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 learn the transfer operator of the system, that 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 context of transfer operator regression. 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其动态的状态表示问题。这对于学习系统的转移算子至关重要，该算子可用于预测和解释系统动力学等众多任务。我们证明，寻找良好表示的问题可以转化为神经网络的优化问题。我们的方法得到统计学习理论最新成果的支持，强调了逼近误差和度量失真在转移算子回归中的作用。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，能够克服度量失真，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先进方法，结果表明我们的方法在所有数据集上均表现更优。