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

TL;DR

This paper develops a neural network framework to approximate operators on labeled conditional distributions, enabling the solution of complex mean-field control problems with heterogeneous agents.

Contribution

It introduces a universal approximation theorem for operators on measure spaces with labels, combining deep neural architectures with measure approximation techniques.

Findings

The method accurately approximates operators on labeled measure spaces.

It effectively solves mean-field control problems with non-exchangeable agents.

Numerical experiments demonstrate computational efficiency and accuracy.

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 $\lambda$, and define $\cal M_\lambda$ as the set of probability measures on $I\times \mathbb{R}^d$ with first marginal $\lambda$. By disintegration, elements of $\cal M_\lambda$ 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 $\cal M_\lambda$. 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 $\cal M_\lambda$, 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.