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.

翻译：本文考虑时间齐次动力系统的一般类别，包括离散系统和连续系统，并研究从观测数据中学习有意义的系统状态表征问题。这对于学习系统前向传递算子（该算子可用于预测未来状态或可观测变量）的任务至关重要。该表征通常通过神经网络参数化，与投影算子相关联，并通过优化类似于典型相关分析的目标函数来学习。然而，与典型相关分析不同，我们的目标函数避免了矩阵求逆，因此通常更加稳定且适用于具有挑战性的场景。我们的目标函数是典型相关分析的紧松弛，并通过提出两种正则化方案进一步增强：一种方案鼓励表征分量的正交性，另一种方案利用查普曼-科尔莫戈罗夫方程。我们将该方法应用于具有挑战性的离散动力系统和连续动力系统，并讨论了相较于先前方法的改进之处。