  In the near-future, dedicated telescopes observe Earth-like exoplanets in
reflected light, allowing their characterization. Because of the huge
distances, every exoplanet will be a single pixel, but temporal variations in
its spectral flux hold information about the planet's surface and atmosphere.
We test convolutional neural networks for retrieving a planet's rotation axis,
surface and cloud map from simulated single-pixel flux and polarization
observations. We investigate the assumption that the planets reflect Lambertian
in the retrieval while their actual reflection is bidirectional, and of
including polarization in retrievals. We simulate observations along a planet's
orbit using a radiative transfer algorithm that includes polarization and
bidirectional reflection by vegetation, desert, oceans, water clouds, and
Rayleigh scattering in 6 spectral bands from 400 to 800 nm, at various photon
noise levels. The surface-types and cloud patterns of the facets covering a
model planet are based on probability distributions. Our networks are trained
with simulated observations of millions of planets before retrieving maps of
test planets. The neural networks can constrain rotation axes with a mean
squared error (MSE) as small as 0.0097, depending on the orbital inclination.
On a bidirectionally reflecting planet, 92% of ocean and 85% of vegetation,
desert, and cloud facets are correctly retrieved, in the absence of noise. With
realistic noise, it should still be possible to retrieve the main map features
with a dedicated telescope. Except for face-on orbits, a network trained with
Lambertian reflecting planets, yields significant retrieval errors when given
observations of bidirectionally reflecting planets, in particular, brightness
artefacts around a planet's pole. Including polarization improves retrieving
the rotation axis and the accuracy of the retrieval of ocean and cloud facets.
