  Model reduction for fluid flow simulation continues to be of great interest
across a number of scientific and engineering fields. Here, we explore the use
of Neural Ordinary Differential Equations, a recently introduced family of
continuous-depth, differentiable networks (Chen et al 2018), as a way to
propagate latent-space dynamics in reduced order models. We compare their
behavior with two classical non-intrusive methods based on proper orthogonal
decomposition and radial basis function interpolation as well as dynamic mode
decomposition. The test problems we consider include incompressible flow around
a cylinder as well as real-world applications of shallow water hydrodynamics in
riverine and estuarine systems. Our findings indicate that Neural ODEs provide
an elegant framework for stable and accurate evolution of latent-space dynamics
with a promising potential of extrapolatory predictions. However, in order to
facilitate their widespread adoption for large-scale systems, significant
effort needs to be directed at accelerating their training times. This will
enable a more comprehensive exploration of the hyperparameter space for
building generalizable Neural ODE approximations over a wide range of system
dynamics.
