# Building symmetries into data-driven manifold dynamics models for complex flows: application to two-dimensional Kolmogorov flow

arXiv: https://arxiv.org/abs/2312.10235 · HTML: https://mlsanity.com/paper/2312.10235/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 50 references that are in the map.
- First posted on arXiv 15 Dec 2023.

## Builds on (most cited first, 18 of 18)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Modern Koopman Theory for Dynamical Systems](https://mlsanity.com/paper/2102.12086/) (2021)
- [On Neural Differential Equations](https://mlsanity.com/paper/2202.02435/) (2022)
- [How to train your neural ODE: the world of Jacobian and kinetic regularization](https://mlsanity.com/paper/2002.02798/) (2020)
- [Learning Differential Equations that are Easy to Solve](https://mlsanity.com/paper/2007.04504/) (2020)
- [Stabilized Neural Ordinary Differential Equations for Long-Time Forecasting of Dynamical Systems](https://mlsanity.com/paper/2203.15706/) (2022)
- [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)
- [Implicit Rank-Minimizing Autoencoder](https://mlsanity.com/paper/2010.00679/) (2020)
- [Chart Auto-Encoders for Manifold Structured Data](https://mlsanity.com/paper/1912.10094/) (2019)
- [Data-Driven Reduced-Order Modeling of Spatiotemporal Chaos with Neural Ordinary Differential Equations](https://mlsanity.com/paper/2109.00060/) (2021)
- [Autoencoders for discovering manifold dimension and coordinates in data from complex dynamical systems](https://mlsanity.com/paper/2305.01090/) (2023)
- [Symmetry reduction for deep reinforcement learning active control of chaotic spatiotemporal dynamics](https://mlsanity.com/paper/2104.05437/) (2021)
- [Data-driven low-dimensional dynamic model of Kolmogorov flow](https://mlsanity.com/paper/2210.16708/) (2022)
- [Symmetry-Aware Autoencoders: s-PCA and s-nlPCA](https://mlsanity.com/paper/2111.02893/) (2021)
- [Dynamics of a data-driven low-dimensional model of turbulent minimal Couette flow](https://mlsanity.com/paper/2301.04638/) (2023)
- [Auto-Encoded Reservoir Computing for Turbulence Learning](https://mlsanity.com/paper/2012.10968/) (2020)
- [An adjoint-based approach for finding invariant solutions of Navier-Stokes equations](https://mlsanity.com/paper/1508.06363/) (2015)

## Built on by (most cited first, 2 of 2)

- [Physically Based Neural Bidirectional Reflectance Distribution Function](https://mlsanity.com/paper/2411.02347/) (2024)
- [Blending data and physics for reduced-order modeling of systems with spatiotemporal chaotic dynamics](https://mlsanity.com/paper/2507.21299/) (2025)

## Abstract (arXiv)

Data-driven reduced-order models of the dynamics of complex flows are important for tasks related to design, understanding, prediction, and control. Many flows obey symmetries, and the present work illustrates how these can be exploited to yield highly efficient low-dimensional data-driven models for chaotic flows. In particular, incorporating symmetries both guarantees that the reduced order model automatically respects them and dramatically increases the effective density of data sampling. Given data for the long-time dynamics of a system, and knowing the set of continuous and discrete symmetries it obeys, the first step in the methodology is to identify a "fundamental chart", a region in the state space of the flow to which all other regions can be mapped by a symmetry operation, and a set of criteria indicating what mapping takes each point in state space into that chart. We then find a low-dimensional coordinate representation of the data in the fundamental chart with the use of an autoencoder architecture that also provides an estimate of the dimension of the invariant manifold where data lie. Finally, we learn dynamics on this manifold with the use of neural ordinary differential equations. We apply this method, denoted "symmetry charting" to simulation data from two-dimensional Kolmogorov flow in a chaotic bursting regime. This system has a continuous translation symmetry, and discrete rotation and shift-reflect symmetries. With this framework we observe that less data is needed to learn accurate data-driven models, more robust estimates of the manifold dimension are obtained, equivariance of the NSE is satisfied, better short-time tracking with respect to the true data is observed, and long-time statistics are correctly captured.