Title: W2GAN: RECOVERING AN OPTIMAL TRANSPORT MAP WITH A GAN. Abstract: Understanding and improving Generative Adversarial Networks (GAN) using notions from Optimal Transport (OT) theory has been a successful area of study, originally established by the introduction of the Wasserstein GAN (WGAN). An increasing number of GANs incorporate OT for improving their discriminators, but that is so far the sole way for the two domains to cross-fertilize. In this work we address the converse question: is it possible to recover an optimal map in a GAN fashion? To achieve this, we build a new model relying on the second Wasserstein distance. This choice enables the use of many results from OT community. In particular, we may completely describe the dynamics of the generator during training. In addition, experiments show that practical uses of our model abide by the rule of evolution we describe. As an application, our generator may be considered as a new way of computing an optimal transport map. It is competitive in low-dimension with standard and deterministic ways to approach the same problem. In high dimension, the fact it is a GAN-style method makes it more powerful than other methods.