Final Oral Public Examination

Mean Field Control on Spaces of Probability Measures: Theory, Computation, and Applications

Advisor: Mete Soner

Abstract:

This thesis studies mean field control (MFC) on spaces of probability measures as a unified framework for analysis, computation, and applications. In control problems with a large population of interacting agents, the underlying particle system can often be approximated by the evolution of a representative particle together with the distribution of the population. The resulting optimization problem is known as a mean field control problem. Its associated value function is naturally defined on a space of probability measures, which leads to Hamilton–Jacobi–Bellman equations in Wasserstein space and raises fundamental questions of well-posedness, approximation, and computation.

On the theoretical side, this thesis studies viscosity solutions of the Hamilton-Jacobi-Bellman equations associated with mean field control problems, providing a characterization of value functions and uniqueness through comparison principles on spaces of probability measures. On the computational side, it develops neural-network-based numerical methods for mean field control by combining particle approximations with feedback control parameterizations adapted to the exchangeable structure of the system. The analysis establishes convergence and approximation results, including a universal approximation theorem on Wasserstein spaces.

This thesis further demonstrates the relevance of this framework to contemporary machine learning. In one direction, the training dynamics of wide neural networks can be interpreted as gradient flows on probability measure spaces and connected to mean field control and Hamilton–Jacobi–Bellman theory, yielding a mathematical perspective on implicit regularization and early stopping. In another direction, a soft-constrained Schrödinger bridge problem is formulated as an unconstrained mean field stochastic control problem. The thesis establishes well-posedness of the associated forward-backward system, proves approximation results based on empirical targets and finite-particle systems, and develops neural numerical schemes for generative modeling tasks.

Overall, this thesis presents mean field control on spaces of probability measures as a common mathematical language linking viscosity solutions, stochastic control, numerical algorithms, and applications in learning and generative AI.