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

TL;DR

This paper introduces a symmetry-aware data-driven modeling approach for complex chaotic flows, leveraging symmetries to improve model efficiency, accuracy, and robustness, demonstrated on two-dimensional Kolmogorov flow.

Contribution

The work presents a novel 'symmetry charting' method that incorporates flow symmetries into autoencoder and neural ODE frameworks for reduced-order modeling.

Findings

Requires less data for accurate models

Achieves more robust manifold dimension estimates

Captures long-term statistics accurately

Abstract

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.