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

On closures for reduced order models $-$ A spectrum of first-principle to machine-learned avenues