  Optimal transport is a notoriously difficult problem to solve numerically,
with current approaches often remaining intractable for very large scale
applications such as those encountered in machine learning. Wasserstein
barycenters -- the problem of finding measures in-between given input measures
in the optimal transport sense -- is even more computationally demanding as it
requires to solve an optimization problem involving optimal transport
distances. By training a deep convolutional neural network, we improve by a
factor of 60 the computational speed of Wasserstein barycenters over the
fastest state-of-the-art approach on the GPU, resulting in milliseconds
computational times on $512\times512$ regular grids. We show that our network,
trained on Wasserstein barycenters of pairs of measures, generalizes well to
the problem of finding Wasserstein barycenters of more than two measures. We
demonstrate the efficiency of our approach for computing barycenters of
sketches and transferring colors between multiple images.
