  Although major advances have been achieved over the past decades for the
reduction and identification of linear systems, deriving nonlinear low-order
models still is a chal- lenging task. In this work, we develop a new
data-driven framework to identify nonlinear reduced-order models of a fluid by
combining dimensionality reductions techniques (e.g. proper orthogonal
decomposition) and sparse regression techniques from machine learn- ing. In
particular, we extend the sparse identification of nonlinear dynamics (SINDy)
algorithm to enforce physical constraints in the regression, namely
energy-preserving quadratic nonlinearities. The resulting models, hereafter
referred to as Galerkin regression models, incorporate many beneficial aspects
of Galerkin projection, but without the need for a full-order or high-fidelity
solver to project the Navier-Stokes equations. Instead, the most parsimonious
nonlinear model is determined that is consistent with observed mea- surement
data and satisfies necessary constraints. Galerkin regression models also
readily generalize to include higher-order nonlinear terms that model the
effect of truncated modes. The effectiveness of Galerkin regression is
demonstrated on two different flow configurations: the two-dimensional flow
past a circular cylinder and the shear-driven cavity flow. For both cases, the
accuracy of the identified models compare favorably against reduced-order
models obtained from a standard Galerkin projection procedure. Present results
highlight the importance of cubic nonlinearities in the construction of
accurate nonlinear low-dimensional approximations of the flow systems,
something which cannot be readily obtained using a standard Galerkin projection
of the Navier-Stokes equations. Finally, the entire code base for our
constrained sparse Galerkin regression algorithm is freely available online.
