Forward Euler for Wasserstein Gradient Flows: Breakdown and Regularization

Wasserstein gradient flows have become a central tool for optimization problems over probability measures. A natural numerical approach is forward-Euler time discretization. We show, however, that even in the simple case where the energy functional is the Kullback-Leibler (KL) divergence against a smooth target density, forward-Euler can fail dramatically: the scheme does not converge to the gradient flow, despite the fact that the first variation $\nabla\frac{δF}{δρ}$ remains formally well defined at every step. We identify the root cause as a loss of regularity induced by the discretization, and prove that a suitable regularization of the functional restores the necessary smoothness, making forward-Euler a viable solver that converges in discrete time to the global minimizer.