  For any finite dimensional algebra $\Lambda$ given by a quiver with
relations, we prove that its dg singularity category is quasi-equivalent to the
perfect dg derived category of a dg Leavitt path algebra. The result might be
viewed as a deformed version of the known description of the dg singularity
category of a radical-square-zero algebra in terms of a Leavitt path algebra
with trivial differential.
  The above result is achieved in two steps. We first introduce the singular
Yoneda dg category of $\Lambda$, which is quasi-equivalent to the dg
singularity category of $\Lambda$. The construction of this new dg category
follows from a general operation for dg categories, namely an explicit dg
localization inverting a natural transformation from the identity functor to a
dg endofunctor. This localization turns out to be quasi-equivalent to a dg
quotient category. Secondly, we prove that the endomorphism algebra of the
quotient of $\Lambda$ modulo its Jacobson radical in the singular Yoneda dg
category is isomorphic to the dg Leavitt path algebra. The appendix is devoted
to an alternative proof of the result using Koszul-Moore duality and derived
localizations.
