Neural computation as operator-theoretic learning on measures.

Abstract

This paper formulates neural computation as the finite-dimensional representation of operators acting on functions and measures. Dense layers, convolutions, attention mechanisms, graph networks, neural fields, and neural operators are treated as instances of kernel-based transformations evaluated through sampling, projection, quadrature, or basis truncation. The aim is to separate the operator being learned from the particular array representation used to compute it. This separation makes architecture design an approximation problem: one must choose a measure space, function space, kernel class, topology, and discretization scheme suited to the target operator. Numerical analysis provides stability and quadrature tools; approximation theory gives representation rates; harmonic analysis identifies bases and transforms adapted to structure; signal processing supplies efficient finite representations; measure theory treats empirical data, weak convergence, transport, and distributional functionals. The result is a unified account of neural architectures as computable approximations of operators over structured domains.