Data-driven dynamics of Kolmogorov flow on the inertial manifold

The high dimensionality of the Navier-Stokes equations (NSEs) hinders controllability of the flows and increases computational costs which in turn motivates the need to find reduced order models (ROMs). In this work we learn high precision low-dimensional models for the NSEs, specifically for two-dimensional Kolmogorov flow. We consider the onset of where chaotic dynamics occurs which exhibits quiescent dynamics that travel near the vicinity of relative periodic orbits followed by bursting events. An undercomplete autoencoder is used to find the inertial manifold dimension and a dense neural network is then used to time map in the reduced space. At a dimension of five, as opposed to the full state dimension of 1024, we see agreement at short times where the predicted trajectory travels close to the true one for approximately two Lyapunov times. Long time statistics related to input power, dissipation, and hibernating/bursting time fractions are also captured. The ROM is also used to predict the phase evolution in the x-direction as well as bursting events, showing agreement and high accuracy at a dimension of five.