Fourier Neural Solver for Large Sparse Linear Algebraic Systems
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...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-10-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/10/21/4014 |
| _version_ | 1797467399382892544 |
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| author | Chen Cui Kai Jiang Yun Liu Shi Shu |
| author_facet | Chen Cui Kai Jiang Yun Liu Shi Shu |
| author_sort | Chen Cui |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-03-09T18:53:05Z |
| format | Article |
| id | doaj.art-9e9e2a4b5987415781ed62f5ff5c7611 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T18:53:05Z |
| publishDate | 2022-10-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-9e9e2a4b5987415781ed62f5ff5c76112023-11-24T05:43:34ZengMDPI AGMathematics2227-73902022-10-011021401410.3390/math10214014Fourier Neural Solver for Large Sparse Linear Algebraic SystemsChen Cui0Kai Jiang1Yun Liu2Shi Shu3Hunan 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, ChinaHunan 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, ChinaHunan 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, ChinaHunan 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, ChinaLarge 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.https://www.mdpi.com/2227-7390/10/21/4014Fourier neural solverfast Fourier transformlocal Fourier analysisconvection–diffusion–reaction equation |
| spellingShingle | Chen Cui Kai Jiang Yun Liu Shi Shu Fourier Neural Solver for Large Sparse Linear Algebraic Systems Mathematics Fourier neural solver fast Fourier transform local Fourier analysis convection–diffusion–reaction equation |
| title | Fourier Neural Solver for Large Sparse Linear Algebraic Systems |
| title_full | Fourier Neural Solver for Large Sparse Linear Algebraic Systems |
| title_fullStr | Fourier Neural Solver for Large Sparse Linear Algebraic Systems |
| title_full_unstemmed | Fourier Neural Solver for Large Sparse Linear Algebraic Systems |
| title_short | Fourier Neural Solver for Large Sparse Linear Algebraic Systems |
| title_sort | fourier neural solver for large sparse linear algebraic systems |
| topic | Fourier neural solver fast Fourier transform local Fourier analysis convection–diffusion–reaction equation |
| url | https://www.mdpi.com/2227-7390/10/21/4014 |
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