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Learning Dissipative Dynamics in Chaotic Systems

arXiv 2106.06898 ↗First posted 13 Jun 2021cs.LG math.DScited by 42

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Abstract (arXiv)

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.

Key facts. Cited by 42 papers in the map (data through 14 Oct 2026). Builds on 16 of its 56 references that are in the map. First posted on arXiv 13 Jun 2021.

Builds on

16 of 56 references.

References · 16

  1. 2015U-Net: Convolutional Networks for Biomedical Image Segmentationcited by 19,713
  2. 2018Neural Ordinary Differential Equationscited by 3,779
  3. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,908
  4. 2014Empirical Evaluation of Gated Recurrent Neural Networks on Sequence Modelingcited by 2,367
  5. 2015Convolutional LSTM Network: A Machine Learning Approach for Precipitation Nowcastingcited by 1,436
  6. 2020Neural Operator: Graph Kernel Network for Partial Differential Equationscited by 734
  7. 2018Data-Driven Forecasting of High-Dimensional Chaotic Systems with Long Short-Term Memory Networkscited by 155
  8. 2017Sobolev Training for Neural Networkscited by 143
  9. 2019Backpropagation Algorithms and Reservoir Computing in Recurrent Neural Networks for the Forecasting of Complex Spatiotemporal Dynamicscited by 134
  10. 2020Improving data-driven global weather prediction using deep convolutional neural networks on a cubed spherecited by 96
  11. 2017Data-driven discovery of Koopman eigenfunctions for controlcited by 64
  12. 2020Hierarchical Deep Learning of Multiscale Differential Equation Time-Stepperscited by 37
4 more
  1. 2020SINDy-BVP: Sparse Identification of Nonlinear Dynamics for Boundary Value Problemscited by 22
  2. 2021Towards physically consistent data-driven weather forecasting: Integrating data assimilation with equivariance-preserving deep spatial transformerscited by 15
  3. 2020Learning Composable Energy Surrogates for PDE Order Reductioncited by 13
  4. 2020Data-Driven Stabilization of Periodic Orbitscited by 7

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Citing papers per year: 2022: 1; 2023: 6; 2024: 10; 2025: 12; 2026: 13.

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  4. 2023Group Equivariant Fourier Neural Operators for Partial Differential Equationscited by 45
  5. 2023Nonlocality and Nonlinearity Implies Universality in Operator Learningcited by 43
  6. 2024State-space models are accurate and efficient neural operators for dynamical systemscited by 28
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  8. 2024Using Uncertainty Quantification to Characterize and Improve Out-of-Domain Learning for PDEscited by 15
  9. 2025TI-DeepONet: Learnable Time Integration for Stable Long-Term Extrapolationcited by 14
  10. 2025Panda: A pretrained forecast model for chaotic dynamicscited by 13
  11. 2024Accelerating Phase Field Simulations Through a Hybrid Adaptive Fourier Neural Operator with U-Net Backbonecited by 10
  12. 2025Enabling Local Neural Operators to perform Equation-Free System-Level Analysiscited by 10
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  9. 2025INC: An Indirect Neural Corrector for Auto-Regressive Hybrid PDE Solverscited by 3
  10. 2026Is Flow Matching Just Trajectory Replay for Sequential Data?cited by 3
  11. 2023Investigating the Ability of PINNs To Solve Burgers' PDE Near Finite-Time BlowUpcited by 2
  12. 2024Implicit factorized transformer approach to fast prediction of turbulent channel flowscited by 2
  13. 2025Physics-informed Reduced Order Modeling of Time-dependent PDEs via Differentiable Solverscited by 2
  14. 2025PICore: Physics-Informed Unsupervised Coreset Selection for Data Efficient Neural Operator Trainingcited by 2
  15. 2026Dynamics-Informed Deep Learning for Predicting Extreme Eventscited by 2
  16. 2023Towards Long-Term predictions of Turbulence using Neural Operatorscited by 1
  17. 2024Fourier neural operators for spatiotemporal dynamics in two-dimensional turbulencecited by 1
  18. 2024Safe PDE Boundary Control with Neural Operatorscited by 1
  19. 2025ChaosNexus: A Foundation Model for ODE-based Chaotic System Forecasting with Hierarchical Multi-scale Awarenesscited by 1
  20. 2025Equation-Free Coarse Control of Distributed Parameter Systems via Local Neural Operatorscited by 1
  21. 2026Neural-HSS: Hierarchical Semi-Separable Neural PDE Solvercited by 1
  22. 2025Ellipsoidal Time Series Forecastingcited by 0
  23. 2026Generalization Bounds for Physics-Informed Neural Networks for the Incompressible Navier-Stokes Equationscited by 0
  24. 2026BHCast: Unlocking Black Hole Plasma Dynamics from a Single Blurry Image with Long-Term Forecastingcited by 0
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  26. 2026Explainable quantum-compressed machine learning for complex fluid flowscited by 0
  27. 2026Flux-form spatiotemporal neural operators for coarse-grained dynamics of multiscale PDEscited by 0
  28. 2026Inertial Manifold Neural Operator for Dissipative Time-Dependent Partial Differential Equationscited by 0
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