Predicting spatiotemporal chaos by learning conjugate tubular neighborhoods

In the mathematical literature on chaos, the study of unstable time-periodic solutions embedded in strange attractors has a long history that can be traced to Poincaré's work on the three-body problem. Following the development of algorithms that enabled the numerical discovery of periodic orbits in high-dimensional chaotic systems, several papers reported such solutions of the two- and three-dimensional Navier--Stokes equations at moderate Reynolds numbers in fully resolved simulations. While most of these papers also illustrated similarities, such as matching dissipation rates, of periodic orbits and chaotic trajectories within their neighborhoods, whether these solutions could form the basis of a reduced-order description of turbulence is a subject of ongoing study. In the present work, I demonstrate that trajectories within the tubular neighborhood of a periodic orbit can be used to train a predictive neural network model. The model consists of an autoencoder that maps the tubular neighborhood onto a three-dimensional latent space, wherein the trajectories evolve according to a linear time evolution that is topologically conjugate to the leading unstable subspace of the periodic orbit. I illustrate the approach by applying it to the spatiotemporally chaotic Kuramoto--Sivashinsky system in one dimension.