Suppose I have a smooth non-proper algebraic variety $X/\mathbb{C}$.

A vector bundle with flat connection (``differential equation'') on $X$ extends, as was noted by Grothendieck, to a coherent crystal on the infinitesimal site of $X$, realising an equivalence between these two types of object.

Deligne's comparison of differential equations on $X$ and $X^{an}$ leads one to restrict one's attention to the subcategory of differential equations on $X$ with regular singularities along the boundary divisor of a good compactification of $X$.

My question: is there a clean way to describe this category as a subcategory of the category of crystals on $Inf(X)$?

My guess (fear?) is the answer might have something to do with log-geometry, but I know very little about such things so would also appreciate a good reference suggestion if indeed it is the required tool.