(422d) 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 as well as for applications such as control for drag reduction. In this work we present a framework for learning low-dimensional models in a symmetric subspace where the dynamics of the flow occur. Symmetries are inherent in many physical and dynamical systems; in the Navier-Stokes equations (NSE) these appear in the form of continuous and discrete symmetry groups. Any dynamic trajectory that is a solution to the NSE will be equivariant with respect to its symmetries, and the result of a symmetry group that acts on a solution will be a solution as well. In general, ROMs will not have information of the group of symmetries. This means that to learn accurate ROMs that describe the flow, the models need access to data in every symmetry subspace which are 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. This system has a continuous symmetry in the flow direction as well as rotation and shift-reflect discrete symmetries. By charting the space into different symmetric sections, tracked with the use of indicators that differentiate these, we are able to map the flow field to a fundamental space and learn dynamics on it. With this framework: equivariance in the NSE is satisfied, less data is needed to learn accurate models, and better short time tracking with respect to the true data is observed. We also comment on the minimal dimension needed to accurately capture the dynamics of the flow.