Quantitative mean-field limits for repulsive Coulomb flows at bounded density and Riesz weak--strong stability

What the math gives to ML

The paper develops a quantitatively controlled repulsive particle flow whose useful transferable ingredients are singular repulsion, a density-dependent interaction scale, and an explicit decay envelope for effective density. These ingredients can be transplanted into neural prototype or embedding training to prevent collapse without keeping repulsion unnecessarily strong late in optimization. The practical adaptation is to add a mollified Coulomb force to embedding updates, decay its coefficient according to the paper's envelope, and enlarge the mollification radius as the estimated density falls. This provides a falsifiable alternative to fixed-distance repulsion for metric learning, prototype learning, vector quantization, and mixture-of-experts router embeddings.