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).

翻译：深度算子网络（DeepONet）框架因其求解高维偏微分方程的能力而得到广泛应用。本文将该框架与近期发展的策略迭代方案相结合，用于数值求解最优控制问题及其对应的Hamilton-Jacobi-Bellman（HJB）方程。本方法的一个显著特点是：得益于算子学习的独特性质，一旦神经网络训练完成，即可快速推演出具有不同终端函数的最优控制问题及HJB方程的解。此外，通过粘性解的比较原理对算法精度进行了定量分析。通过包括10维线性二次调节器问题（LQRs）在内的多个算例验证了该方法的有效性。