  Koopman spectral analysis has attracted attention for understanding nonlinear
dynamical systems by which we can analyze nonlinear dynamics with a linear
regime by lifting observations using a nonlinear function. For analysis, we
need to find an appropriate lift function. Although several methods have been
proposed for estimating a lift function based on neural networks, the existing
methods train neural networks without spectral analysis. In this paper, we
propose neural dynamic mode decomposition, in which neural networks are trained
such that the forecast error is minimized when the dynamics is modeled based on
spectral decomposition in the lifted space. With our proposed method, the
forecast error is backpropagated through the neural networks and the spectral
decomposition, enabling end-to-end learning of Koopman spectral analysis. When
information is available on the frequencies or the growth rates of the
dynamics, the proposed method can exploit it as regularizers for training. We
also propose an extension of our approach when observations are influenced by
exogenous control time-series. Our experiments demonstrate the effectiveness of
our proposed method in terms of eigenvalue estimation and forecast performance.
