  We obtain a remainder estimate for the truncated Taylor expansion for
differential equations driven by weakly geometric $\Pi $-rough paths for $\Pi
=\left( p_{1},\cdots ,p_{k}\right) $, $p_{i}\geq 1$. When there exists $ p\geq
1$ such that $p_{i}=pk_{i}^{-1}\ $for some $k_{i}\in \left\{ 1,\dots , \left[
p\right] \right\} $, we obtain a refined Taylor remainder estimate that
contains a factorial decay component. The remainder estimates are in the right
order as they are comparable to the next term in the Taylor expansion.
