Model reduction methods aim to describe complex dynamic phenomena using only relevant dynamical variables, decreasing computational cost, and potentially highlighting key dynamical mechanisms. In the absence of special dynamical features such as scale separation or symmetries, the time evolution of these variables typically exhibits memory effects. Recent work has found a variety of data-driven model reduction methods to be effective for representing such non-Markovian dynamics, but their scope and dynamical underpinning remain incompletely understood. Here, we study data-driven model reduction from a dynamical systems perspective. For both chaotic and randomly-forced systems, we show the problem can be naturally formulated within the framework of Koopman operators and the Mori-Zwanzig projection operator formalism. We give a heuristic derivation of a NARMAX (Nonlinear Auto-Regressive Moving Average with eXogenous input) model from an underlying dynamical model. The derivation is based on a simple construction we call Wiener projection, which links Mori-Zwanzig theory to both NARMAX and to classical Wiener filtering. We apply these ideas to the Kuramoto-Sivashinsky model of spatiotemporal chaos and a viscous Burgers equation with stochastic forcing.

模型简化方法旨在仅使用相关的动力学变量来描述复杂的动力学现象，从而降低计算成本，并可能突出关键的动力学机制。在没有特殊的动力学特征（例如尺度分离或对称性）的情况下，这些变量的时间演变通常表现出记忆效应。最近的工作发现，各种数据驱动的模型简化方法可以有效地表示此类非马尔可夫动力学，但它们的范围和动力学基础仍未完全理解。在这里，我们从动态系统的角度研究数据驱动的模型约简。对于混沌系统和随机强迫系统，我们都表明问题可以自然地在Koopman算子和Mori-Zwanzig投影算子形式主义的框架内提出。我们从基础动力学模型中给出了NARMAX（带有外源输入的非线性自回归移动平均）模型的启发式推导。该推导基于我们称为维纳投影的简单构造，它将Mori-Zwanzig理论与NARMAX和经典的Wiener滤波相结合。我们将这些思想应用于时空混沌的Kuramoto-Sivashinsky模型和具有随机强迫的粘性Burgers方程。