  We introduce a new paradigm for scaling simulations with projected
entangled-pair states (PEPS) for critical strongly-correlated systems, allowing
for reliable extrapolations of PEPS data with relatively small bond dimensions
$D$. The key ingredient consists of using the effective correlation length
$\chi$ for inducing a collapse of data points, $f(D,\chi)=f(\xi(D,\chi))$, for
arbitrary values of $D$ and the environment bond dimension $\chi$. As such we
circumvent the need for extrapolations in $\chi$ and can use many distinct data
points for a fixed value of $D$. Here, we need that the PEPS has been optimized
using a fixed-$\chi$ gradient method, which can be achieved using a novel
tensor-network algorithm for finding fixed points of 2-D transfer matrices, or
by using the formalism of backwards differentiation. We test our hypothesis on
the critical 3-D dimer model, the 3-D classical Ising model, and the 2-D
quantum Heisenberg model.
