mlsanityData to 14 Oct 2026

Constraining Chaos: Enforcing dynamical invariants in the training of recurrent neural networks

arXiv 2304.12865 ↗First posted 24 Apr 2023cs.LG math.DS physics.geo-phcited by 6

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Abstract (arXiv)

Drawing on ergodic theory, we introduce a novel training method for machine learning based forecasting methods for chaotic dynamical systems. The training enforces dynamical invariants--such as the Lyapunov exponent spectrum and fractal dimension--in the systems of interest, enabling longer and more stable forecasts when operating with limited data. The technique is demonstrated in detail using the recurrent neural network architecture of reservoir computing. Results are given for the Lorenz 1996 chaotic dynamical system and a spectral quasi-geostrophic model, both typical test cases for numerical weather prediction.

Key facts. Cited by 6 papers in the map (data through 14 Oct 2026). Builds on 16 of its 67 references that are in the map. First posted on arXiv 24 Apr 2023.

Builds on

16 of 67 references.

References · 16

  1. 2022FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operatorscited by 776
  2. 2019Hamiltonian Neural Networkscited by 712
  3. 2020B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Datacited by 335
  4. 2022GraphCast: Learning skillful medium-range global weather forecastingcited by 267
  5. 2022Pangu-Weather: A 3D High-Resolution Model for Fast and Accurate Global Weather Forecastcited by 166
  6. 2019Symplectic Recurrent Neural Networkscited by 150
  7. 2019Backpropagation Algorithms and Reservoir Computing in Recurrent Neural Networks for the Forecasting of Complex Spatiotemporal Dynamicscited by 134
  8. 2020AI Poincar'e: Machine Learning Conservation Laws from Trajectoriescited by 77
  9. 2019Forecasting Chaotic Systems with Very Low Connectivity Reservoir Computerscited by 43
  10. 2022A Systematic Exploration of Reservoir Computing for Forecasting Complex Spatiotemporal Dynamicscited by 30
  11. 2021Noether Networks: Meta-Learning Useful Conserved Quantitiescited by 26
  12. 2021Integrating Recurrent Neural Networks with Data Assimilation for Scalable Data-Driven State Estimationcited by 21
4 more
  1. 2020Physics-Informed Echo State Networkscited by 16
  2. 2020Towards Physically-consistent, Data-driven Models of Convectioncited by 13
  3. 2020Automatic-differentiated Physics-Informed Echo State Network (API-ESN)cited by 8
  4. 2021Discovering conservation laws from trajectories via machine learningcited by 8

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Cited by year

Citing papers per year: 2023: 1; 2024: 3; 2025: 2.

6 papers.

  1. 2024DySLIM: Dynamics Stable Learning by Invariant Measure for Chaotic Systemscited by 30
  2. 2024A probabilistic framework for learning non-intrusive corrections to long-time climate simulations from short-time training datacited by 5
  3. 2024Invariant Measures in Time-Delay Coordinates for Unique Dynamical System Identificationcited by 5
  4. 2023Temporal Subsampling Diminishes Small Spatial Scales in Recurrent Neural Network Emulators of Geophysical Turbulencecited by 3
  5. 2025Detecting Invariant Manifolds in ReLU-Based RNNscited by 3
  6. 2025Calibrating Geophysical Predictions under Constrained Probabilistic Distributionscited by 2