论文标题
数据驱动的变异多尺度减少订单模型
Data-Driven Variational Multiscale Reduced Order Models
论文作者
论文摘要
我们提出了一个新的数据驱动的减少订单模型（ROM）框架，该框架围绕变异多尺度（VMS）方法的层次结构，并利用数据以适度的计算成本提高ROM的准确性。 VMS方法是对ROM基础的层次结构的自然拟合：在第一步中，我们使用ROM投影将量表分为三类：（i）解决的大尺度，（ii）解析的小尺度和（iii）未解决的量表。在第二步中，我们明确识别VMS-ROM闭合项，即表示三种类型量表之间相互作用的术语。在第三步中，我们使用可用数据来对VMS-ROM闭合项进行建模。因此，我们利用可用的数据来构建新的结构VMS-ROM闭合模型，而不是用于标准数值离散化的VMS中使用的现象学模型。具体来说，我们构建了最接近使用可用数据评估的真实ROM闭合项的ROM操作员（向量，矩阵和张量）。我们在四个测试用例的数值模拟中测试了新的数据驱动的VMS-ROM：（i）具有粘度系数$ν= 10^{ - 3} $的1D汉堡方程； （ii）在雷诺数$ re = 100 $，$ re = 500 $和$ re = 1000 $的圆形圆柱体上经过的2D流。 （iii）Reynolds Number $ re = 450 $和Rossby Number $ ro = 0.0036 $的准晶状体方程式$ re = 450 $; （iv）在雷诺数$ re = 1000 $上的向后面向步骤上的2D流。数值结果表明，数据驱动的VMS-ROM比标准ROM明显更准确。
We propose a new data-driven reduced order model (ROM) framework that centers around the hierarchical structure of the variational multiscale (VMS) methodology and utilizes data to increase the ROM accuracy at a modest computational cost. The VMS methodology is a natural fit for the hierarchical structure of the ROM basis: In the first step, we use the ROM projection to separate the scales into three categories: (i) resolved large scales, (ii) resolved small scales, and (iii) unresolved scales. In the second step, we explicitly identify the VMS-ROM closure terms, i.e., the terms representing the interactions among the three types of scales. In the third step, we use available data to model the VMS-ROM closure terms. Thus, instead of phenomenological models used in VMS for standard numerical discretizations (e.g., eddy viscosity models), we utilize available data to construct new structural VMS-ROM closure models. Specifically, we build ROM operators (vectors, matrices, and tensors) that are closest to the true ROM closure terms evaluated with the available data. We test the new data-driven VMS-ROM in the numerical simulation of four test cases: (i) the 1D Burgers equation with viscosity coefficient $ν= 10^{-3}$; (ii) a 2D flow past a circular cylinder at Reynolds numbers $Re=100$, $Re=500$, and $Re=1000$; (iii) the quasi-geostrophic equations at Reynolds number $Re=450$ and Rossby number $Ro=0.0036$; and (iv) a 2D flow over a backward facing step at Reynolds number $Re=1000$. The numerical results show that the data-driven VMS-ROM is significantly more accurate than standard ROMs.
