Building symmetries into data-driven manifold dynamics models for complex flows

Symmetries in a dynamical system provide an opportunity to dramatically improve the performance of data-driven models. For fluid flows, such models are needed for tasks related to design, understanding, prediction, and control. In this work we exploit the symmetries of the Navier-Stokes equations (NSE) and use simulation data to find the manifold where the long-time dynamics live, which has many fewer degrees of freedom than the full state representation, and the evolution equation for the dynamics on that manifold. We call this method ''symmetry charting''. The first step is to map to a ''fundamental chart'', which is a region in the state space of the flow to which all other regions can be mapped by a symmetry operation. To map to the fundamental chart we identify a set of indicators from the Fourier transform that uniquely identify the symmetries of the system. We then find a low-dimensional coordinate representation of the data in the fundamental chart with the use of an autoencoder. We use a variation called an implicit rank minimizing autoencoder with weight decay, which in addition to compressing the dimension of the data, also gives estimates of how many dimensions are needed to represent the data: i.e. the dimension of the invariant manifold of the long-time dynamics. Finally, we learn dynamics on this manifold with the use of neural ordinary differential equations. We apply symmetry charting to 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.