arXiv:2607.12579 2026 Training 2 ideas extracted · analyzed Aug 30, 2026
The paper gives a constructive Wasserstein gradient-flow mechanism for reducing a Coulomb-kernel discrepancy between a learned source distribution and a target distribution. Its transferable asset is the explicit density-weighted transport field: particles move along the negative gradient of the Coulomb potential generated by the source-minus-target measure, producing repulsion in addition to target attraction and therefore counteracting mode collapse. On a torus this field and its energy can be computed efficiently by solving a Poisson equation or by FFT, making the construction practical for generative-model training. The main caveat is that global PL-type guarantees require bounded, sufficiently positive targets; vacuum regions and spatial separation destroy uniform rates, so experiments should explicitly test these failure modes.
Train a generator with a Coulomb discrepancy rather than, or in addition to, a local adversarial or reconstruction loss. The induced force attracts generated mass toward the target while repelling excess source mass, giving a geometry-aware anti-collapse regularizer.
Use one or a few explicit Coulomb transport steps on generated particles as a differentiable or detached corrector, then train the generator to imitate the corrected particles. This separates global distribution matching from the generator parameterization and can reduce adversarial-gradient noise and mode collapse.