mlsanityData to 14 Oct 2026

Data-Driven Reduced-Order Modeling of Spatiotemporal Chaos with Neural Ordinary Differential Equations

arXiv 2109.00060 ↗First posted 31 Aug 2021cs.LG nlin.CDcited by 30

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Abstract (arXiv)

Dissipative partial differential equations that exhibit chaotic dynamics tend to evolve to attractors that exist on finite-dimensional manifolds. We present a data-driven reduced order modeling method that capitalizes on this fact by finding the coordinates of this manifold and finding an ordinary differential equation (ODE) describing the dynamics in this coordinate system. The manifold coordinates are discovered using an undercomplete autoencoder -- a neural network (NN) that reduces then expands dimension. Then the ODE, in these coordinates, is approximated by a NN using the neural ODE framework. Both of these methods only require snapshots of data to learn a model, and the data can be widely and/or unevenly spaced. We apply this framework to the Kuramoto-Sivashinsky for different domain sizes that exhibit chaotic dynamics. With this system, we find that dimension reduction improves performance relative to predictions in the ambient space, where artifacts arise. Then, with the low-dimensional model, we vary the training data spacing and find excellent short- and long-time statistical recreation of the true dynamics for widely spaced data (spacing of ~0.7 Lyapunov times). We end by comparing performance with various degrees of dimension reduction, and find a "sweet spot" in terms of performance vs. dimension.

Key facts. Cited by 30 papers in the map (data through 14 Oct 2026). Builds on 9 of its 57 references that are in the map. First posted on arXiv 31 Aug 2021.

Builds on

9 of 57 references.

References · 9

  1. 2019PyTorch: An Imperative Style, High-Performance Deep Learning Librarycited by 20,305
  2. 2018Neural Ordinary Differential Equationscited by 3,779
  3. 2014On the Properties of Neural Machine Translation: Encoder-Decoder Approachescited by 1,304
  4. 2017Deep learning for universal linear embeddings of nonlinear dynamicscited by 494
  5. 2018Data-Driven Forecasting of High-Dimensional Chaotic Systems with Long Short-Term Memory Networkscited by 155
  6. 2019Backpropagation Algorithms and Reservoir Computing in Recurrent Neural Networks for the Forecasting of Complex Spatiotemporal Dynamicscited by 134
  7. 2018Multistep Neural Networks for Data-driven Discovery of Nonlinear Dynamical Systemscited by 89
  8. 2019Deep learning to discover and predict dynamics on an inertial manifoldcited by 43
  9. 2018Rapid Time Series Prediction with a Hardware-Based Reservoir Computercited by 14

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2022: 4; 2023: 8; 2024: 10; 2025: 4; 2026: 4.

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  1. 2022Stabilized Neural Ordinary Differential Equations for Long-Time Forecasting of Dynamical Systemscited by 50
  2. 2022Data-driven control of spatiotemporal chaos with reduced-order neural ODE-based models and reinforcement learningcited by 17
  3. 2024When are dynamical systems learned from time series data statistically accurate?cited by 15
  4. 2023Neural Dynamical Operator: Continuous Spatial-Temporal Model with Gradient-Based and Derivative-Free Optimization Methodscited by 14
  5. 2023Autoencoders for discovering manifold dimension and coordinates in data from complex dynamical systemscited by 13
  6. 2025Binned Spectral Power Loss for Improved Prediction of Chaotic Systemscited by 10
  7. 2022Data-driven low-dimensional dynamic model of Kolmogorov flowcited by 9
  8. 2022Deep learning delay coordinate dynamics for chaotic attractors from partial observable datacited by 8
  9. 2023Dynamics of a data-driven low-dimensional model of turbulent minimal Couette flowcited by 8
  10. 2023Nonlinear dimensionality reduction then and now: AIMs for dissipative PDEs in the ML eracited by 7
  11. 2024Stability analysis of chaotic systems in latent spacescited by 7
  12. 2023Turbulence control in plane Couette flow using low-dimensional neural ODE-based models and deep reinforcement learningcited by 5
18 more
  1. 2023Enhancing Predictive Capabilities in Data-Driven Dynamical Modeling with Automatic Differentiation: Koopman and Neural ODE Approachescited by 5
  2. 2024On the relationship between Koopman operator approximations and neural ordinary differential equations for data-driven time-evolution predictionscited by 3
  3. 2023Building symmetries into data-driven manifold dynamics models for complex flows: application to two-dimensional Kolmogorov flowcited by 2
  4. 2024On instabilities in neural network-based physics simulatorscited by 2
  5. 2024Semi-Implicit Neural Ordinary Differential Equationscited by 2
  6. 2023Unified Long-Term Time-Series Forecasting Benchmarkcited by 1
  7. 2024Modeling chaotic Lorenz ODE System using Scientific Machine Learningcited by 1
  8. 2025Robust Moment Identification for Nonlinear PDEs via a Neural ODE Approachcited by 1
  9. 2024Machine-Learned Closure of URANS for Stably Stratified Turbulence: Connecting Physical Timescales & Data Hyperparameters of Deep Time-Series Modelscited by 0
  10. 2024Data-driven low-dimensional model of a sedimenting flexible fibercited by 0
  11. 2024Thinner Latent Spaces: Detecting Dimension and Imposing Invariance with Conformal Autoencoderscited by 0
  12. 2024Inferring stability properties of chaotic systems on autoencoders' latent spacescited by 0
  13. 2025Blending data and physics for reduced-order modeling of systems with spatiotemporal chaotic dynamicscited by 0
  14. 2025A Novel Multi-Timescale Stability-Preserving Hierarchical Reinforcement Learning Controller Framework for Adaptive Control in High-Dimensional Dynamical Systemscited by 0
  15. 2026Generative prediction of laser-induced rocket ignition with dynamic latent space representationscited by 0
  16. 2026Hyperfastrl: Hypernetwork-based reinforcement learning for unified control of parametric chaotic PDEscited by 0
  17. 2026Learning Control-Affine Reduced-Order Models via Autoencoderscited by 0
  18. 2026CasEm: A Cascade Architecture for Long-Horizon Neural Emulationcited by 0