  For over a century, reduced order models (ROMs) have been a fundamental
discipline of theoretical fluid mechanics. Early examples include Galerkin
models inspired by the Orr-Sommerfeld stability equation and numerous vortex
models, of which the von K\'arm\'an vortex street is one of the most prominent.
Subsequent ROMs typically relied on first principles, like mathematical
Galerkin models, weakly nonlinear stability theory, and two- and
three-dimensional vortex models. Aubry et al. [N. Aubry, P. Holmes, J. Lumley,
and E. Stone, Journal of Fluid Mechanics, 192, 115-173 (1988)] pioneered
data-driven proper orthogonal decomposition (POD) modeling. In early POD
modeling, available data was used to build an optimal basis, which was then
utilized in a classical Galerkin procedure to construct the ROM. But data has
made a profound impact on ROMs beyond the Galerkin expansion. In this paper, we
take a modest step and illustrate the impact of data-driven modeling on one
significant ROM area. Specifically, we focus on ROM closures, which are
correction terms that are added to classical ROMs in order to model the effect
of the discarded ROM modes in under-resolved simulations. Through simple
examples, we illustrate the main modeling principles used to construct
classical ROMs, motivate and introduce modern ROM closures, and show how
data-driven modeling, artificial intelligence, and machine learning have
changed the standard ROM methodology over the last two decades. Finally, we
outline our vision on how state-of-the-art data-driven modeling can continue to
reshape the field of reduced order modeling.
