Machine-aided initial guesses for unstable periodic orbits

Unstable periodic orbits (UPOs) are believed to be the underlying dynamical structures of turbulence. Loop convergence algorithms deform entire space-time fields (loops) until they satisfy the evolution equations, and initial guesses are thus space-time fields in a high-dimensional space, rendering their identification highly challenging. We use a convolutional autoencoder to obtain a low-dimensional latent representation of the discretized physical space for the one-dimensional Kuramoto-Sivashinksy equation. In this latent space, we construct loops, which are decoded to physical space and used as initial guesses. They are found to be realistic initial guesses, and together with variational convergence algorithms, these guesses help us to quickly converge to UPOs. These initial loops are constructed both through random guesses, and by 'gluing' known UPOs to create longer ones.