Learning the dynamics of symmetry-reduced chaotic attractors from data

Recently, a variety of dimensionality reduction and modeling techniques utilizing deep neural networks have been successfully applied to chaotic systems. Many such systems of interest exhibit symmetries under which the rules governing the dynamics preserve their form. Consequently, the data produced by these systems exhibit redundancies due to the symmetries which are undesirable for data-driven modeling. Here, we formulate a discrete symmetry reduction method by means of complex invariant polynomials that is applicable to high-dimensional truncations of formally infinite-dimensional systems such as those arising in fluid simulations. By applying our method to the Lorenz system, the simulations of a periodic cylinder wake, as well as periodic and chaotic Kolmogorov flows, we show that discrete symmetry reduction effectively reduces the amount of data required to learn the dynamics.