Critical fractional Hardy inequalities

What the math gives to ML

The paper develops a ground-state representation for nonlocal fractional energies and identifies critical Hardy weights together with an additional nonnegative remainder obtained by optimizing an asymmetric ground state across the two half-lines. The transferable asset is not the one-dimensional Hardy inequality itself, but the combination of a fractional graph energy, scale-singular reference measure, and ground-state reparameterization that converts a difficult singular regularity problem into a weighted difference penalty. This suggests a practical regularizer and parameterization for coordinate MLPs and PINNs whose targets have boundary or interface singularities. The idea is niche rather than a general-purpose optimizer, but it is concrete and falsifiable on singular neural-field benchmarks.