用FNO模拟二维柯尔莫戈洛夫-希瓦辛斯基方程混沌动力，发现高阶傅里叶截断是关键。
A Numerical Study of Chaotic Dynamics of K-S Equation with FNOs
- 用傅里叶神经算子（FNO）求解二维柯氏方程混沌态
- 高阶傅里叶模式截断下FNO结果与传统方法一致
- 提出归一化误差谱，量化模型输出误差
求解具有混沌动力的非线性偏微分方程在气象极端预测和金融风险分析等领域有广泛应用。傅里叶神经算子（FNO）已被证明在求解偏微分方程方面高效。本文展示使用FNO模拟二维柯尔莫戈洛夫-希瓦辛斯基（2d K-S）方程混沌区域的动力学行为。特别地，分析了傅里叶模式截断对FNO结果与传统PDE求解器结果的影响。通过二维功率谱和径向功率谱等指标进行对比。此外，提出归一化误差功率谱，用于衡量FNO模型输出中的相对误差。结论表明，当傅里叶模式截断足够高时，FNO能准确捕捉2d K-S方程混沌状态的动力学特征。
原文摘要 · Abstract (English)
Solving non-linear partial differential equations which exhibit chaotic dynamics is an important problem with a wide-range of applications such as predicting weather extremes and financial market risk. Fourier neural operators (FNOs) have been shown to be efficient in solving partial differential equations (PDEs). In this work we demonstrate simulation of dynamics in the chaotic regime of the two-dimensional (2d) Kuramoto-Sivashinsky equation using FNOs. Particularly, we analyze the effect of Fourier mode cutoff on the results obtained by using FNOs vs those obtained using traditional PDE solvers. We compare the outputs using metrics such as the 2d power spectrum and the radial power spectrum. In addition we propose the normalised error power spectrum which measures the percentage error in the FNO model outputs. We conclude that FNOs capture the dynamics in the chaotic regime of the 2d K-S equation, provided the Fourier mode cutoff is kept sufficiently high.
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