# Forward-Euler time-discretization for Wasserstein gradient flows can be wrong

arXiv: https://arxiv.org/abs/2406.08209 · HTML: https://mlsanity.com/paper/2406.08209/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 24 references that are in the map.
- First posted on arXiv 12 Jun 2024.

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

- [Learning with a Wasserstein Loss](https://mlsanity.com/paper/1506.05439/) (2015)
- [Distributionally Robust Stochastic Optimization with Wasserstein Distance](https://mlsanity.com/paper/1604.02199/) (2016)
- [Large-Scale Wasserstein Gradient Flows](https://mlsanity.com/paper/2106.00736/) (2021)
- [Projected Wasserstein gradient descent for high-dimensional Bayesian inference](https://mlsanity.com/paper/2102.06350/) (2021)
- [Dataset Dynamics via Gradient Flows in Probability Space](https://mlsanity.com/paper/2010.12760/) (2020)
- [Wasserstein Gradient Flows for Moreau Envelopes of f-Divergences in Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/2402.04613/) (2024)
- [MonoFlow: Rethinking Divergence GANs via the Perspective of Wasserstein Gradient Flows](https://mlsanity.com/paper/2302.01075/) (2023)
- [Estimating the Rate-Distortion Function by Wasserstein Gradient Descent](https://mlsanity.com/paper/2310.18908/) (2023)
- [High order spatial discretization for variational time implicit schemes: Wasserstein gradient flows and reaction-diffusion systems](https://mlsanity.com/paper/2303.08950/) (2023)
- [Wasserstein gradient flow for optimal probability measure decomposition](https://mlsanity.com/paper/2406.00914/) (2024)

## Built on by (most cited first, 3 of 3)

- [Linear convergence of proximal descent schemes on the Wasserstein space](https://mlsanity.com/paper/2411.15067/) (2024)
- [Using Linearized Optimal Transport to Predict the Evolution of Stochastic Particle Systems](https://mlsanity.com/paper/2408.01857/) (2024)
- [Difference of Convex Programming in the Wasserstein Space with Applications to MMD Optimization](https://mlsanity.com/paper/2606.27767/) (2026)

## Abstract (arXiv)

In this note, we examine the forward-Euler discretization for simulating Wasserstein gradient flows. We provide two counter-examples showcasing the failure of this discretization even for a simple case where the energy functional is defined as the KL divergence against some nicely structured probability densities. A simple explanation of this failure is also discussed.