Data-driven approximations of the Koopman operator are promising for predicting the time evolution of systems characterized by complex dynamics. Among these methods, the approach known as extended dynamic mode decomposition with dictionary learning (EDMD-DL) has garnered significant attention. Here we present a modification of EDMD-DL that concurrently determines both the dictionary of observables and the corresponding approximation of the Koopman operator. This innovation leverages automatic differentiation to facilitate gradient descent computations through the pseudoinverse. We also address the performance of several alternative methodologies. We assess a 'pure' Koopman approach, which involves the direct time-integration of a linear, high-dimensional system governing the dynamics within the space of observables. Additionally, we explore a modified approach where the system alternates between spaces of states and observables at each time step -- this approach no longer satisfies the linearity of the true Koopman operator representation. For further comparisons, we also apply a state space approach (neural ODEs). We consider systems encompassing two and three-dimensional ordinary differential equation systems featuring steady, oscillatory, and chaotic attractors, as well as partial differential equations exhibiting increasingly complex and intricate behaviors. Our framework significantly outperforms EDMD-DL. Furthermore, the state space approach offers superior performance compared to the 'pure' Koopman approach where the entire time evolution occurs in the space of observables. When the temporal evolution of the Koopman approach alternates between states and observables at each time step, however, its predictions become comparable to those of the state space approach.

翻译：基于数据驱动的Koopman算子近似方法在预测具有复杂动力学特性系统的时间演化方面具有广阔前景。其中，基于字典学习的扩展动态模态分解（EDMD-DL）方法备受关注。本文提出一种改进的EDMD-DL方法，可同时确定可观测函数字典及对应的Koopman算子近似。该创新利用自动微分技术，通过伪逆运算实现梯度下降计算。我们还评估了若干替代方法的性能：包括"纯"Koopman方法（直接在可观测函数空间中对线性高维系统进行时间积分），以及一种改进方法（系统在每个时间步交替切换状态空间与可观测函数空间），后者不再满足真实Koopman算子表示的线性性质。为进一步比较，我们同时应用了状态空间方法（神经ODE）。研究涵盖具有稳态、振荡和混沌吸引子的二维及三维常微分方程组，以及呈现日益复杂精细行为的偏微分方程系统。我们的框架显著优于EDMD-DL方法。此外，状态空间方法比"纯"Koopman方法（完全在可观测函数空间进行时间演化）表现出更优性能。然而，当Koopman方法的时间演化在每个时间步交替切换状态与可观测函数时，其预测能力可达与状态空间方法相当的水平。