Discovering Governing Equations from Partial Measurements with Deep Delay Autoencoders

A central challenge in data-driven 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 full-state 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 demonstrate this approach on the Lorenz, R"ossler, and Lotka-Volterra systems, learning dynamics from a single measurement variable. As a challenging example, we learn a Lorenz analogue from a single scalar variable extracted from a video of a chaotic waterwheel experiment. The resulting modeling framework combines deep learning to uncover effective coordinates and the sparse identification of nonlinear dynamics (SINDy) for interpretable modeling. Thus, we show that it is possible to simultaneously learn a closed-form model and the associated coordinate system for partially observed dynamics.