  Proper orthogonal decomposition (POD) allows reduced-order modeling of
complex dynamical systems at a substantial level, while maintaining a high
degree of accuracy in modeling the underlying dynamical systems. Advances in
machine learning algorithms enable learning POD-based dynamics from data and
making accurate and fast predictions of dynamical systems. In this paper, we
leverage the recently proposed heavy-ball neural ODEs (HBNODEs) [Xia et al.
NeurIPS, 2021] for learning data-driven reduced-order models (ROMs) in the POD
context, in particular, for learning dynamics of time-varying coefficients
generated by the POD analysis on training snapshots generated from solving full
order models. HBNODE enjoys several practical advantages for learning POD-based
ROMs with theoretical guarantees, including 1) HBNODE can learn long-term
dependencies effectively from sequential observations and 2) HBNODE is
computationally efficient in both training and testing. We compare HBNODE with
other popular ROMs on several complex dynamical systems, including the von
K\'{a}rm\'{a}n Street flow, the Kurganov-Petrova-Popov equation, and the
one-dimensional Euler equations for fluids modeling.
