Born Discrete, Made Smooth: Variational Formulation of Shallow Neural Networks

arXiv:2607.02003v1 Announce Type: new Abstract: Although neural networks are remarkably effective, their underlying optimization principles remain theoretically elusive, often characterized by non-convex landscapes and stochastic heuristics. In this work, we propose a paradigm shift by replacing the discrete training problem of shallow neural networks with a well-posed continuum variational surrogate. We identify a family of $\lambda$-convex functionals over parameter densities in weighted Sobolev spaces and prove that these variational problems are globally well-posed, stable, and exhibit une