  We study the unitary dynamics of randomly or quasi-periodically driven tilted
Bose-Hubbard (tBH) model in one dimension deep inside its Mott phase starting
from a $\mathbb{Z}_2$ symmetry-broken state. The randomness is implemented via
a telegraph noise protocol in the drive period while the quasi-periodic drive
is chosen to correspond to a Thue-Morse sequence. The periodically driven tBH
model (with a square pulse protocol characterized by a time period $T$) is
known to exhibit transitions from dynamical regimes with long-time coherent
oscillations to those with rapid thermalization. Here we show that starting
from a regime where the periodic drive leads to rapid thermalization, a random
drive, which consists of a random sequence of square pulses with period
$T+\alpha dT$, where $\alpha=\pm 1$ is a random number and $dT$ is the
amplitude of the noise, restores long-time coherent oscillations for special
values of $dT$. A similar phenomenon can be seen for a quasi-periodic drive
following a Thue-Morse sequence where such coherent behavior is shown to occur
for a larger number of points in the $(T, dT)$ plane due to the additional
structure of the drive protocol. We chart out the dynamics of the system in the
presence of such aperiodic drives, provide a qualitative analytical
understanding of this phenomenon, point out the role of quantum scars behind
it, and discuss experiments which can test our theory.
