  Accurate and inexpensive Reduced Order Models (ROMs) for forecasting
turbulent flows can facilitate rapid design iterations and thus prove critical
for predictive control in engineering problems. Galerkin projection based
Reduced Order Models (GP-ROMs), derived by projecting the Navier-Stokes
equations on a truncated Proper Orthogonal Decomposition (POD) basis, are
popular because of their low computational costs and theoretical foundations.
However, the accuracy of traditional GP-ROMs degrades over long time prediction
horizons. To address this issue, we extend the recently proposed Neural
Galerkin Projection (NeuralGP) data driven framework to
compressibility-dominated transonic flow, considering a prototypical problem of
a buffeting NACA0012 airfoil governed by the full Navier-Stokes equations. The
algorithm maintains the form of the ROM-ODE obtained from the Galerkin
projection; however coefficients are learned directly from the data using
gradient descent facilitated by differentiable programming. This blends the
strengths of the physics driven GP-ROM and purely data driven neural
network-based techniques, resulting in a computationally cheaper model that is
easier to interpret. We show that the NeuralGP method minimizes a more rigorous
full trajectory error norm compared to a linearized error definition optimized
by the calibration procedure. We also find that while both procedures stabilize
the ROM by displacing the eigenvalues of the linear dynamics matrix of the
ROM-ODE to the complex left half-plane, the NeuralGP algorithm adds more
dissipation to the trailing POD modes resulting in its better long-term
performance. The results presented highlight the superior accuracy of the
NeuralGP technique compared to the traditional calibrated GP-ROM method.
