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The non-linear sewing lemma constructs flows of rough differential equationsfrom a braod class of approximations called almost flows. We consider a classof almost flows that could be approximated by solutions of ordinarydifferential equations, in the spirit of the backward error analysis. Mixingalgebra and analysis, a Taylor formula with remainder and a composition formulaare central in the expansion analysis. With a suitable algebraic structure onthe non-smooth vector fields to be integrated, we recover in a single frameworkseveral results regarding high-order expansions for various kind of drivingpaths. We also extend the notion of driving rough path. We also introduce as anexample a new family of branched rough paths, called aromatic rough pathsmodeled after aromatic Butcher series.

Constructing general rough differential equations through flow approximations