Discrete and Continuous Symmetry Reduction for Minimal Parametrizations of Chaotic Kolmogorov Flows

Mathematical laws that govern fluid motion preserve their shape under translation, rotation, and reflection of coordinates. Consequently, most hydrodynamic systems of interest exhibit a set of symmetries, the action of which on the fluid states commutes with the dynamics. In complex flows, typical non-laminar fluid states are not invariant under these symmetries. Thus, each solution of the system has many dynamically equivalent symmetry copies. For data-driven model reduction methods, such as undercomplete Autoencoders, this multiplicity is not desired since it results in an artificial inflation of the training data which does not yield any physical insight. We consider this problem in the sinusoidally-driven Navier-Stokes equations in two dimensions, i.e. Kolmogorov flow, which is symmetric under continuous translations as well as discrete rotations and reflections.

We formulate a symmetry reduction that combines first Fourier mode slicing with invariant polynomials that yields a fully invariant formulation of the corresponding dynamical system. Through this symmetry reduction, we are able to find a minimal approximation to the inertial manifold of this system as well as

ordinary differential equations on this manifold that describe the dynamics.