各向同性Turbulence中Mori-Zwanzig算子的数据驱动学习
Data Driven Learning of Mori-Zwanzig Operators for Isotropic Turbulence
摘要：
湍流包含了大范围的动力学过程，发展降阶湍流模型是一个极具挑战性的问题。在统计力学中，Mori-Zwanzig（MZ）形式主义提供了一个数学形式化的过程，用于构造高维动力系统的降阶表示，其中由于未解决的动力学的影响被捕获在记忆核和正交动力学中。基于MZ形式的湍流模型一直很稀少，由于有限的知识的MZ运营商，这源于在复杂的非线性动力系统的MZ核推导困难。在这项工作中，我们应用最近开发的数据驱动的学习算法，这是基于Koopman的动力系统和Mori的线性投影算子的描述，在一组完全解决各向同性湍流数据集提取Mori-Zwanzig运营商。使用已知的湍流对称性的数据增强，提取的马尔可夫项，记忆内核，和正交动态统计收敛和广义波动耗散关系可以验证。的内存内核和正交动力学的属性，以及它们对观测量的选择的依赖性进行了研究，以解决通常使用的基于MZ的模型的建模假设。一系列的数值实验，然后使用提取的内核来评估记忆效应的预测。结果表明，观测量的选择对预测误差有很大的影响，通过在记忆核中加入观测量的历史，可以进一步减小预测误差。
英文摘要：
Developing reduced-order models for turbulent flows, which contain dynamics over a wide range of scales, is an extremely challenging problem. In statistical mechanics, the Mori-Zwanzig (MZ) formalism provides a mathematically formal procedure for constructing reduced-order representations of high-dimensional dynamical systems, where the effect due to the unresolved dynamics are captured in the memory kernel and orthogonal dynamics. Turbulence models based on MZ formalism have been scarce due to the limited knowledge of the MZ operators, which originates from the difficulty in deriving MZ kernels for complex nonlinear dynamical systems. In this work, we apply a recently developed data-driven learning algorithm, which is based on Koopman's description of dynamical systems and Mori's linear projection operator, on a set of fully-resolved isotropic turbulence datasets to extract the Mori-Zwanzig operators. With data augmentation using known turbulence symmetries, the extracted Markov term, memory kernel, and orthogonal dynamics are statistically converged and the Generalized Fluctuation-Dissipation Relation can be verified. The properties of the memory kernel and orthogonal dynamics, and their dependence on the choices of observables are investigated to address the modeling assumptions that are commonly used in MZ-based models. A series of numerical experiments are then constructed using the extracted kernels to evaluate the memory effects on predictions. Results show that the prediction errors are strongly affected by the choice of observables and can be further reduced by including the past history of the observables in the memory kernel.
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