Deep learning to discover the dimension of an inertial manifold and predict dynamics on it

One of the senior author’s last conversations with Bruno Eckhardt concerned the connection of machine learning tools and ideas to dynamical systems and turbulence. This talk concerns one such connection. Many flow geometries, including pipe, channel and boundary layer, have a continuous translation symmetry. As a model for such systems we consider the Kuramoto-Sivashinsky equation (KSE) in a periodic domain. We describe a method to map the dynamics onto a translationally invariant low-dimensional manifold and time-evolve them using neural networks (NN). Dimensionality reduction is achieved by phase-aligning the spatial structures at each time, then putting them into an undercomplete autoencoder that maps the original dynamics onto a lower-dimensional inertial manifold where the long-time dynamics live. We infer the dimension of the manifold by tracking the autoencoder error vs. dimension—this drops by orders of magnitude once the proper dimension is reached. The spatial structure and phase are then integrated forward in time using a NN. This approach significantly outperforms Principal Components Analysis.