  We introduce time-dependent variational Monte Carlo for continuous-space Bose
gases. Our approach is based on the systematic expansion of the many-body
wave-function in terms of multi-body correlations and is essentially exact up
to adaptive truncation. The method is benchmarked by comparison to exact
Bethe-ansatz or existing numerical results for the integrable Lieb-Liniger
model. We first show that the many-body wave-function achieves high precision
for ground-state properties, including energy and first-order as well as
second-order correlation functions. Then, we study the out-of-equilibrium,
unitary dynamics induced by a quantum quench in the interaction strength. Our
time-dependent variational Monte Carlo results are benchmarked by comparison to
exact Bethe ansatz results available for a small number of particles, and also
compared to quench action results available for non-interacting initial states.
Moreover, our approach allows us to study large particle numbers and general
quench protocols, previously inaccessible beyond the mean-field level. Our
results suggest that it is possible to find correlated initial states for which
the long-term dynamics of local density fluctuations is close to the
predictions of a simple Boltzmann ensemble.
