PODNO: Proper Orthogonal Decomposition Neural Operators
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866916707673571328 |
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| author | Cheng, Zilan Wang, Zhongjian Wang, Li-Lian Azaiez, Mejdi |
| author_facet | Cheng, Zilan Wang, Zhongjian Wang, Li-Lian Azaiez, Mejdi |
| contents | In this paper, we introduce Proper Orthogonal Decomposition Neural Operators (PODNO) for solving partial differential equations (PDEs) dominated by high-frequency components. Building on the structure of Fourier Neural Operators (FNO), PODNO replaces the Fourier transform with (inverse) orthonormal transforms derived from the Proper Orthogonal Decomposition (POD) method to construct the integral kernel. Due to the optimality of POD basis, the PODNO has potential to outperform FNO in both accuracy and computational efficiency for high-frequency problems. From analysis point of view, we established the universality of a generalization of PODNO, termed as Generalized Spectral Operator (GSO). In addition, we evaluate PODNO's performance numerically on dispersive equations such as the Nonlinear Schrodinger (NLS) equation and the Kadomtsev-Petviashvili (KP) equation. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_18513 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | PODNO: Proper Orthogonal Decomposition Neural Operators Cheng, Zilan Wang, Zhongjian Wang, Li-Lian Azaiez, Mejdi Numerical Analysis Machine Learning Computational Physics 68T07, 65M12, 41A35, 65N99 In this paper, we introduce Proper Orthogonal Decomposition Neural Operators (PODNO) for solving partial differential equations (PDEs) dominated by high-frequency components. Building on the structure of Fourier Neural Operators (FNO), PODNO replaces the Fourier transform with (inverse) orthonormal transforms derived from the Proper Orthogonal Decomposition (POD) method to construct the integral kernel. Due to the optimality of POD basis, the PODNO has potential to outperform FNO in both accuracy and computational efficiency for high-frequency problems. From analysis point of view, we established the universality of a generalization of PODNO, termed as Generalized Spectral Operator (GSO). In addition, we evaluate PODNO's performance numerically on dispersive equations such as the Nonlinear Schrodinger (NLS) equation and the Kadomtsev-Petviashvili (KP) equation. |
| title | PODNO: Proper Orthogonal Decomposition Neural Operators |
| topic | Numerical Analysis Machine Learning Computational Physics 68T07, 65M12, 41A35, 65N99 |
| url | https://arxiv.org/abs/2504.18513 |