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

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 λ, and define cal Mλ as the set of probability measures on I× ℝd with first marginal λ. By disintegration, elements of cal Mλ 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 cal Mλ. 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 cal Mλ, 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.