{"ID":2870545,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2509.13260","arxiv_id":"2509.13260","title":"Forward Euler for Wasserstein Gradient Flows: Breakdown and Regularization","abstract":"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.","short_abstract":"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...","url_abs":"https://arxiv.org/abs/2509.13260","url_pdf":"https://arxiv.org/pdf/2509.13260v2","authors":"[\"Yewei Xu\",\"Qin Li\"]","published":"2025-09-16T17:14:40Z","proceeding":"math.NA","tasks":"[\"math.NA\",\"math.OC\"]","methods":"[]","has_code":false}