Is there a resolution of singularities for flat families?

More precisely, if $X \rightarrow \mathbb{A} ^n$ is a flat map, does there exist a map $Y \rightarrow X$ such that, for every $p \in \mathbb{A} ^n$, the fiber map $Y_p \rightarrow X_p$ is a resolution of singularities? Can one require, moreover, that the map $Y \rightarrow \mathbb{A} ^n$ is smooth?