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

arXiv: https://arxiv.org/abs/2603.21683 · HTML: https://mlsanity.com/paper/2603.21683/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 39 references that are in the map.
- First posted on arXiv 23 Mar 2026.

## Builds on (most cited first, 19 of 19)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [KAN: Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2404.19756/) (2024)
- [Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations](https://mlsanity.com/paper/1706.04702/) (2017)
- [A Machine Learning Framework for Solving High-Dimensional Mean Field Game and Mean Field Control Problems](https://mlsanity.com/paper/1912.01825/) (2019)
- [Mean-field neural networks: learning mappings on Wasserstein space](https://mlsanity.com/paper/2210.15179/) (2022)
- [Graphon Mean Field Games and the GMFG Equations](https://mlsanity.com/paper/2008.10216/) (2020)
- [DeepSets and their derivative networks for solving symmetric PDEs](https://mlsanity.com/paper/2103.00838/) (2021)
- [A fast iterative PDE-based algorithm for feedback controls of nonsmooth mean-field control problems](https://mlsanity.com/paper/2108.06740/) (2021)
- [Mean-field neural networks-based algorithms for McKean-Vlasov control problems *](https://mlsanity.com/paper/2212.11518/) (2022)
- [Ito's formula for flows of measures on semimartingales](https://mlsanity.com/paper/2010.05288/) (2020)
- [A label-state formulation of stochastic graphon games and approximate equilibria on large networks](https://mlsanity.com/paper/2204.09642/) (2022)
- [P1-KAN: an effective Kolmogorov-Arnold network with application to hydraulic valley optimization](https://mlsanity.com/paper/2410.03801/) (2024)
- [Probabilistic Analysis of Graphon Mean Field Control](https://mlsanity.com/paper/2505.19664/) (2025)
- [Non-Exchangeable Mean Field Markov Decision Processes with common noise : from Bellman equation to quantitative propagation of chaos](https://mlsanity.com/paper/2603.00009/) (2026)
- [Mean-field control of non exchangeable systems](https://mlsanity.com/paper/2407.18635/) (2024)
- [Linear-quadratic optimal control for non-exchangeable mean-field SDEs and applications to systemic risk](https://mlsanity.com/paper/2503.03318/) (2025)
- [Contracting a crowd of heterogeneous agents](https://mlsanity.com/paper/2507.09415/) (2025)
- [Optimal Control of Heterogeneous Mean-Field Stochastic Differential Equations with Common Noise and Applications](https://mlsanity.com/paper/2511.18636/) (2025)

## Built on by (most cited first, 3 of 3)

- [Actor-Critic Learning for Extended Mean Field Control with Deterministic Policies](https://mlsanity.com/paper/2607.11005/) (2026)
- [Universal Approximation of Nonlinear Operators and Their Derivatives](https://mlsanity.com/paper/2605.15285/) (2026)
- [Randomized Transport Maps for Model-Free Policy-Gradient Mean-Field Control](https://mlsanity.com/paper/2610.11619/) (2026)

## Abstract (arXiv)

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.