Solving nonconvex Hamilton--Jacobi--Isaacs equations with PINN-based policy iteration
| Source: arXiv AI
Tags: PINN, differential games, robust control, Hamilton-Jacobi, numerical methods
A mesh-free framework combining PINNs with policy iteration solves high-dimensional nonconvex Hamilton-Jacobi-Isaacs equations with proven convergence guarantees, achieving relative L2 errors below 1% on 2D benchmarks and outperforming direct PINN solvers in 5D and 10D game problems.
Details
Hamilton-Jacobi-Isaacs (HJI) equations govern optimal strategies in two-player adversarial differential games and robust control problems. They are notoriously difficult to solve numerically in high dimensions: conventional finite-difference grid methods hit the curse of dimensionality, and existing PINN approaches often lack convergence guarantees for nonconvex cases.\n\nThe proposed method alternates between two steps: (1) solving a linearized PDE under fixed control policies using PINNs, and (2) updating the controls via pointwise minimax optimization through automatic differentiation. Under standard Lipschitz and uniform ellipticity assumptions, the authors prove convergence of the value function iterates to the unique viscosity solution — without requiring convexity of the Hamiltonian.\n\nNumerically, the method achieves relative L2 errors below 1% on a 2D stochastic path-planning game with a moving obstacle, matching finite-difference benchmarks. In 5D and 10D publisher-subscriber games with anisotropic noise, the PINN-policy-iteration approach produces smoother value functions and lower residuals than direct PINN solvers.