A priori estimation of memory effects in reduced-order models of nonlinear systems using the Mori-Zwanzig formalism
A priori estimation of memory effects in reduced-order models of nonlinear systems using the Mori-Zwanzig formalism
复制标题
DOI：
10.1098/rspa.2017.0385
复制
发表时间：
2017-09-01
影响因子：
3.5
通讯作者：
Duraisamy, Karthik
中科院分区：
文献类型：
--
作者：
Gouasmi, Ayoub;Parish, Eric J.;Duraisamy, Karthik
非线性动力系统的简化模型需要闭包，或对未分解的模式进行建模。Mori-zwanzig程序可以用来推导形式上封闭的演化方程。在这些方程中，未闭合项被重塑为包含所解析变量的时间历史的记忆积分。虽然此过程不会降低原始系统的复杂性，但这些公式可以作为开发基于内存近似的闭包的数学一致性基础。在这种情况下，内存内核的知识对于评估内存近似的有效性至关重要。解开记忆内核需要求解正交动力学，这是一个高维偏微分方程式，通常是难以处理的。提出了一种基于全阶解快照的内存核先验估计方法。其关键思想是求解一个伪正交动力学方程，该方程具有方便的Liouville形式。这是因为假设正交动力学的半群是一个能观量的复合算子。该方法对线性系统是精确的。对Burgers方程和Kuramoto-Sivashinsky方程的数值结果表明，该方法可以提供关于记忆核的有价值的信息。
Reduced models of nonlinear dynamical systems require closure, or the modelling of the unresolved modes. The Mori-Zwanzig procedure can be used to derive formally closed evolution equations for the resolved physics. In these equations, the unclosed terms are recast as a memory integral involving the time history of the resolved variables. While this procedure does not reduce the complexity of the original system, these equations can serve as a mathematically consistent basis to develop closures based on memory approximations. In this scenario, knowledge of the memory kernel is paramount in assessing the validity of a memory approximation. Unravelling the memory kernel requires solving the orthogonal dynamics, which is a high-dimensional partial differential equation that is intractable, in general. A method to estimate the memory kernel a priori, using full-order solution snapshots, is proposed. The key idea is to solve a pseudo orthogonal dynamics equation, which has a convenient Liouville form, instead. This ersatz arises from the assumption that the semi-group of the orthogonal dynamics is a composition operator for one observable. The method is exact for linear systems. Numerical results on the Burgers and Kuramoto-Sivashinsky equations demonstrate that the proposed technique can provide valuable information about the memory kernel.