Divergence-free interpolation of tangential vector fields via matrix-valued kernels

What the math gives to ML

The paper provides a constructive way to build tangent-valued, surface-divergence-free matrix kernels from scalar zonal kernels, while avoiding repeated explicit surface differential operators in the final representation. The transferable asset is a hard architectural constraint: every output channel is tangent to a manifold and has exactly zero intrinsic divergence, rather than merely penalizing divergence during training. Positive-definiteness, explicit spectral multipliers, and interpolation-matrix eigenvalue bounds also suggest numerically stable kernel-attention layers on spherical or approximately spherical data. The most direct neural-network experiment is to replace an unconstrained kernel or graph aggregation layer for sphere-valued vector fields with a divergence-free matrix-kernel aggregation layer.