A Numerical Study of Chaotic Dynamics of K-S Equation with FNOs

TL;DR

This paper demonstrates that Fourier neural operators can effectively simulate the chaotic dynamics of the 2D Kuramoto-Sivashinsky equation, highlighting the importance of Fourier mode cutoff in capturing complex behaviors.

Contribution

The study shows how FNOs can be used to model chaotic PDEs and analyzes the impact of Fourier mode cutoff on simulation accuracy compared to traditional solvers.

Findings

FNOs accurately capture chaotic dynamics with sufficient Fourier modes.

The normalized error power spectrum quantifies FNO performance.

Higher Fourier mode cutoff improves FNO accuracy.

Abstract

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.