  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 $\nu = 10^{-3}$; (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.
