Forward-Euler time-discretization for Wasserstein gradient flows can be wrong
This highlights a critical flaw in a common numerical method for computational optimal transport, affecting researchers and practitioners in machine learning and applied mathematics.
The paper demonstrates that forward-Euler time-discretization can fail when simulating Wasserstein gradient flows, even for simple cases like KL divergence energies, by providing two counter-examples and an explanation.
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.