Deep Koopman-layered Model with Universal Property Based on Toeplitz Matrices

Abstract

We propose deep Koopman-layered models with learnable parameters in the form of Toeplitz matrices for analyzing the transition of the dynamics of time-series data. The proposed model has both theoretical solidness and flexibility. By virtue of the universal property of Toeplitz matrices and the reproducing property underlying the model, we show its universality and generalization property. In addition, the flexibility of the proposed model enables the model to fit time-series data coming from nonautonomous dynamical systems. When training the model, we apply Krylov subspace methods for efficient computations, which establish a new connection between Koopman operators and numerical linear algebra. We also empirically demonstrate that the proposed model outperforms existing methods on eigenvalue estimation of multiple Koopman operators for nonautonomous systems.