Tracking and forecasting oscillatory data streams using Koopman autoencoders and Kalman filtering

Data-driven modelling techniques provide a method for deriving models of dynamical systems directly from complicated data streams. However, tracking and forecasting such data streams poses a significant challenge to most methods, as they assume the underlying process and model does not change over time. In this paper, we apply one such data-driven method, the Koopman autoencoder (KAE), to high-dimensional oscillatory data to generate a low-dimensional latent space and model, where the system's dynamics appear linear. This allows one to accurately track and forecast systems where the underlying model may change over time. States and the model in the reduced order latent space can then be efficiently updated as new data becomes available, using data assimilation techniques such as the ensemble Kalman filter (EnKF), in a technique we call the KAE EnKF. We demonstrate that this approach is able to effectively track and forecast time-varying, nonlinear dynamical systems in synthetic examples. We then apply the KAE EnKF to a video of a physical pendulum, and achieve a significant improvement over current state-of-the-art methods. By generating effective latent space reconstructions, we find that we are able to construct accurate short-term forecasts and efficient adaptations to externally forced changes to the pendulum's frequency.