  Weak lensing maps contain information beyond two-point statistics on small
scales. Much recent work has tried to extract this information through a range
of different observables or via nonlinear transformations of the lensing field.
Here we train and apply a 2D convolutional neural network to simulated
noiseless lensing maps covering 96 different cosmological models over a range
of {$\Omega_m,\sigma_8$}. Using the area of the confidence contour in the
{$\Omega_m,\sigma_8$} plane as a figure-of-merit, derived from simulated
convergence maps smoothed on a scale of 1.0 arcmin, we show that the neural
network yields $\approx 5 \times$ tighter constraints than the power spectrum,
and $\approx 4 \times$ tighter than the lensing peaks. Such gains illustrate
the extent to which weak lensing data encode cosmological information not
accessible to the power spectrum or even other, non-Gaussian statistics such as
lensing peaks.
