An Almgren-type formula for planar $p$-harmonic functions

What the math gives to ML

The paper supplies a nonlinear, gradient-weighted analogue of Almgren frequency that measures the local homogeneity degree of a function without requiring a linear PDE. Its transferable asset is the scale-invariant ratio between interior p-energy and a boundary amplitude normalization weighted by |Du|^{p-2}; unlike an ordinary gradient penalty, it directly tests whether a learned field has a prescribed power-law behavior around a point. The most practical neural-network use is as a local scale-consistency regularizer for neural implicit fields, signed-distance models, or multiscale coordinate networks, with the ratio evaluated on differentiable concentric patches. This should be treated as an empirical regularizer rather than assuming exact monotonicity, because the paper's theorem is specific to planar p-harmonic functions.