  We propose an Euler particle transport (EPT) approach for generative
learning. The proposed approach is motivated by the problem of finding an
optimal transport map from a reference distribution to a target distribution
characterized by the Monge-Ampere equation. Interpreting the infinitesimal
linearization of the Monge-Ampere equation from the perspective of gradient
flows in measure spaces leads to a stochastic McKean-Vlasov equation. We use
the forward Euler method to solve this equation. The resulting forward Euler
map pushes forward a reference distribution to the target. This map is the
composition of a sequence of simple residual maps, which are computationally
stable and easy to train. The key task in training is the estimation of the
density ratios or differences that determine the residual maps. We estimate the
density ratios (differences) based on the Bregman divergence with a gradient
penalty using deep density-ratio (difference) fitting. We show that the
proposed density-ratio (difference) estimators do not suffer from the "curse of
dimensionality" if data is supported on a lower-dimensional manifold. Numerical
experiments with multi-mode synthetic datasets and comparisons with the
existing methods on real benchmark datasets support our theoretical results and
demonstrate the effectiveness of the proposed method.
