PODNO: Proper Orthogonal Decomposition Neural Operators

Zilan Cheng, Zhongjian Wang, Li-Lian Wang, Mejdi Azaiez
4/25/2025
math.NAcs.LGphysics.comp-ph

No official code found. Best implementation: Zilan-Cheng/PODNO (1 star, last commit 2025-10-24).

Abstract

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.

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Code Implementations(1)

Zilan-Cheng/PODNOCommunity68%

Here are the codes for PODNO: Proper Orthogonal Decomposition Neural Operators.

10MATLAB, PythonOct 23, 202512 months ago

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Cite this paper

@article{cheng2025podno,
  title  = {PODNO: Proper Orthogonal Decomposition Neural Operators},
  author = {Zilan Cheng and Zhongjian Wang and Li-Lian Wang and Mejdi Azaiez},
  year   = {2025},
  eprint = {2504.18513},
  archivePrefix = {arXiv},
  url    = {http://arxiv.org/abs/2504.18513v1}
}

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