  The dynamics of ensembles of phase oscillators are usually described
considering their infinite-size limit. In practice, however, this limit is
fully accessible only if the Ott-Antonsen theory can be applied, and the
heterogeneity is distributed following a rational function. In this work we
demonstrate the usefulness of a moment-based scheme to reproduce the dynamics
of infinitely many oscillators. Our analysis is particularized for Gaussian
heterogeneities, leading to a FourierHermite decomposition of the oscillator
density. The Fourier-Hermite moments obey a set of hierarchical ordinary
differential equations. As a preliminary experiment, the effects of truncating
the moment system and implementing different closures are tested in the
analytically solvable Kuramoto model. The moment-based approach proves to be
much more efficient than the direct simulation of a large oscillator ensemble.
The convenience of the moment-based approach is exploited in two illustrative
examples: (i) the Kuramoto model with bimodal frequency distribution, and (ii)
the 'enlarged Kuramoto model' (endowed with nonpairwise interactions). In both
systems we obtain new results inaccessible through direct numerical integration
of populations.
