Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Learning Dissipative Dynamics in Chaotic Systems

Zongyi Li, Miguel Liu-Schiaffini, Nikola Kovachki, Burigede Liu, Kamyar Azizzadenesheli, Kaushik Bhattacharya, Andrew Stuart, Anima Anandkumar

TL;DR

<3-5 sentence high-level summary> Chaotic dissipative systems are difficult to forecast over long horizons, but their long-time behavior concentrates on a global attractor with an invariant measure. The authors introduce a Markov neural operator (MNO) to learn the time-h step solution operator $S_h$ by approximating it with a mesh-insensitive neural operator $\hat{S}_h$, then generate long-horizon statistics by composing the learned map. They provide a theoretical approximation guarantee and demonstrate accurate short-term dynamics and invariant statistics for Lorenz-63, Kuramoto-Sivashinsky, and Kolmogorov flows up to $Re=5000$, including turbulent Navier–Stokes-like regimes, by enforcing dissipativity via regularization and post-processing. The approach enables stable, data-efficient learning of dissipative chaotic dynamics and practical estimation of attractor statistics, with potential benefits for partially observed systems and bifurcation analysis.

Abstract

Chaotic systems are notoriously challenging to predict because of their sensitivity to perturbations and errors due to time stepping. Despite this unpredictable behavior, for many dissipative systems the statistics of the long term trajectories are governed by an invariant measure supported on a set, known as the global attractor; for many problems this set is finite dimensional, even if the state space is infinite dimensional. For Markovian systems, the statistical properties of long-term trajectories are uniquely determined by the solution operator that maps the evolution of the system over arbitrary positive time increments. In this work, we propose a machine learning framework 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.

Learning Dissipative Dynamics in Chaotic Systems

TL;DR

<3-5 sentence high-level summary> Chaotic dissipative systems are difficult to forecast over long horizons, but their long-time behavior concentrates on a global attractor with an invariant measure. The authors introduce a Markov neural operator (MNO) to learn the time-h step solution operator by approximating it with a mesh-insensitive neural operator , then generate long-horizon statistics by composing the learned map. They provide a theoretical approximation guarantee and demonstrate accurate short-term dynamics and invariant statistics for Lorenz-63, Kuramoto-Sivashinsky, and Kolmogorov flows up to , including turbulent Navier–Stokes-like regimes, by enforcing dissipativity via regularization and post-processing. The approach enables stable, data-efficient learning of dissipative chaotic dynamics and practical estimation of attractor statistics, with potential benefits for partially observed systems and bifurcation analysis.

Abstract

Chaotic systems are notoriously challenging to predict because of their sensitivity to perturbations and errors due to time stepping. Despite this unpredictable behavior, for many dissipative systems the statistics of the long term trajectories are governed by an invariant measure supported on a set, known as the global attractor; for many problems this set is finite dimensional, even if the state space is infinite dimensional. For Markovian systems, the statistical properties of long-term trajectories are uniquely determined by the solution operator that maps the evolution of the system over arbitrary positive time increments. In this work, we propose a machine learning framework 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.

Paper Structure

This paper contains 47 sections, 1 theorem, 28 equations, 13 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Machine learning methods for chaotic systems.
  3. Invariants in chaos.
  4. Neural operators.
  5. Our contributions.
  6. Problem setting
  7. Dissipativity.
  8. Global Attractors.
  9. Data distribution.
  10. Ergodic systems.
  11. Learning the Markov neural operator in chaotic dynamics
  12. Sobolev norm as loss
  13. Long-term predictions.
  14. Invariant statistics.
  15. Encouraging dissipativity via regularization.
  16. ...and 32 more sections

Key Result

Theorem 1

Let $K \subset \mathcal{U}$ be a compact set and assume that, for some $h > 0$, the solution operator $S_h : \mathcal{U} \to \mathcal{U}$ associated to the dynamic eq:pde is locally Lipschitz. Then, for any $n \in \mathbb{N}$ and $\epsilon > 0$ there exists a neural operator $\hat{S}_h : \mathcal{U}

Figures (13)

  • Figure 1: Dynamic evolution of the Markov neural operator for Kolmogorov flow systems
  • Figure 2: Markov neural operator (MNO): learn global dynamics from local data
  • Figure 3: Dissipativity regularization on the Lorenz 63 system -- flow maps.
  • Figure 4: Choice of time step (a), Fourier spectrum (b), and spatial correlation (c) for KS equations.
  • Figure 5: The learned attractor of Kolmogorov flow ($Re=500$) and choice of Sobolev loss for KF.
  • ...and 8 more figures

Theorems & Definitions (2)

  • Theorem 1
  • proof : Proof of Theorem \ref{['thm:existance']}