Identifying recurrent flows in high-dimensional dissipative chaos from low-dimensional embeddings

Unstable periodic orbits (UPOs) are the non-chaotic, dynamical building blocks of spatio-temporal chaos, motivating a first-principles based theory for turbulence ever since the discovery of deterministic chaos. Despite their key role in the ergodic theory approach to fluid turbulence, identifying UPOs is challenging for two reasons: chaotic dynamics and the high-dimensionality of the spatial discretization. We address both issues at once by proposing a loop convergence algorithm for UPOs directly within a low-dimensional embedding of the chaotic attractor. The convergence algorithm circumvents time-integration, hence avoiding instabilities from exponential error amplification, and operates on a latent dynamics obtained by pulling back the physical equations using automatic differentiation through the learned embedding function. The interpretable latent dynamics is accurate in a statistical sense, and, crucially, the embedding preserves the internal structure of the attractor, which we demonstrate through an equivalence between the latent and physical UPOs of both a model PDE and the 2D Navier-Stokes equations. This allows us to exploit the collapse of high-dimensional dissipative systems onto a lower dimensional manifold, and identify UPOs in the low-dimensional embedding.