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

• URL: http://arxiv.org/abs/2603.21683v1
• Date: Mon, 23 Mar 2026 08:13:21 GMT
• Title: Learning operators on labelled conditional distributions with applications to mean field control of non exchangeable systems
• Abstract summary: We study the approximation of operators acting on probability measures on a product space with prescribed marginal.<n>Our main theoretical result establishes a universal approximation theorem for continuous operators on $cal M_$.<n>We introduce a sampling strategy for generating training measures in $cal M_$, enabling practical learning of such conditional mean-field operators.
• Score: 3.441021278275805
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 $λ$, and define $\cal M_λ$ as the set of probability measures on $I\times \mathbb{R}^d$ with first marginal $λ$. By disintegration, elements of $\cal M_λ$ 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 $\cal M_λ$. 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 $\cal M_λ$, 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.

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