  Despite the advances provided by large-scale photometric surveys, stellar
features - such as metallicity - generally remain limited to spectroscopic
observations often of bright, nearby low-extinction stars. To rectify this, we
present a neural network approach for estimating the metallicities and
distances of red giant stars with 8-band photometry and parallaxes from Gaia
EDR3 and the 2MASS and WISE surveys. The algorithm accounts for uncertainties
in the predictions arising from the range of possible outputs at each input and
from the range of models compatible with the training set (through drop-out). A
two-stage procedure is adopted where an initial network to estimate
photo-astrometric parallaxes is trained using a large sample of noisy parallax
data from Gaia EDR3 and then a secondary network is trained using spectroscopic
metallicities from the APOGEE and LAMOST surveys and an augmented feature space
utilising the first-stage parallax estimates. The algorithm produces
metallicity predictions with an average uncertainty of $\pm0.19$ dex. The
methodology is applied to stars within the Galactic bar/bulge with particular
focus on a sample of 1.69 million objects with Gaia radial velocities. We
demonstrate the use and validity of our approach by inspecting both spatial and
kinematic gradients with metallicity in the Galactic bar/bulge recovering
previous results on the vertical metallicity gradient ($-0.528 \pm 0.002$
dex/kpc) and the vertex deviation of the bar ($-21.29 \pm 2.74$ deg).
