Learning operators on labelled conditional distributions with applications to mean field control of non exchangeable systems

Abstract

We study the approximation of operators acting on probability measures on a product space with prescribed marginal. Let $I$ be a label space endowed with a reference measure $\hat{I}»$, and define ${\mathcal{M}}_{\hat{\mathcal{I}}}»$ as the set of probability measures on $I\times {\mathbb{R}}^d$ with first marginal $\hat{I}»$. By disintegration, elements of ${\mathcal{M}}_{\hat{\mathcal{I}}}»$ correspond to families of labeled conditional distributions. Operators defined on this constrained measure space arise naturally in mean-field control problems with heterogeneous, non-exchangeable agents. Our main theoretical result establishes a universal approximation theorem for continuous operators on ${\mathcal{M}}_{\hat{\mathcal{I}}}»$. The proof combines cylindrical approximations of probability measures with DeepONet-type branch-trunk neural architecture, yielding finite-dimensional representations of such operators. We further introduce a sampling strategy for generating training measures in ${\mathcal{M}}_{\hat{\mathcal{I}}}»$, enabling practical learning of such conditional mean-field operators. We apply the method to the numerical resolution of mean-field control problems with heterogeneous interactions, thereby extending previous neural approaches developed for homogeneous (exchangeable) systems. Numerical experiments illustrate the accuracy and computational effectiveness of the proposed framework.