시간: 12:30~13:30

Neural Policy Iteration for High-Dimensional Hamilton--Jacobi--Bellman Equations: Monotone Schemes, Operator Learning, and Error Estimates

Policy iteration replaces a nonlinear Hamilton–Jacobi–Bellman (HJB) equation by a sequence of linear policy-evaluation problems followed by pointwise greedy updates. In high dimension, the evaluation step is naturally carried out by neural networks or neural operators, trained on a bounded box and typically without boundary data. This talk asks what such inexact evaluations guarantee for the iteration as a whole. For deterministic control, where the value function is only Lipschitz, a monotone semi-discrete scheme restores the comparison principle and makes policy improvement well defined; combined with a physics-informed DeepONet, one training covers a family of terminal costs. We then show how the missing boundary data are controlled by the equation itself: through elliptic regularity and exponential interior attenuation for second-order equations, and through a Markov-chain structure of the monotone stencil for first-order equations. Both yield closed error estimates for inexact policy iteration, which numerical experiments measure and test for sharpness. Based on joint work with Jae Yong Lee, Yeongjong Kim, Minseok Kim, and Namkyeong Cho.