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

Samy Mekkaoui, Huyên Pham, Xavier Warin

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.

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

Abstract

We study the approximation of operators acting on probability measures on a product space with prescribed marginal. Let be a label space endowed with a reference measure , and define as the set of probability measures on with first marginal . By disintegration, elements of 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 . 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 , 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.
Paper Structure (27 sections, 5 theorems, 104 equations, 33 figures, 9 tables, 1 algorithm)

This paper contains 27 sections, 5 theorems, 104 equations, 33 figures, 9 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Operator learning of conditional mean field functionals
  3. Neural network approximation
  4. Proof of the universal approximation theorem
  5. Data generation
  6. Training the mean-field operator
  7. Numerical experiments
  8. Application to optimal control of non exchangeable mean field systems
  9. Background on controlled non exchangeable mean field systems
  10. Stochastic maximum principle and FBSDE equation
  11. Dynamic programming and HJB equation.
  12. The class of linear quadratic control problem
  13. Algorithms
  14. Riccati solver
  15. Deep Graphon
  16. ...and 12 more sections

Key Result

Theorem 2.1

Let $\rho$ be a probability measure on the Borel space ${\cal M}_{\lambda}$ and $V$ a continuous map from $I \times \mathbb{R}^d \times {\cal M}_{\lambda}$ into $\mathbb{R}^q$ such that $\lVert V \rVert^2_{L^2(\rho)} := \int_{{\cal M}_{\lambda}} |V(\cdot,\cdot,\mu)|^2_{\mu} \rho(\mathrm{d} \mu) < \i where $\Phi_J(\mu) := ( \langle \varphi_1,\mu \rangle, \ldots, \langle \varphi_J,\mu \rangle ) \in

Figures (33)

  • Figure : T1
  • Figure : T1
  • Figure : V1-T1
  • Figure : T1
  • Figure : T1
  • ...and 28 more figures

Theorems & Definitions (15)

  • Theorem 2.1
  • Remark 2.2
  • Lemma 2.3
  • Remark 2.4
  • Theorem 4.2
  • Remark 4.4
  • Proposition 4.6
  • Remark 4.7
  • Remark 4.8
  • Theorem 4.9
  • ...and 5 more