Sharp Barron Regularity Results for Coulombic Many-Electron Wave Functions
arXiv:2608.22252
2026
Architecture
1 ideas extracted · analyzed Aug 29, 2026
What the math gives to ML
The paper provides a concrete representation-level preconditioner for Coulombic wavefunctions: analytically remove electron-nucleus and electron-electron cusp factors before approximating the remaining function with a neural network. The resulting quotient belongs to Barron regularity classes of every order s less than 2, with an explicit endpoint deterioration proportional to epsilon^{-2}. This suggests that neural quantum-state models should learn a smoother residual while a fixed Jastrow envelope handles singular derivatives. The transferable principle is analytic singularity extraction, which can improve approximation efficiency and stabilize derivative-based physics losses.
Ideas from this paper
Unverified
Re-invented
2026
Represent a many-electron wavefunction as a fixed analytic cusp envelope multiplied by a neural residual, instead of forcing the network to learn Coulomb singularities. The paper predicts that the residual has Barron smoothness arbitrarily close to order two, which should reduce approximation difficulty and noisy local-energy derivatives near particle coalescences.
Useful6/10
Difficulty4/10
Novelty4/10