Data-driven model reduction, Wiener projections, and the Koopman-Mori-Zwanzig formalism
Data-driven model reduction, Wiener projections, and the Koopman-Mori-Zwanzig formalism
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DOI：
10.1016/j.jcp.2020.109864
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发表时间：
2021-01-01
影响因子：
4.1
通讯作者：
Lu, Fei
中科院分区：
文献类型：
--
作者：
Lin, Kevin K.;Lu, Fei
模型降阶方法的目标是只使用相关的动力学变量来描述复杂的动力学现象，降低计算成本，并潜在地突出关键的动力学机制。在没有尺度分离或对称性等特殊动力学特征的情况下，这些变量的时间演化通常表现出记忆效应。最近的工作已经发现了各种数据驱动的模型简化方法是有效的代表这样的非马尔可夫动态，但其范围和动力学基础仍然不完全理解。在这里，我们从动力系统的角度研究数据驱动的模型简化。对于混沌和随机强迫系统，我们表明这个问题可以自然地制定的Koopman运营商和Mori-Zwanzig投影算子形式主义的框架内。我们给出了一个启发式推导的NARMAX（非线性自回归移动平均与eXogenous输入）模型从一个基本的动力学模型。推导是基于一个简单的构造，我们称之为维纳投影，其中链接Mori-Zwanzig理论的NARMAX和经典的维纳滤波。我们将这些想法应用到Kuramoto-Sivashinsky时空混沌模型和随机强迫粘性Burgers方程。(C)2020爱思唯尔公司All rights reserved.
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. (C) 2020 Elsevier Inc. All rights reserved.