  A comprehensive microscopic understanding of ambient liquid water is a major
challenge for $ab$ $initio$ simulations as it simultaneously requires an
accurate quantum mechanical description of the underlying potential energy
surface (PES) as well as extensive sampling of configuration space. Due to the
presence of light atoms (e.g., H or D), nuclear quantum fluctuations lead to
observable changes in the structural properties of liquid water (e.g., isotope
effects), and therefore provide yet another challenge for $ab$ $initio$
approaches. In this work, we demonstrate that the combination of
dispersion-inclusive hybrid density functional theory (DFT), the Feynman
discretized path-integral (PI) approach, and machine learning (ML) constitutes
a versatile $ab$ $initio$ based framework that enables extensive sampling of
both thermal and nuclear quantum fluctuations on a quite accurate underlying
PES. In particular, we employ the recently developed deep potential molecular
dynamics (DPMD) model---a neural-network representation of the $ab$ $initio$
PES---in conjunction with a PI approach based on the generalized Langevin
equation (PIGLET) to investigate how isotope effects influence the structural
properties of ambient liquid H$_2$O and D$_2$O. Through a detailed analysis of
the interference differential cross sections as well as several radial and
angular distribution functions, we demonstrate that this approach can furnish a
semi-quantitative prediction of these subtle isotope effects.
