Simultaneous calibration of spectro-photometric distances and the Gaia
  DR2 parallax zero-point offset with deep learning
  Gaia measures the five astrometric parameters for stars in the Milky Way, but
only four of them (positions and proper motion, but not parallax) are well
measured beyond a few kpc from the Sun. Modern spectroscopic surveys such as
APOGEE cover a large area of the Milky Way disk and we can use the relation
between spectra and luminosity to determine distances to stars beyond Gaia's
parallax reach. Here, we design a deep neural network trained on stars in
common between Gaia and APOGEE that determines spectro-photometric distances to
APOGEE stars, while including a flexible model to calibrate parallax zero-point
biases in Gaia DR2. We determine the zero-point offset to be $-52.3 \pm 2.0uas$
when modeling it as a global constant, but also train a multivariate zero-point
offset model that depends on $G$, $G_{BP} - G_{RP}$ color, and $T_\mathrm{eff}$
and that can be applied to all 139 million stars in Gaia DR2 within APOGEE's
color--magnitude range. Our spectro-photometric distances are more precise than
Gaia at distances $\approx 2kpc$ from the Sun. We release a catalog of
spectro-photometric distances for the entire APOGEE DR14 data set which covers
Galactocentric radii $2kpc\lesssim R \lesssim19kpc$; $\approx 150,000$ stars
have <10% uncertainty, making this a powerful sample to study the
chemo-dynamical structure of the disk. We use this sample to map the mean
[Fe/H] and 15 abundance ratios [X/Fe] from the Galactic center to the edge of
the disk. Among many interesting trends, we find that the bulge and bar region
at $R \lesssim 5kpc$ clearly stands out in [Fe/H] and most abundance ratios.
