Data-driven low-dimensional dynamic model of Kolmogorov flow

Reduced order models (ROMs) that capture flow dynamics are of interest for decreasing computational costs for simulation as well as for model-based control approaches. This work presents a data-driven framework for minimal-dimensional models that effectively capture the dynamics and properties of the flow. We apply this to Kolmogorov flow in a regime consisting of chaotic and intermittent behavior, which is common in many flow processes and is challenging to model. The trajectory of the flow travels near relative peri-odic orbits (RPOs), interspersed with sporadic bursting events corresponding to excursions between the regions containing the RPOs. The first step in development of the models is use of an undercomplete autoencoder to map from the full state data down to a latent space of dramatically lower dimension. Then models of the discrete-time evolution of the dynamics in the latent space are developed. By analyzing the model performance as a function of latent space dimension, we can estimate the minimum number of dimensions required to capture the system dynamics. To further reduce the dimension of the dynamical model, we factor out a phase variable in the direction of translational invariance for the flow, leading to separate evolution equations for the pattern and phase dynamics. At a model dimension of five for the pattern dynamics, as opposed to the full state dimension of 1024 (i.e., a 32 x 32 grid), accurate predictions are found for individual trajectories out to about two Lyapunov times, as well as for long-time statistics. Further small improvements in the results occur as dimension is increased to nine, beyond which the statistics of the model and the true system are in very good agreement. The nearly heteroclinic connections between the different RPOs, including the quiescent and bursting timescales, are well captured. We also capture key features of the phase dynamics. Finally, we use the low-dimensional representation to predict future bursting events, finding good success.