# Data-driven low-dimensional dynamic model of Kolmogorov flow

arXiv: https://arxiv.org/abs/2210.16708 · HTML: https://mlsanity.com/paper/2210.16708/

## Key facts

- Cited by 9 papers in the map (data through 7 Oct 2026).
- Builds on 6 of its 40 references that are in the map.
- First posted on arXiv 29 Oct 2022.

## Builds on (most cited first, 6 of 6)

- [Data-driven discovery of intrinsic dynamics](https://mlsanity.com/paper/2108.05928/) (2021)
- [Deep learning to discover and predict dynamics on an inertial manifold](https://mlsanity.com/paper/2001.04263/) (2019)
- [Data-Driven Reduced-Order Modeling of Spatiotemporal Chaos with Neural Ordinary Differential Equations](https://mlsanity.com/paper/2109.00060/) (2021)
- [Data-driven control of spatiotemporal chaos with reduced-order neural ODE-based models and reinforcement learning](https://mlsanity.com/paper/2205.00579/) (2022)
- [Symmetry-Aware Autoencoders: s-PCA and s-nlPCA](https://mlsanity.com/paper/2111.02893/) (2021)
- [Auto-Encoded Reservoir Computing for Turbulence Learning](https://mlsanity.com/paper/2012.10968/) (2020)

## Built on by (most cited first, 9 of 9)

- [Grasping Extreme Aerodynamics on a Low-Dimensional Manifold](https://mlsanity.com/paper/2305.08024/) (2023)
- [Autoencoders for discovering manifold dimension and coordinates in data from complex dynamical systems](https://mlsanity.com/paper/2305.01090/) (2023)
- [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)
- [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)
- [Data-driven low-dimensional model of a sedimenting flexible fiber](https://mlsanity.com/paper/2405.10442/) (2024)
- [Blending data and physics for reduced-order modeling of systems with spatiotemporal chaotic dynamics](https://mlsanity.com/paper/2507.21299/) (2025)
- [Inertial Manifold Neural Operator for Dissipative Time-Dependent Partial Differential Equations](https://mlsanity.com/paper/2608.23546/) (2026)

## Abstract (arXiv)

Reduced order models (ROMs) that capture flow dynamics are of interest for decreasing computational costs for simulation as well as for model-based control approaches. This work presents a data-driven framework for minimal-dimensional models that effectively capture the dynamics and properties of the flow. We apply this to Kolmogorov flow in a regime consisting of chaotic and intermittent behavior, which is common in many flows processes and is challenging to model. The trajectory of the flow travels near relative periodic orbits (RPOs), interspersed with sporadic bursting events corresponding to excursions between the regions containing the RPOs. The first step in development of the models is use of an undercomplete autoencoder to map from the full state data down to a latent space of dramatically lower dimension. Then models of the discrete-time evolution of the dynamics in the latent space are developed. By analyzing the model performance as a function of latent space dimension we can estimate the minimum number of dimensions required to capture the system dynamics. To further reduce the dimension of the dynamical model, we factor out a phase variable in the direction of translational invariance for the flow, leading to separate evolution equations for the pattern and phase. At a model dimension of five for the pattern dynamics, as opposed to the full state dimension of 1024 (i.e. a 32x32 grid), accurate predictions are found for individual trajectories out to about two Lyapunov times, as well as for long-time statistics. Further small improvements in the results occur at a dimension of nine. The nearly heteroclinic connections between the different RPOs, including the quiescent and bursting time scales, are well captured. We also capture key features of the phase dynamics. Finally, we use the low-dimensional representation to predict future bursting events, finding good success.