这篇论文教你如何用一条轨迹学透一个动力系统，从预测函数到Koopman算子都有数学保证，适合想做理论深度的人。
该论文研究了从单个有限轨迹中学习遍历随机动力系统的问题，针对时间齐次马尔可夫过程。通过非线性最小二乘估计最优一步预测函数，并给出了关于过程不变测度的高概率保证。结果揭示了轨迹数据的非独立同分布特性如何改变经典统计学习分析。框架扩展至高阶系统和有限状态空间，并证明相同的最小二乘和浓度参数可自然延伸至学习Koopman算子。方法结合了统计学习理论和马尔可夫链定量遍历理论，依赖于Hilbert空间值加性泛函的一致几何遍历马尔可夫链的浓度不等式。
Learning Ergodic Dynamical Systems from a Finite Trajectory
We consider the problem of learning from a single finite trajectory of an ergodic stochastic dynamical system. More precisely, we study discrete-time autonomous stochastic systems defining time-homogeneous Markov processes. We first focus on estimating the optimal one-step prediction function by nonlinear least squares, and derive high-probability guarantees measured with respect to the invariant measure of the process. These results make explicit how the non-independent and non-identically distributed nature of trajectory data modifies the classical statistical learning analysis. We then extend the framework to higher-order systems and finite-state spaces. Finally, we show that the same least squares and concentration arguments naturally extend to learning Koopman operators. Our approach combines tools from statistical learning theory and quantitative ergodic theory for Markov chains. It relies, in particular, on a concentration inequality for Hilbert-space-valued additive functionals of uniformly geometrically ergodic Markov chains.