Gaussian process policy iteration with additive Schwarz acceleration for forward and inverse HJB and mean field game problems

We propose a Gaussian Process (GP)-based policy iteration framework for addressing both forward and inverse problems in Hamilton--Jacobi--Bellman (HJB) equations and mean field games (MFGs). Policy iteration is formulated as an alternating procedure between solving the value function under a fixed control policy and updating the policy based on the resulting value function. By exploiting the linear structure of GPs for function approximation, each policy evaluation step admits an explicit closed-form solution, eliminating the need for numerical optimization. To improve convergence, we incorporate the additive Schwarz acceleration as a preconditioning step following each policy update. Numerical experiments demonstrate the effectiveness of Schwarz acceleration in improving computational efficiency.