An indefinite Coulomb interaction from the Steklov spectrum of perforated manifolds

What the math gives to ML

The paper’s transferable asset is a signed discrepancy-energy construction: the mismatch between a discrete boundary measure and a smooth background density is evaluated through a reduced Green kernel, producing a nonlocal Coulomb-type interaction rather than a purely pairwise repulsion. In neural networks, this can become a regularizer for learned prototypes, token embeddings, or MoE router assignments that simultaneously discourages clustering and enforces a prescribed spatial or semantic density. The important distinction from ordinary repulsive losses is that the signed background term and the zero-mode-removed Green operator make the energy measure discrepancy relative to a target distribution, while the kernel may be indefinite rather than positive-definite. A practical first transfer is a differentiable latent-space distribution-matching regularizer using a truncated Green-kernel approximation.