  We prove that a general class of nonlinear, non-autonomous ODEs in Fr\'echet
spaces are close to ODEs in a specific normal form, where closeness means that
solutions of the normal form ODE satisfy the original ODE up to a residual that
vanishes up to any desired order. In this normal form, the centre, stable and
unstable coordinates of the ODE are clearly separated, which allows us to
define invariant manifolds of such equations in a robust way. In particular,
our method empowers us to study approximate centre manifolds, given by
solutions of ODEs that are central up to a desired, possibly nonzero precision.
The main motivation is the case where the Fr\'echet space in question is a
suitable function space, and the maps involved in an ODE in this space are
defined in terms of derivatives of the functions, so that the
infinite-dimensional ODE is a finite-dimensional PDE. We show that our methods
apply to a relevant class of nonlinear, non-autonomous PDEs in this way.
