  We propose a data-driven filtered reduced order model (DDF-ROM) framework for
the numerical simulation of fluid flows. The novel DDF-ROM framework consists
of two steps: (i) In the first step, we use explicit ROM spatial filtering of
the nonlinear PDE to construct a filtered ROM. This filtered ROM is
low-dimensional, but is not closed (because of the nonlinearity in the given
PDE). (ii) In the second step, we use data-driven modeling to close the
filtered ROM, i.e., to model the interaction between the resolved and
unresolved modes. To this end, we use a quadratic ansatz to model this
interaction and close the filtered ROM. To find the new coefficients in the
closed filtered ROM, we solve an optimization problem that minimizes the
difference between the full order model data and our ansatz. We emphasize that
the new DDF-ROM is built on general ideas of spatial filtering and optimization
and is independent of (restrictive) phenomenological arguments.
  We investigate the DDF-ROM in the numerical simulation of a 2D channel flow
past a circular cylinder at Reynolds number $Re=100$. The DDF-ROM is
significantly more accurate than the standard projection ROM. Furthermore, the
computational costs of the DDF-ROM and the standard projection ROM are similar,
both costs being orders of magnitude lower than the computational cost of the
full order model. We also compare the new DDF-ROM with modern ROM closure
models in the numerical simulation of the 1D Burgers equation. The DDF-ROM is
more accurate and significantly more efficient than these ROM closure models.
