  Weak gravitational lensing is one of the most promising cosmological probes
of the late universe. Several large ongoing (DES, KiDS, HSC) and planned (LSST,
EUCLID, WFIRST) astronomical surveys attempt to collect even deeper and larger
scale data on weak lensing. Due to gravitational collapse, the distribution of
dark matter is non-Gaussian on small scales. However, observations are
typically evaluated through the two-point correlation function of galaxy shear,
which does not capture non-Gaussian features of the lensing maps. Previous
studies attempted to extract non-Gaussian information from weak lensing
observations through several higher-order statistics such as the three-point
correlation function, peak counts or Minkowski-functionals. Deep convolutional
neural networks (CNN) emerged in the field of computer vision with tremendous
success, and they offer a new and very promising framework to extract
information from 2 or 3-dimensional astronomical data sets, confirmed by recent
studies on weak lensing. We show that a CNN is able to yield significantly
stricter constraints of ($\sigma_8, \Omega_m$) cosmological parameters than the
power spectrum using convergence maps generated by full N-body simulations and
ray-tracing, at angular scales and shape noise levels relevant for future
observations. In a scenario mimicking LSST or Euclid, the CNN yields 2.4-2.8
times smaller credible contours than the power spectrum, and 3.5-4.2 times
smaller at noise levels corresponding to a deep space survey such as WFIRST. We
also show that at shape noise levels achievable in future space surveys the CNN
yields 1.4-2.1 times smaller contours than peak counts, a higher-order
statistic capable of extracting non-Gaussian information from weak lensing
maps.
