Trajectory-Informed versus Physics-Informed Machine Learning Methods for Dynamic Zero-Sum Games

In this paper, we introduce a trajectory-informed, machine-learning framework designed to address two-player zero-sum games for uncertain nonlinear systems. Our approach approximates the optimal value function that solves the Hamilton-Jacobi-Isaacs (HJI) equation within the context of infinite-horizon zero-sum games. We employ trajectory-informed machine learning, inspired by the principles of physics-informed neural networks (PINNs). Remarkably, our method does not require explicit knowledge of the drift term in the system dynamics. Additionally, it provides guarantees for a unique solution for the finite-horizon zero-sum game variant, which PINNs cannot theoretically guarantee. We offer rigorous mathematical justification, demonstrating uniform convergence and satisfactory approximation of the saddle point policies for sufficiently large time horizons. This holds true regardless of whether the system dynamics are fully known or only partially known. Our proposed approach is validated through simulations, comparing scenarios where the system dynamics are fully known with those where the saddle point policies are learned through interaction with the system.