We propose deep Koopman-layered models with learnable parameters in the form of Toeplitz matrices for analyzing 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 underlined in the model, we can show its universality and the 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. In addition, the proposed model can be regarded as a neural ODE-based model. In this sense, the proposed model establishes a new connection among Koopman operators, neural ODEs, and numerical linear algebraic methods.

翻译：我们提出了一种深度Koopman分层模型，其可学习参数以Toeplitz矩阵的形式表示，用于分析时间序列数据的动力学特性。该模型兼具理论坚实性与灵活性。凭借Toeplitz矩阵的普适性质以及模型所强调的再生性质，我们可以证明其普适性与泛化特性。此外，所提模型的灵活性使其能够拟合来自非自治动力系统的时间序列数据。在训练模型时，我们应用Krylov子空间方法以实现高效计算。同时，该模型可被视为一种基于神经ODE的模型。在此意义上，所提模型在Koopman算子、神经ODE与数值线性代数方法之间建立了一种新的联系。