Hamilton-Jacobi Based Policy-Iteration via Deep Operator Learning
This work addresses computational challenges in high-dimensional control for researchers and engineers, but it is incremental as it combines existing methods (DeepONet and policy iteration) rather than introducing a new paradigm.
The authors tackled the numerical solution of high-dimensional optimal control problems and Hamilton-Jacobi-Bellman equations by integrating DeepONet with a policy iteration scheme, achieving efficient inference for different terminal functions and verifying effectiveness on examples like 10-dimensional linear quadratic regulators.
The framework of deep operator network (DeepONet) has been widely exploited thanks to its capability of solving high dimensional partial differential equations. In this paper, we incorporate DeepONet with a recently developed policy iteration scheme to numerically solve optimal control problems and the corresponding Hamilton--Jacobi--Bellman (HJB) equations. A notable feature of our approach is that once the neural network is trained, the solution to the optimal control problem and HJB equations with different terminal functions can be inferred quickly thanks to the unique feature of operator learning. Furthermore, a quantitative analysis of the accuracy of the algorithm is carried out via comparison principles of viscosity solutions. The effectiveness of the method is verified with various examples, including 10-dimensional linear quadratic regulator problems (LQRs).