  The paper presents a framework for online learning of the Koopman operator
using streaming data. Many complex systems for which data-driven modeling and
control are sought provide streaming sensor data, the abundance of which can
present computational challenges but cannot be ignored. Streaming data can
intermittently sample dynamically different regimes or rare events which could
be critical to model and control. Using ideas from subspace identification, we
present a method where the Grassmannian distance between the subspace of an
extended observability matrix and the streaming segment of data is used to
assess the `novelty' of the data. If this distance is above a threshold, it is
added to an archive and the Koopman operator is updated if not it is discarded.
Therefore, our method identifies data from segments of trajectories of a
dynamical system that are from different dynamical regimes, prioritizes
minimizing the amount of data needed in updating the Koopman model and
furthermore reduces the number of basis functions by learning them adaptively.
Therefore, by dynamically adjusting the amount of data used and learning basis
functions, our method optimizes the model's accuracy and the system order.
