Physics-Informed Policy Iteration for High-Dimensional Hamilton--Jacobi--Bellman Equations: Interior Error Bounds without Boundary Data

Abstract

We develop a physics-informed policy-iteration method for stationary second-order Hamilton--Jacobi--Bellman equations arising in continuous-time stochastic control. Each policy-evaluation step is a linear elliptic PDE and is approximated by a mesh-free neural residual solver; policy improvement is then performed pointwise from the surrogate gradient. The analysis addresses bounded-domain training without prescribed boundary data. We prove well-posedness of the PDE-defined evaluation for every Borel Markov policy, establish Lipschitz stability of the greedy map on bounded gradient ranges, and derive an exponential attenuation estimate for unresolved boundary information. These ingredients yield a closed finite-step interior error bound whose floors are determined by a continuous $L^p$ residual, with a finite exponent $p>d$, and an attenuated amplitude term. The experiments measure the quantities in the estimate. On a linear--quadratic testbed with an exact reference, the gradient-error floor scales nearly linearly with a fresh-sample $L^p$ residual estimate, and finite-difference probes identify when a training buffer is beneficial. At matched architecture and budget, linear fixed-policy training becomes markedly more reliable than direct minimization of the nonlinear HJB residual as the tested problems become more difficult. The method also produces effective feedback on an inverted pendulum, a planar quadrotor, and a 100-dimensional posterior-seeking problem. These nonlinear tests expose two limitations not visible from sampled residuals alone: the collocation distribution may miss the region visited by the learned closed loop, and evaluation without boundary data may leave undetermined the gradient component on which the greedy update depends. On-policy collocation, rollout-anchored evaluation, and rollout-based stopping provide effective model-only safeguards.