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

TL;DR

This paper introduces neural dynamic mode decomposition, an end-to-end learning method that combines neural networks with spectral analysis to improve modeling and forecasting of nonlinear dynamical systems.

Contribution

It proposes a novel neural network training approach that incorporates spectral decomposition for Koopman analysis, enabling end-to-end learning with regularization from frequency or growth rate information.

Findings

Effective eigenvalue estimation demonstrated

Improved forecast accuracy shown in experiments

Extension to systems with exogenous control

Abstract

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.