  Machine learning can play a powerful role in inferring missing line-of-sight
velocities from astrometry in surveys such as Gaia. In this paper, we apply a
neural network to Gaia Early Data Release 3 (EDR3) and obtain line-of-sight
velocities and associated uncertainties for ~92 million stars. The network,
which takes as input a star's parallax, angular coordinates, and proper
motions, is trained and validated on ~6.4 million stars in Gaia with complete
phase-space information. The network's uncertainty on its velocity prediction
is a key aspect of its design; by properly convolving these uncertainties with
the inferred velocities, we obtain accurate stellar kinematic distributions. As
a first science application, we use the new network-completed catalog to
identify candidate stars that belong to the Milky Way's most recent major
merger, Gaia-Sausage-Enceladus (GSE). We present the kinematic, energy, angular
momentum, and spatial distributions of the ~450,000 GSE candidates in this
sample, and also study the chemical abundances of those with cross matches to
GALAH and APOGEE. The network's predictive power will only continue to improve
with future Gaia data releases as the training set of stars with complete
phase-space information grows. This work provides a first demonstration of how
to use machine learning to exploit high-dimensional correlations on data to
infer line-of-sight velocities, and offers a template for how to train,
validate and apply such a neural network when complete observational data is
not available.
