  We analyze accuracy of different low-dimensional reductions of the collective
dynamics in large populations of coupled phase oscillators with intrinsic
noise. Three approximations are considered: (i) the Ott-Antonsen ansatz, (ii)
the Gaussian ansatz, and (iii) a two-cumulant truncation of the circular
cumulant representation of the original system's dynamics. For the latter we
suggest a closure, which makes the truncation, for small noise, a rigorous
first-order correction to the Ott-Antonsen ansatz, and simultaneously is a
generalization of the Gaussian ansatz. The Kuramoto model with intrinsic noise,
and the population of identical noisy active rotators in excitable states with
the Kuramoto-type coupling, are considered as examples to test validity of
these approximations. For all considered cases, the Gaussian ansatz is found to
be more accurate than the Ott-Antonsen one for high-synchrony states only. The
two-cumulant approximation is always superior to both other approximations.
