标题： 动量加速Richardson(m)及其多级神经求解器 显示英文标题

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

摘要： 最近，为大规模线性方程组设计神经求解器在科学和工程计算中已成为一种有前景的方法。 本文首先通过采用元网络来预测长步Richardson迭代方法的权重，介绍了Richardson(m)神经求解器。 接下来，通过结合动量和预处理技术，我们进一步提高了收敛性。 对各向异性二阶椭圆方程的数值实验表明，这些新的求解器相比具有最优权重的切比雪夫迭代方法和切比雪夫半迭代方法，实现了更快的收敛速度和更低的计算复杂度。 为了解决上述单级神经求解器对PDE参数和网格大小的强依赖性，我们将它们与近年来开发的两种多级神经求解器相结合。 使用交替优化技术，我们构建了针对各向异性方程的Richardson(m)-FNS和针对Helmholtz方程的NAG-Richardson(m)-WANS。 数值实验表明，这两种多级神经求解器有效克服了单级方法的缺点，提供了更好的鲁棒性和计算效率。 显示英文摘要

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.