Fourier Neural Solver for Large Sparse Linear Algebraic Systems

Chen Cui, Kai Jiang (), Yun Liu and Shi Shu ()
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Chen Cui: Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education, School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, China
Kai Jiang: Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education, School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, China
Yun Liu: Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education, School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, China
Shi Shu: Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education, School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, China

Mathematics, 2022, vol. 10, issue 21, 1-16

Abstract: 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 a fast Fourier transform. Because the error between the iterative solution and the ground truth involves a wide range of frequency modes, the FNS combines a stationary iterative method and frequency space correction to eliminate different components of the error. Local Fourier analysis shows 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 anisotropic diffusion equation, convection–diffusion equation, and Helmholtz equation show that the FNS is more efficient and more robust than the state-of-the-art neural solver.

Keywords: Fourier neural solver; fast Fourier transform; local Fourier analysis; convection–diffusion–reaction equation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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