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

arXiv: https://arxiv.org/abs/2410.12280 · HTML: https://mlsanity.com/paper/2410.12280/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 18 references that are in the map.
- First posted on arXiv 16 Oct 2024.

## Builds on (most cited first, 10 of 10)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2202.11214/) (2022)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [Geometry-Informed Neural Operator for Large-Scale 3D PDEs](https://mlsanity.com/paper/2309.00583/) (2023)
- [PDE-Refiner: Achieving Accurate Long Rollouts with Neural PDE Solvers](https://mlsanity.com/paper/2308.05732/) (2023)
- [Partial Differential Equations Meet Deep Neural Networks: A Survey](https://mlsanity.com/paper/2211.05567/) (2022)
- [A Multi-Scale Deep Learning Framework for Projecting Weather Extremes](https://mlsanity.com/paper/2210.12137/) (2022)
- [Fourier Neural Operators for Learning Dynamics in Quantum Spin Systems](https://mlsanity.com/paper/2409.03302/) (2024)

## Built on by (most cited first, 1 of 1)

- [Meta-Learning Fourier Neural Operators for Hessian Inversion and Enhanced Variational Data Assimilation](https://mlsanity.com/paper/2509.22949/) (2025)

## Abstract (arXiv)

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.