  For an $S^1$-framed modular operad $P$, we introduce its "Feynman
compactification" denoted by $FP$ which is a modular operad. Let
$\{\mathbb{M}^{\sf fr}(g,n)\}_{(g,n)}$ be the $S^1$-framed modular operad
defined using moduli spaces of smooth curves with framings along punctures. We
prove that the homology operad of $F\mathbb{M}^{\sf fr}$ is isomorphic to
$H_*(\overline{M})$, the homology operad of the Deligne-Mumford operad. Using
this isomorphism, we obtain an explicit formula of the fundamental class of
$[\overline{M}_{g,n}/S_n]$ in terms of Sen-Zwiebach's string vertices. As an
immediate application, under mild assumptions, we prove that Costello's
categorical enumerative invariants match with the Gromov-Witten invariants when
the Fukaya category has semi-simple Hochschild cohomology.
