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

2026-03-23

AI Summary

This work addresses the mean-field control problem for non-exchangeable, heterogeneous multi-agent systems by proposing a novel approach based on approximating continuous operators over constrained spaces of probability measures. By treating families of labeled conditional distributions as measure-valued elements under fixed marginal constraints, the study establishes, for the first time, a universal approximation theorem for continuous operators in this space and extends the DeepONet architecture to accommodate structured measure inputs. The method integrates cylindrical approximation with a branch-trunk network design and employs a tailored sampling strategy to generate training data. Experimental results demonstrate that the proposed framework achieves superior accuracy and computational efficiency compared to existing neural approaches designed for homogeneous systems when solving heterogeneous mean-field control problems.

Technology Category

• Multiagent Systems: Adversarial Agents
• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

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.

Problem

• mean field control
• non-exchangeable systems
• conditional distributions
• operator learning
• heterogeneous agents

Innovation

• conditional mean-field operators
• non-exchangeable systems
• DeepONet architecture
• cylindrical approximation
• heterogeneous mean-field control