Physics-Informed Optimal Control by Control-Only Supervision with Error Guarantees on Value and Policy

We propose a physics-informed method for solving high-dimensional optimal control problems that requires only control labels, not value function labels. A neural network parameterizes the value function, and the control policy is derived from its gradient. The network is trained by minimizing a weighted sum of the standard Hamilton–Jacobi–Bellman (HJB) residual and the mismatch between the network’s control output and control labels, which are precomputed by a state-dependent Riccati equation (SDRE) solver. Theoretically, with the help of the viscous HJB equation, we establish error bounds for the value function and its gradient, providing a rigorous performance guarantee for the resulting policy. Numerical experiments show that the learned model accurately approximates the optimal value function, closely matches the cost-to-go of the SDRE reference, and yields a neural controller that is significantly faster while exhibiting excellent stability.