用POD分解替代傅里叶变换，提升高频偏微分方程求解精度与效率。
PODNO: Proper Orthogonal Decomposition Neural Operators
- 以POD正交基替代FNO中的傅里叶变换构造积分核
- 在NLS和KP方程上实现比FNO更优的精度与速度
- 理论证明了广义谱算子的通用性，适合高频物理模拟
本文提出针对高频主导型偏微分方程的PODNO方法。基于傅里叶神经算子（FNO）结构，用来自本征正交分解（POD）的正交变换取代傅里叶变换，构建积分核。由于POD基的最优性，该方法在高频问题上具有更高的准确性和计算效率。从理论角度，我们建立了广义谱算子（GSO）的通用性。数值实验表明，PODNO在非线性薛定谔（NLS）方程和卡多姆采夫-佩蒂亚什维利（KP）方程等色散方程上表现优异。
原文摘要 · Abstract (English)
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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