  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.
