Mouhcine Assouli (XLIM - Universitè de Limoges) - Initialization-driven neural generation and training for high-dimensional optimal control and first order mean field games

mercredi 29 avr. 2026, 10:00 → 11:00 Europe/Paris

We introduce a neural-network-based method for approximating the value function of high-dimensional deterministic optimal control problems. The proposed approach exploits the relationship between Pontryagin's Maximum Principle (PMP) and the value function, which is characterized as the unique viscosity solution of the Hamilton–Jacobi–Bellman (HJB) equation. A neural network is first trained to obtain a coarse approximation of the value function by minimizing the residual of the HJB equation. The gradient of this approximation is then used to initialize the numerical solution of the two-point boundary value problem arising from PMP, enabling the generation of reliable optimal trajectories, adjoint states, and costs. This dataset is then used to further train the neural network through a loss function enforcing the HJB equation.

We next address the computation of equilibria in first-order Mean Field Game (MFG) problems by integrating the proposed methodology with the fictitious play algorithm. Such equilibria are characterized by a coupled system consisting of an HJB equation and a continuity equation. To approximate the solution of the continuity equation, we introduce a second neural network that learns the flow map transporting the initial agent distribution along optimal trajectories. This network is trained using data obtained by solving the ordinary differential equations (ODEs) associated with the controlled dynamics. The iterative procedure is initialized via joint training of the value function and the flow map, based on a composite loss involving both HJB and ODE residuals. Several numerical experiments demonstrate the accuracy and robustness of the proposed method.