Let $R$ be a local Gorenstein ring of Krull dimension $d$ with an isolated singularity. Defined $D_{sing}(R)$ as the Verdier quotient $D^b(R)/Perf(R)$). Then, a famous result of Auslander says that the shift by $[d-1]$ is a Serre functor for $D_{sing}(R)$.

Let $A$ be a $\mathbb{N}$-graded Gorenstein algebra satisfying some "nice finiteness" conditions and let $$D_{sing}^{gr}(A) = D^b(gr-A)/Perf(gr-A),$$ the graded derived category of singularities of $A$. I was wondering if there is a general description of the Serre functor of $D_{sing}^{gr}(A)$? Or Perhaps under some extra-hypotheses on A?

I know that various results of Orlov and others show that $D_{sing}^{gr}(A)$ can be sometimes seen as a semi-orthogonal component of $D^b(Coh(X))$, where $X = Proj(A)$. But unless we are in the very special case where $D_{sing}^{gr}(A) = D^b(Coh(X))$, I haven't been able to find a description of the Serre functor of $D_{gr}^{sing}(A)$.

Thanks in advance!