  We demonstrate the feasibility of the time-linear scaling formulation of the
$GW$ method [Phys. Rev. Lett. {\bf 124}, 076601 (2020)] for {\it ab initio}
simulations of optically driven two-dimensional materials. The time-dependent
$GW$ equations are derived and solved numerically in the basis of Bloch states.
We address carrier multiplication and relaxation in photo-excited graphene and
find deviations from the typical exponential behavior predicted by the
Markovian Boltzmann approach. For resonantly pumped semiconductor we discover a
self-sustained screening cascade leading to the Mott transition of coherent
excitons. Our results draw attention to the importance of non-Markovian and
dynamical screening effects in out-of-equilibrium phenomena.
