvers achieve faster convergence and lower computational complexity compared to both the Chebyshev iterative method with optimal weights and the Chebyshev semi-iteration method. To address the strong dependence of the aforementioned single-level neural solvers on PDE parameters and grid size, we integrate them with two multilevel neural solvers developed in recent years. Using alternating optimization techniques, we construct Richardson(m)-FNS for anisotropic equations and NAG-Richardson(m)-WANS for the Helmholtz equation. Numerical experiments show that these two multilevel neural solvers effectively overcome the drawback of single-level methods, providing better robustness and computational efficiency.