Data-driven symmetry-aware low-dimensional models for predicting turbulent fluid flows

Reduced-order models (ROMs) that capture flow dynamics are important for decreasing computational cost in simulations and for practical applications such as control for drag reduction. In this work we present a framework for developing low-dimensional models that take advantage of discrete and continuous symmetries in the Navier-Stokes equations (NSE). In general, ROMs will not have information of the symmetries. This means that to learn accurate ROMs, the models need access to data in every symmetry subspace which is populated in the long-time dynamics. To overcome this, we learn ROMs with the use of neural networks in a subspace of the symmetries and apply this to the case of two-dimensional Kolmogorov flow in a chaotic bursting regime. By charting the space into different symmetric sections related by the symmetries of the system, tracked with the use of indicators that distinguish these, we can map the flow field to a fundamental space and learn dynamics on it. With this framework: equivariance is satisfied, less data is needed to learn accurate models, better short-time tracking with respect to the true data is observed, and long-time statistics are captured.