论文标题
通过koopman操作员回归学习动力系统，重现内核希尔伯特空间
Learning Dynamical Systems via Koopman Operator Regression in Reproducing Kernel Hilbert Spaces
论文作者
论文摘要
我们研究以马尔可夫链建模的一类动力系统，这些系统通过相应的转移或操作员Koopman接收不变分布。尽管重建此类运算符的数据驱动算法是众所周知的，但它们与统计学习的关系在很大程度上尚未探索。我们正式化了一个框架，以从动态系统的有限数据轨迹中学习Koopman操作员。我们考虑将该操作员的限制限制在繁殖内核希尔伯特空间，并引入风险概念，自然会出现不同的估计量。我们将风险与Koopman操作员光谱分解的估计联系起来。这些观察结果激发了降低的运算符回归（RRR）估计器。我们为拟议的估计器得出了学习界限，同时在I.I.D.和非I.I.D.设置，后者在混合系数方面。我们的结果表明，RRR可能比其他广泛使用的估计器有益，如数值实验中的预测和模式分解。
We study a class of dynamical systems modelled as Markov chains that admit an invariant distribution via the corresponding transfer, or Koopman, operator. While data-driven algorithms to reconstruct such operators are well known, their relationship with statistical learning is largely unexplored. We formalize a framework to learn the Koopman operator from finite data trajectories of the dynamical system. We consider the restriction of this operator to a reproducing kernel Hilbert space and introduce a notion of risk, from which different estimators naturally arise. We link the risk with the estimation of the spectral decomposition of the Koopman operator. These observations motivate a reduced-rank operator regression (RRR) estimator. We derive learning bounds for the proposed estimator, holding both in i.i.d. and non i.i.d. settings, the latter in terms of mixing coefficients. Our results suggest RRR might be beneficial over other widely used estimators as confirmed in numerical experiments both for forecasting and mode decomposition.
