耦合动力系统的降阶模型：数据驱动方法和Koopman算子
Reduced-Order Models for Coupled Dynamical Systems: Data-driven Methods and the Koopman Operator
摘要：
为模型降阶提供有效和准确的参数化是许多科学和技术领域的关键目标。在这里，我们提出了数据驱动和理论方法之间的紧密联系，以实现这一目标。形式的扰动扩展的Koopman算子，使我们能够得到一般的随机参数化的弱耦合动力系统。这样的参数化产生一组随机积分微分方程与显式噪声和记忆内核公式来描述未解决的变量的影响。我们表明，所涉及的扰动展开不需要截断时，耦合是添加剂。笨重的积分微分方程可以重铸为一个更简单的多层次马尔可夫模型，我们建立了一个直观的连接与广义朗之万方程。这种连接有助于建立自上而下的，基于方程的方法之间的并行性，在这里和完善的经验模型简化（EMR）的方法，已被证明提供有效的动态关闭部分观察到的系统。因此，我们的研究结果支持，一方面，EMR方法的物理基础和鲁棒性，另一方面，说明了用于推导参数化的微扰扩展的实际相关性。
英文摘要：
Providing efficient and accurate parametrizations for model reduction is a key goal in many areas of science and technology. Here we present a strong link between data-driven and theoretical approaches to achieving this goal. Formal perturbation expansions of the Koopman operator allow us to derive general stochastic parametrizations of weakly coupled dynamical systems. Such parametrizations yield a set of stochastic integro-differential equations with explicit noise and memory kernel formulas to describe the effects of unresolved variables. We show that the perturbation expansions involved need not be truncated when the coupling is additive. The unwieldy integro-differential equations can be recast as a simpler multilevel Markovian model, and we establish an intuitive connection with a generalized Langevin equation. This connection helps setting up a parallelism between the top-down, equations-based methodology herein and the well-established empirical model reduction (EMR) methodology that has been shown to provide efficient dynamical closures to partially observed systems. Hence, our findings support, on the one hand, the physical basis and robustness of the EMR methodology and, on the other hand, illustrate the practical relevance of the perturbative expansion used for deriving the parametrizations.
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