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Symmetries in a dynamical system provide an opportunity to dramaticallyimprove the performance of data-driven models. For fluid flows, such models areneeded for tasks related to design, understanding, prediction, and control. Inthis work we exploit the symmetries of the Navier-Stokes equations (NSE) anduse simulation data to find the manifold where the long-time dynamics live,which has many fewer degrees of freedom than the full state representation, andthe 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 canbe mapped by a symmetry operation. To map to the fundamental chart we identifya set of indicators from the Fourier transform that uniquely identify thesymmetries of the system. We then find a low-dimensional coordinaterepresentation of the data in the fundamental chart with the use of anautoencoder. We use a variation called an implicit rank minimizing autoencoderwith 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 ordinarydifferential equations. We apply symmetry charting to two-dimensionalKolmogorov flow in a chaotic bursting regime. This system has a continuoustranslation symmetry, and discrete rotation and shift-reflect symmetries. Withthis framework we observe that less data is needed to learn accuratedata-driven models, more robust estimates of the manifold dimension areobtained, equivariance of the NSE is satisfied, better short-time tracking withrespect to the true data is observed, and long-time statistics are correctlycaptured.

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