The PAU Survey: Photometric redshifts using transfer learning from
  simulations
  In this paper we introduce the \textsc{Deepz} deep learning photometric
redshift (photo-$z$) code. As a test case, we apply the code to the PAU survey
(PAUS) data in the COSMOS field. \textsc{Deepz} reduces the $\sigma_{68}$
scatter statistic by 50\% at $i_{\rm AB}=22.5$ compared to existing algorithms.
This improvement is achieved through various methods, including transfer
learning from simulations where the training set consists of simulations as
well as observations, which reduces the need for training data. The redshift
probability distribution is estimated with a mixture density network (MDN),
which produces accurate redshift distributions. Our code includes an
autoencoder to reduce noise and extract features from the galaxy SEDs. It also
benefits from combining multiple networks, which lowers the photo-$z$ scatter
by 10 percent. Furthermore, training with randomly constructed coadded fluxes
adds information about individual exposures, reducing the impact of photometric
outliers. In addition to opening up the route for higher redshift precision
with narrow bands, these machine learning techniques can also be valuable for
broad-band surveys.
