动量加速的Richardson(m)及其多层神经求解器
Momentum-Accelerated Richardson(m) and Their Multilevel Neural Solvers
AI总结：
该研究提出Richardson(m)神经求解器及动量加速版本，结合多层神经求解器与交替优化构建两类求解器，在椭圆、Helmholtz方程上验证了更快收敛、更低复杂度与更强鲁棒性。
摘要：
近年来，为大规模线性方程组设计神经求解器已成为科学与工程计算领域一种颇具前景的方法。本文首先通过采用元网络预测长步Richardson迭代法的权重，提出了Richardson(m)神经求解器。随后，我们通过引入动量与预处理技术，进一步提升了收敛性。在各向异性二阶椭圆方程上的数值实验表明，与采用最优权重的Chebyshev迭代法以及Chebyshev半迭代法相比，这些新型求解器收敛速度更快、计算复杂度更低。为解决上述单层神经求解器对PDE参数和网格尺寸的强依赖问题，我们将其与近年来开发的两种多层神经求解器相结合。利用交替优化技术，我们构建了适用于各向异性方程的Richardson(m)-FNS，以及适用于Helmholtz方程的NAG-Richardson(m)-WANS。数值实验表明，这两种多层神经求解器有效克服了单层方法的缺陷，具备更好的鲁棒性和计算效率。
英文摘要：
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.
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