This repository contains the code for the paper "Learning Dissipative Dynamics in Chaotic Systems," published in NeurIPS 2022.

In this work, we propose a machine learning framework, which we call the Markov Neural Operator (MNO), to learn the underlying solution operator for dissipative chaotic systems, showing that the resulting learned operator accurately captures short-time trajectories and long-time statistical behavior. Using this framework, we are able to predict various statistics of the invariant measure for the turbulent Kolmogorov Flow dynamics with Reynolds numbers up to 5000.

• Neural operator code is based on the Fourier Neural Operator (FNO), which requires PyTorch 1.8.0 or later.

• utilities.py: basic utilities including a reader for .mat files and Sobolev (Hk) and Lp losses.
• dissipative_utils.py: helper functions for encouraging (regularization loss) and enforcing dissipative dynamics (postprocessing).
• models/: model architectures
  • densenet.py: simple feedforward neural network
  • fno_2d.py: FNO architecture for operators acting on a function space with two spatial dimensions.
• data_generation/: directory containing data generation code for our toy Lorenz-63 dataset and the 1D Kuramoto–Sivashinsky PDE.
• scripts/: scripts for training Lorenz-63 model, 1D KS, and 2D NS equations.
  • NS_fno_baseline.py: FNO baseline trained on 2D NS with Reynolds number 500. No dissipativity or Sobolev loss.
  • NS_mno_dissipative.py: MNO model built on FNO architecture with dissipativity encouraged and Sobolev loss.
  • lorenz_densenet.py: simple feedforward neural network learning Markovian solution operator for Lorenz-63 system.
  • lorenz_dissipative_densenet.py: simple feedforward neural network with dissipativity encouraged trained on Lorenz-63 system.
• lorenz.ipynb: Jupyter notebook with examples to reproduce plots and figures for our Lorenz-63 examples in the paper.
• visualize_navier_stokes2d.ipynb : Jupyter notebook with examples to reproduce plots and figures for our 2D Navier-Stokes case study in the paper.

In our work, we train and evaluate on datasets from the Lorenz-63 system (finite-dimensional ODE), Kuramoto–Sivashinsky equation (1D PDE system), and the 2D Navier-Stokes equations (Kolmogorov flow, 2D PDE). Our datasets can be found online under DOI 10.5281/zenodo.74955555.

• Lorenz: Can be found in the data_generation directory.
• KS: Can be found in the data_generation directory.
• Data generation for 2D Navier-Stokes is based on the data generation scripts in the FNO repository.

In our work, we use three different models to learn the Markovian solution operator. These can be found under the models/ folder in the repository.

• Lorenz: Since the Lorenz-63 system is a finite-dimensional ODE system, we use a standard feedforward neural network to learn the Markov solution operator.
• 1D KS and 2D NS equations: We interpret PDEs as function-space ODEs, and we adopt the 1D and 2D FNO architecture (resp.) to learn the Markov solution operator for the 1D KS and 2D NS equations.

@inproceedings{MNO,
  title={Learning Dissipative Dynamics in Chaotic Systems},
  author={Zongyi Li and Miguel Liu-Schiaffini and Nikola B. Kovachki and Burigede Liu and Kamyar Azizzadenesheli and Kaushik Bhattacharya and Andrew Stuart and Anima Anandkumar},
  year={2022}
}