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.

翻译：我们研究在具有给定边际分布的乘积空间上作用于概率测度的算子的逼近问题。设$I$为带参考测度$λ$的标签空间，定义$\cal M_λ$为第一边际为$λ$的$I\times \mathbb{R}^d$上概率测度的集合。通过分解，$\cal M_λ$中的元素对应于标记条件分布族。定义在此约束测度空间上的算子自然地出现在具有异质、非可交换智能体的平均场控制问题中。我们的主要理论结果建立了$\cal M_λ$上连续算子的通用逼近定理。证明结合了概率测度的柱面近似与DeepONet型分支-主干神经网络架构，从而得到此类算子的有限维表示。我们进一步引入一种在$\cal M_λ$中生成训练测度的采样策略，使得此类条件平均场算子的实际学习成为可能。我们将该方法应用于具有异质相互作用的平均场控制问题的数值求解，从而扩展了先前为均质（可交换）系统开发的神经方法。数值实验展示了所提出框架的准确性和计算效率。