NEURAL DYNAMIC MODE DECOMPOSITION FOR END-TO-END MODELING OF NONLINEAR DYNAMICS

Abstract

Koopman spectral analysis has attracted attention for understand- ing 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 nd an appropriate lift function. Although several meth- ods have been proposed for estimating a lift function based on neural net- works, 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 dy- namics is modeled based on spectral decomposition in the lifted space. With our proposed method, the forecast error is backpropagated through the neu- ral 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 inuenced by exogenous control time-series. Our experiments demonstrate the e ectiveness of our proposed method in terms of eigenvalue estimation and forecast performance.