Strict Concavity of the Torsion Function for the Restricted Half-Laplacian in Bounded Convex Domains

What the math gives to ML

The paper supplies a strong structural fact for a nonlocal boundary-value problem: on every bounded convex domain, the restricted half-Laplacian torsion function has a strictly negative-definite Hessian everywhere. This is transferable as a mathematically motivated prior for scalar neural fields representing value functions, energies, potentials, or PDE solutions on convex domains, especially when global concavity is more useful than pointwise smoothness. The most practical adaptation is to combine a Monte Carlo residual for the restricted half-Laplacian with a negative-Hessian barrier, then test whether the resulting model is more stable and sample-efficient than a standard MLP or a local Laplacian regularizer.