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.

中文翻译：

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数据驱动的模型简化、Wiener 投影和 Koopman-Mori-Zwanzig 形式主义

模型缩减方法旨在仅使用相关的动态变量来描述复杂的动态现象，降低计算成本，并可能突出关键的动态机制。在没有特殊的动态特征（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对于表示这种非马尔可夫动力学是有效的，但它们的范围和动力学基础仍未完全了解。在这里，我们从动态系统的角度研究数据驱动的模型简化。对于混沌和随机受迫系统，我们表明问题可以在 Koopman 算子和 Mori-Zwanzig 投影算子形式主义的框架内自然地表述。我们从基础动力学模型中启发式推导了 NARMAX（具有外生输入的非线性自回归移动平均线）模型。推导基于我们称为 Wiener 投影的简单结构，它将 Mori-Zwanzig 理论与 NARMAX 和经典 Wiener 滤波联系起来。我们将这些想法应用于时空混沌的 Kuramoto-Sivashinsky 模型和具有随机强迫的粘性 Burgers 方程。