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.

翻译：本文考察了用于模拟Wasserstein梯度流的前向欧拉离散化方法。我们提供了两个反例，表明即使在能量泛函定义为与某些结构良好的概率密度之间的KL散度这一简单情形下，该离散化方法也会失效。此外，本文还讨论了这一失效现象的简单解释。