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

arXiv: https://arxiv.org/abs/2109.00060 · HTML: https://mlsanity.com/paper/2109.00060/

## 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 (most cited first, 9 of 9)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [On the Properties of Neural Machine Translation: Encoder-Decoder Approaches](https://mlsanity.com/paper/1409.1259/) (2014)
- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Data-Driven Forecasting of High-Dimensional Chaotic Systems with Long Short-Term Memory Networks](https://mlsanity.com/paper/1802.07486/) (2018)
- [Backpropagation Algorithms and Reservoir Computing in Recurrent Neural Networks for the Forecasting of Complex Spatiotemporal Dynamics](https://mlsanity.com/paper/1910.05266/) (2019)
- [Multistep Neural Networks for Data-driven Discovery of Nonlinear Dynamical Systems](https://mlsanity.com/paper/1801.01236/) (2018)
- [Deep learning to discover and predict dynamics on an inertial manifold](https://mlsanity.com/paper/2001.04263/) (2019)
- [Rapid Time Series Prediction with a Hardware-Based Reservoir Computer](https://mlsanity.com/paper/1807.07627/) (2018)

## Built on by (most cited first, 30 of 30)

- [Stabilized Neural Ordinary Differential Equations for Long-Time Forecasting of Dynamical Systems](https://mlsanity.com/paper/2203.15706/) (2022)
- [Data-driven control of spatiotemporal chaos with reduced-order neural ODE-based models and reinforcement learning](https://mlsanity.com/paper/2205.00579/) (2022)
- [When are dynamical systems learned from time series data statistically accurate?](https://mlsanity.com/paper/2411.06311/) (2024)
- [Neural Dynamical Operator: Continuous Spatial-Temporal Model with Gradient-Based and Derivative-Free Optimization Methods](https://mlsanity.com/paper/2311.11798/) (2023)
- [Autoencoders for discovering manifold dimension and coordinates in data from complex dynamical systems](https://mlsanity.com/paper/2305.01090/) (2023)
- [Binned Spectral Power Loss for Improved Prediction of Chaotic Systems](https://mlsanity.com/paper/2502.00472/) (2025)
- [Data-driven low-dimensional dynamic model of Kolmogorov flow](https://mlsanity.com/paper/2210.16708/) (2022)
- [Deep learning delay coordinate dynamics for chaotic attractors from partial observable data](https://mlsanity.com/paper/2211.11061/) (2022)
- [Dynamics of a data-driven low-dimensional model of turbulent minimal Couette flow](https://mlsanity.com/paper/2301.04638/) (2023)
- [Nonlinear dimensionality reduction then and now: AIMs for dissipative PDEs in the ML era](https://mlsanity.com/paper/2310.15816/) (2023)
- [Stability analysis of chaotic systems in latent spaces](https://mlsanity.com/paper/2410.00480/) (2024)
- [Turbulence control in plane Couette flow using low-dimensional neural ODE-based models and deep reinforcement learning](https://mlsanity.com/paper/2301.12098/) (2023)
- [Enhancing Predictive Capabilities in Data-Driven Dynamical Modeling with Automatic Differentiation: Koopman and Neural ODE Approaches](https://mlsanity.com/paper/2310.06790/) (2023)
- [On the relationship between Koopman operator approximations and neural ordinary differential equations for data-driven time-evolution predictions](https://mlsanity.com/paper/2411.12940/) (2024)
- [Building symmetries into data-driven manifold dynamics models for complex flows: application to two-dimensional Kolmogorov flow](https://mlsanity.com/paper/2312.10235/) (2023)
- [On instabilities in neural network-based physics simulators](https://mlsanity.com/paper/2406.13101/) (2024)
- [Semi-Implicit Neural Ordinary Differential Equations](https://mlsanity.com/paper/2412.11301/) (2024)
- [Unified Long-Term Time-Series Forecasting Benchmark](https://mlsanity.com/paper/2309.15946/) (2023)
- [Modeling chaotic Lorenz ODE System using Scientific Machine Learning](https://mlsanity.com/paper/2410.06452/) (2024)
- [Robust Moment Identification for Nonlinear PDEs via a Neural ODE Approach](https://mlsanity.com/paper/2506.05245/) (2025)
- [Machine-Learned Closure of URANS for Stably Stratified Turbulence: Connecting Physical Timescales & Data Hyperparameters of Deep Time-Series Models](https://mlsanity.com/paper/2404.16141/) (2024)
- [Data-driven low-dimensional model of a sedimenting flexible fiber](https://mlsanity.com/paper/2405.10442/) (2024)
- [Thinner Latent Spaces: Detecting Dimension and Imposing Invariance with Conformal Autoencoders](https://mlsanity.com/paper/2408.16138/) (2024)
- [Inferring stability properties of chaotic systems on autoencoders' latent spaces](https://mlsanity.com/paper/2410.18003/) (2024)
- [Blending data and physics for reduced-order modeling of systems with spatiotemporal chaotic dynamics](https://mlsanity.com/paper/2507.21299/) (2025)
- [A Novel Multi-Timescale Stability-Preserving Hierarchical Reinforcement Learning Controller Framework for Adaptive Control in High-Dimensional Dynamical Systems](https://mlsanity.com/paper/2510.22420/) (2025)
- [Generative prediction of laser-induced rocket ignition with dynamic latent space representations](https://mlsanity.com/paper/2603.07525/) (2026)
- [Hyperfastrl: Hypernetwork-based reinforcement learning for unified control of parametric chaotic PDEs](https://mlsanity.com/paper/2604.06497/) (2026)
- [Learning Control-Affine Reduced-Order Models via Autoencoders](https://mlsanity.com/paper/2606.05045/) (2026)
- [CasEm: A Cascade Architecture for Long-Horizon Neural Emulation](https://mlsanity.com/paper/2609.34246/) (2026)

## 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.