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

Surbhi Khetrapal, Jaswin Kasi

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.

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

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.

Paper Structure

This paper contains 16 sections, 7 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Physics-Informed Neural Networks
  3. Data-Driven Approaches
  4. Fourier Neural Operators (FNOs)
  5. Chaotic dynamics and Kuramoto-Sivashinsky equation
  6. Methodology
  7. Data generation
  8. Fourier modes truncation in FNOs
  9. Experimental Results
  10. $2d$ Power Spectrum
  11. Radial power spectrum for prediction error
  12. Conclusion
  13. Future Work
  14. Appendix
  15. Spectral analysis of training data
  16. ...and 1 more sections

Figures (5)

  • Figure 1: A comparison of output from FNO models vs ground truth.
  • Figure 2: The two-dimensional log power spectrum of FNO model outputs vs ground truth.
  • Figure 3: A comparison of error power spectrum (left) and normalised error power spectrum (right) vs wavenumber for FNO modes-12 and FNO modes-24.
  • Figure 4: Radial power spectrum of ground truth vs radial wavenumber (left) and power spectrum of error between FNO model outputs and ground truth vs wavenumber (right).
  • Figure 5: A comparison of training and validation loss of the two FNO models