We consider the general class of time-homogeneous stochastic dynamical systems, both discrete and continuous, and study the problem of learning a representation of the state that faithfully captures its dynamics. This is instrumental to learning the transfer operator or the generator of the system, which in turn can be used for numerous tasks, such as forecasting and interpreting the system dynamics. We show that the search for a good representation can be cast as an optimization problem over neural networks. Our approach is supported by recent results in statistical learning theory, highlighting the role of approximation error and metric distortion in the learning problem. The objective function we propose is associated with projection operators from the representation space to the data space, overcomes metric distortion, and can be empirically estimated from data. In the discrete-time setting, we further derive a relaxed objective function that is differentiable and numerically well-conditioned. We compare our method against state-of-the-art approaches on different datasets, showing better performance across the board.

翻译：我们考虑离散和连续时间齐次随机动力系统的一般类别，并研究学习能够忠实捕捉系统动态的状态表示的问题。这对于学习系统的传输算子或生成元至关重要，进而可用于多种任务，例如预测和解释系统动态。我们证明，良好表示的搜索可转化为神经网络上的优化问题。我们的方法得到了统计学习理论最新成果的支持，突出了近似误差和度量失真在学习问题中的作用。我们提出的目标函数与从表示空间到数据空间的投影算子相关联，克服了度量失真，并可从数据中经验估计。在离散时间设置下，我们进一步推导出一个可微且数值条件良好的松弛目标函数。我们在不同数据集上比较了我们的方法与现有最优方法，结果显示我们的方法在所有方面均表现更优。