We consider the general class of time-homogeneous dynamical systems, both discrete and continuous, and study the problem of learning a meaningful representation of the state from observed data. This is instrumental for the task of learning a forward transfer operator of the system, that in turn can be used for forecasting future states or observables. The representation, typically parametrized via a neural network, is associated with a projection operator and is learned by optimizing an objective function akin to that of canonical correlation analysis (CCA). However, unlike CCA, our objective avoids matrix inversions and therefore is generally more stable and applicable to challenging scenarios. Our objective is a tight relaxation of CCA and we further enhance it by proposing two regularization schemes, one encouraging the orthogonality of the components of the representation while the other exploiting Chapman-Kolmogorov's equation. We apply our method to challenging discrete dynamical systems, discussing improvements over previous methods, as well as to continuous dynamical systems.

我们考虑时齐动力系统的一般类别，包括离散和连续性，并研究从观测数据中学习状态的有意义表示的问题。这对于学习系统的前向传递算符非常关键，从而可以用于预测未来状态或可观测量。该表示通常通过神经网络参数化，并通过优化类似于规范相关分析（CCA）的目标函数来学习。然而，与CCA不同，我们的目标避免矩阵求逆，因此通常更稳定且适用于挑战性的情况。我们的目标是CCA的一个紧松弛，并通过提出两种正则化方案进一步增强，一种鼓励表示的分量正交而另一种利用Chapman-Kolmogorov方程。我们将该方法应用于具有挑战性的离散动力系统，讨论其相对先前方法的改进，并应用于连续动力系统。

深度投影网络用于学习时间齐次动力系统