$Γ$-Convergence of Weak-Type Nonlocal Functionals on Bounded Domains

What the math gives to ML

The paper identifies a thresholded weak-type nonlocal interaction whose large-threshold limit is a local Sobolev gradient energy or BV total variation. The transferable asset is that this regularizer can be estimated from sampled pairs of neural-field evaluations, while the Gamma-limit explains why the thresholded construction approaches a meaningful local smoothness prior. A practical adaptation is a differentiable stochastic regularizer for coordinate MLPs, neural fields, or spatially indexed predictors, with the threshold increased during training and the original distance weighting retained.