# A Differential Equation Approach for Wasserstein GANs and Beyond

arXiv: https://arxiv.org/abs/2405.16351 · HTML: https://mlsanity.com/paper/2405.16351/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 26 references that are in the map.
- First posted on arXiv 25 May 2024.

## Builds on (most cited first, 10 of 10)

- [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022)
- [Improved Training of Wasserstein GANs](https://mlsanity.com/paper/1704.00028/) (2017)
- [OT-Flow: Fast and Accurate Continuous Normalizing Flows via Optimal Transport](https://mlsanity.com/paper/2006.00104/) (2020)
- [Large-Scale Optimal Transport and Mapping Estimation](https://mlsanity.com/paper/1711.02283/) (2017)
- [On the regularization of Wasserstein GANs](https://mlsanity.com/paper/1709.08894/) (2017)
- [Variational Wasserstein gradient flow](https://mlsanity.com/paper/2112.02424/) (2021)
- [Scalable Wasserstein Gradient Flow for Generative Modeling through Unbalanced Optimal Transport](https://mlsanity.com/paper/2402.05443/) (2024)
- [Adversarial Computation of Optimal Transport Maps](https://mlsanity.com/paper/1906.09691/) (2019)
- [GANs as Gradient Flows that Converge](https://mlsanity.com/paper/2205.02910/) (2022)
- [Faster SGD training by minibatch persistency](https://mlsanity.com/paper/1806.07353/) (2018)

## Abstract (arXiv)

This paper proposes a new theoretical lens to view Wasserstein generative adversarial networks (WGANs). To minimize the Wasserstein-1 distance between the true data distribution and our estimate of it, we derive a distribution-dependent ordinary differential equation (ODE) which represents the gradient flow of the Wasserstein-1 loss, and show that a forward Euler discretization of the ODE converges. This inspires a new class of generative models that naturally integrates persistent training (which we call W1-FE). When persistent training is turned off, we prove that W1-FE reduces to WGAN. When we intensify persistent training, W1-FE is shown to outperform WGAN in training experiments from low to high dimensions, in terms of both convergence speed and training results. Intriguingly, one can reap the benefits only when persistent training is carefully integrated through our ODE perspective. As demonstrated numerically, a naive inclusion of persistent training in WGAN (without relying on our ODE framework) can significantly worsen training results.