论文标题
大型稀疏线性代数系统的傅立叶神经求解器
Fourier Neural Solver for large sparse linear algebraic systems
论文作者
论文摘要
可以在各种科学和工程领域中找到较大的稀疏线性代数系统，许多科学家努力以有效而健壮的方式解决它们。在本文中，我们提出了一个可解释的神经求解器，即傅立叶神经求解器（FNS），以解决它们。 FNS基于深度学习和快速的傅立叶变换。由于迭代解决方案和地面真相之间的误差涉及多种频率模式，因此FNS结合了一种固定的迭代方法和频率空间校正，以消除误差的不同组件。局部傅立叶分析表明，FNS可以在频率空间中的误差组件中拾取，这些误差组件用固定方法挑战就可以消除。在各向异性扩散方程，对流扩散方程和Helmholtz方程上的数值实验表明，FNS比最先进的神经求解器更有效，更强大。
Large sparse linear algebraic systems can be found in a variety of scientific and engineering fields, and many scientists strive to solve them in an efficient and robust manner. In this paper, we propose an interpretable neural solver, the Fourier Neural Solver (FNS), to address them. FNS is based on deep learning and Fast Fourier transform. Because the error between the iterative solution and the ground truth involves a wide range of frequency modes, FNS combines a stationary iterative method and frequency space correction to eliminate different components of the error. Local Fourier analysis reveals that the FNS can pick up on the error components in frequency space that are challenging to eliminate with stationary methods. Numerical experiments on the anisotropy diffusion equation, convection-diffusion equation, and Helmholtz equation show that FNS is more efficient and more robust than the state-of-the-art neural solver.
