We propose an Euler particle transport (EPT) approach for generative learning. EPT is motivated by the problem of constructing the 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 characteristic ODEs. We use the forward Euler method to solve this equation.

The resulting forward Euler map deppends on unkown target via the density ratio/difference. The key task in training is the estimation of them based on the Bregman divergence using deep density-ratio/ difference fitting. We show that the proposed deep 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.