We propose a potential flow generator with L 2 optimal transport regularity, which can be easily integrated into a wide range of generative models including different versions of GANs and normalizing flow models. With only a slight augmentation to the original generator loss functions, our generator not only tries to transport the input distribution to the target one, but also aims to find the one with minimum L 2 transport cost. We show the effectiveness of our method in several 2D problems, and illustrate the concept of "proximity" due to the L 2 optimal transport regularity. Subsequently, we demonstrate the effectiveness of the potential flow generator in image translation tasks with unpaired training data from the MNIST dataset and the CelebA dataset with a comparison against vanilla WGAN-GP and CycleGAN.
The paper proposes a ‘potential flow generator’ that can be seen as a regularizer for traditional GAN losses. It is based on the idea that samples flowing from one distribution to another should follow a minimum travel cost path. This regularization is expressed as an optimal transport problem with a squared Euclidean cost. Authors rely on the dynamic formulation of OT proposed by Benamou and Brenier, 2000. They propose to learn a time-dependent potential field which gradient defines the velocity fields used to drive samples from a source distribution toward a target one. Experiments on a simple 1D case (where the optimal transport map is known), and on images with an MNIST / CelebA qualitative example.
The use of this dynamic formulation is well known in the OT community. See as a good examples:
Trigila, G., & Tabak, E. G. (2016). Data‐driven optimal transport. Communications on Pure and Applied Mathematics, 69(4), 613-648.
and more generally Chapter 7 of the ‘Computational Optimal Transport’ book by Peyré and Cuturi.
The novelty arises from the use of neural networks to represent the potentials. However, the claim that the obtained map is the optimal transport map seems wrong to me, because:
The class of potential functions over which the optimization is performed is not the whole class of functions, leading to approximations;
The optimality conditions (a.k.a continuity or preservation of mass equations) are only enforced on sampled trajectories, not on the entire space. 
While this claim should definitely be lowered, it is nonetheless still acceptable provided that the proposed model is performing good. On this part, the paper strength could be improved provided that comparisons with existing methods computing a Monge map could be given. Notably, a comparison with the approach from Seguy et al. 2018 is missing. On a same level, a qualitative comparison with cycleGAN  in Figure 4 is missing. 
There are some unclear elements in the paper. The final, total, optimization problem is never clearly expressed. I believe a general algorithm presentation could help in understanding the general picture of the method. Notably, for instance, It is still not clear if the generator G is disconnected from the potential definition (following Eq. 10 I assume not). How are the trajectories sampled ? Is the discriminator trained on the same sampled trajectories or different ones ?  In the potential generator, it seems that the time is considered the same way as the feature space. Can you comment on this point ? Is it possible to evaluate the flow of different time stamps that the ones used for training ? 
As a summary,
Pros:
An interesting way to represent time-dependent potentials with a network for regularizing generative models
Cons:
Not much theoretical novelties in the paper, nor a good analysis on the source of errors of the model (e.g. impact of discretization on the problem)
There are some unclear aspects in the paper (see comments)
The potential benefits of the approach over the state-of-the-art should be more clearly discussed. 
Minor comment:
 P3. Uniqueness of Monge problem for the squared Euclidean cost should be attributed to Brenier 91 and his polar factorization theorem. McCann generalized it to Riemannian manifold.