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Koopman spectral analysis has attracted attention for understanding nonlineardynamical systems by which we can analyze nonlinear dynamics with a linearregime by lifting observations using a nonlinear function. For analysis, weneed to find an appropriate lift function. Although several methods have beenproposed for estimating a lift function based on neural networks, the existingmethods train neural networks without spectral analysis. In this paper, wepropose neural dynamic mode decomposition, in which neural networks are trainedsuch that the forecast error is minimized when the dynamics is modeled based onspectral decomposition in the lifted space. With our proposed method, theforecast error is backpropagated through the neural networks and the spectraldecomposition, enabling end-to-end learning of Koopman spectral analysis. Wheninformation is available on the frequencies or the growth rates of thedynamics, the proposed method can exploit it as regularizers for training. Wealso propose an extension of our approach when observations are influenced byexogenous control time-series. Our experiments demonstrate the effectiveness ofour proposed method in terms of eigenvalue estimation and forecast performance.

Neural Dynamic Mode Decomposition for End-to-End Modeling of Nonlinear Dynamics