  The non-linear sewing lemma constructs flows of rough differential equations
from a braod class of approximations called almost flows. We consider a class
of almost flows that could be approximated by solutions of ordinary
differential equations, in the spirit of the backward error analysis. Mixing
algebra and analysis, a Taylor formula with remainder and a composition formula
are central in the expansion analysis. With a suitable algebraic structure on
the non-smooth vector fields to be integrated, we recover in a single framework
several results regarding high-order expansions for various kind of driving
paths. We also extend the notion of driving rough path. We also introduce as an
example a new family of branched rough paths, called aromatic rough paths
modeled after aromatic Butcher series.
