  Koopman spectral analysis has attracted attention for nonlinear dynamical
systems since we can analyze nonlinear dynamics with a linear regime by
embedding data into a Koopman space by a nonlinear function. For the analysis,
we need to find appropriate embedding functions. Although several neural
network-based methods have been proposed for learning embedding functions,
existing methods require long time-series for training neural networks. This
limitation prohibits performing Koopman spectral analysis in applications where
only short time-series are available. In this paper, we propose a meta-learning
method for estimating embedding functions from unseen short time-series by
exploiting knowledge learned from related but different time-series. With the
proposed method, a representation of a given short time-series is obtained by a
bidirectional LSTM for extracting its properties. The embedding function of the
short time-series is modeled by a neural network that depends on the
time-series representation. By sharing the LSTM and neural networks across
multiple time-series, we can learn common knowledge from different time-series
while modeling time-series-specific embedding functions with the time-series
representation. Our model is trained such that the expected test prediction
error is minimized with the episodic training framework. We experimentally
demonstrate that the proposed method achieves better performance in terms of
eigenvalue estimation and future prediction than existing methods.
