Learning Dynamical Systems via Koopman Operator Regression in Reproducing Kernel Hilbert Spaces
Learning Dynamical Systems via Koopman Operator Regression in Reproducing Kernel Hilbert Spaces
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DOI：
10.48550/arxiv.2205.14027
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发表时间：
2022-05
期刊：
影响因子：
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通讯作者：
V. Kostić;P. Novelli;Andreas Maurer;C. Ciliberto;L. Rosasco;M. Pontil
中科院分区：
文献类型：
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作者：
V. Kostić;P. Novelli;Andreas Maurer;C. Ciliberto;L. Rosasco;M. Pontil
我们研究了一类被建模为马氏链的动力系统，它通过相应的转移或Koopman算子接受不变分布。虽然重构这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还没有被探索。我们形式化了一个框架，以便从动力系统的有限数据轨迹中学习Koopman算子。我们考虑了这个算子对再生核Hilbert空间的限制，并引入了风险的概念，由此自然产生了不同的估计量。我们将风险与Koopman算子的谱分解估计联系起来。这些观察结果激发了降阶算子回归(RRR)估计器。我们推导了所提出的估计量的学习界，两者都在I.I.D.中成立。而且没有身份证。设置，后者以混合系数表示。我们的结果表明，RRR可能比其他广泛使用的估计器更有利，正如数值实验所证实的那样，在预测和模式分解方面都是如此。
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.