We propose a new theoretical lens to view Wasserstein generative adversarial networks (WGANs). In our framework, we define a discretization inspired by a distribution-dependent ordinary differential equation (ODE). We show that such a discretization is convergent and propose a viable class of adversarial training methods to implement this discretization, which we call W1 Forward Euler (W1-FE). In particular, the ODE framework allows us to implement persistent training, a novel training technique that cannot be applied to typical WGAN algorithms without the ODE interpretation. Remarkably, when we do not implement persistent training, we prove that our algorithms simplify to existing WGAN algorithms; when we increase the level of persistent training appropriately, our algorithms outperform existing WGAN algorithms in both low- and high-dimensional examples.

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