  Using the truncated Wigner approximation (TWA) we study quench dynamics of
two-dimensional lattice systems consisting of interacting spinless fermions
with potential disorder. First, we demonstrate that the semiclassical dynamics
generally relaxes faster than the full quantum dynamics. We obtain this result
by comparing the semiclassical dynamics with exact diagonalization and Lanczos
propagation of one-dimensional chains. Next, exploiting the TWA capabilities of
simulating large lattices, we investigate how the relaxation rates depend on
the dimensionality of the studied system. We show that strongly disordered
one-dimensional and two-dimensional systems exhibit a transient,
logarithmic-in-time relaxation, which was recently established for
one-dimensional chains. Such relaxation corresponds to the infamous $1/f$-noise
at strong disorder.
