  Deep learning is a powerful analysis technique that has recently been
proposed as a method to constrain cosmological parameters from weak lensing
mass maps. Due to its ability to learn relevant features from the data, it is
able to extract more information from the mass maps than the commonly used
power spectrum, and thus achieve better precision for cosmological parameter
measurement. We explore the advantage of Convolutional Neural Networks (CNN)
over the power spectrum for varying levels of shape noise and different
smoothing scales applied to the maps. We compare the cosmological constraints
from the two methods in the $\Omega_M-\sigma_8$ plane for sets of 400 deg$^2$
convergence maps. We find that, for a shape noise level corresponding to 8.53
galaxies/arcmin$^2$ and the smoothing scale of $\sigma_s = 2.34$ arcmin, the
network is able to generate 45% tighter constraints. For smaller smoothing
scale of $\sigma_s = 1.17$ the improvement can reach $\sim 50 \%$, while for
larger smoothing scale of $\sigma_s = 5.85$, the improvement decreases to 19%.
The advantage generally decreases when the noise level and smoothing scales
increase. We present a new training strategy to train the neural network with
noisy data, as well as considerations for practical applications of the deep
learning approach.
