Research

We study numerical methods for solving mean-field optimal control problems by leveraging machine learning techniques. A convergence proof based on propagation of chaos and Rademacher complexity is provided. We also discuss a universal approximation theorem for neural networks on Wasserstein spaces.

We study Markov perfect equilibria in continuous-time dynamic games with finitely many symmetric players. We prove the convergence of both Picard and weighted Picard iterations, yielding efficient computational methods. Numerical experiments confirm the effectiveness of algorithms based on this approach.

We prove comparison principles for viscosity solutions of Hamilton–Jacobi equations on the Wasserstein space.

We prove comparison principles for viscosity solutions of Hamilton–Jacobi equations on the Wasserstein space.

We propose a policy gradient method for entropy-regularized mean-field control in the discounted infinite-horizon setting. We consider randomized feedback policies and a coupled representative-particle/population system, in which the representative state evolves jointly with a population law governed by a McKean–Vlasov equation.

We provide a theoretical formulation to explain the generalization property of overparameterized neural networks through implicit regularization of the training process. We consider the mean-field formulation of wide neural networks, and propose an equivalent mean-field control formulation for the loss function and the corresponding gradient flows.

We investigate particle systems with singular local interactions through hitting times on sparse graphs, motivated by problems from systemic risk in financial networks.