Deep Delay Autoencoders Discover Dynamical Systems w Latent Variables: Deep Learning meets Dynamics

Video abstract for Discovering Governing Equations from Partial Measurements with Deep Delay Autoencoders by Joseph Bakarji, Kathleen Champion, J. Nathan Kutz, Steven L. Brunton A central challenge in datadriven model discovery is the presence of hidden, or latent, variables that are not directly measured but are dynamically important. Takens theorem provides conditions for when it is possible to augment these partial measurements with time delayed information, resulting in an attractor that is diffeomorphic to that of the original fullstate system. However, the coordinate transformation back to the original attractor is typically unknown, and learning the dynamics in the embedding space has remained an open challenge for decades. Here, we design a custom deep autoencoder network to learn a coordinate transformation from the delay embedded space into a new space where it is possible to represent the dynamics in a sparse, closed form. We