A Numerical Study of Chaotic Dynamics of K-S Equation with FNOs
This work addresses the problem of solving chaotic non-linear PDEs for applications like weather prediction and financial risk, but it is incremental as it builds on existing FNO methods by analyzing cutoff effects.
The study tackled simulating chaotic dynamics in the 2D Kuramoto-Sivashinsky equation using Fourier neural operators (FNOs), finding that FNOs capture the dynamics effectively when the Fourier mode cutoff is kept high, as measured by metrics like the 2D power spectrum and normalized error power spectrum.
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.