Actor-Critic Method for High Dimensional Static Hamilton–Jacobi–Bellman Partial Differential Equations based on Neural Networks
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We propose a novel numerical method for high dimensional Hamilton–Jacobi–Bellman (HJB) type elliptic partial differential equations (PDEs). The HJB PDEs, reformulated as optimal control problems, are tackled by the actor-critic framework inspired by reinforcement learning, based on neural network parametrization of the value and control functions. Within the actor-critic framework, we employ a policy gradient approach to improve the control, while for the value function, we derive a variance reduced least square temporal difference method (VR-LSTD) using stochastic calculus. To numerically discretize the stochastic control problem, we employ an adaptive stepsize scheme to improve the accuracy near the domain boundary. Numerical examples up to 20 spatial dimensions including the linear quadratic regulators, the stochastic Van der Pol oscillators, and the diffusive Eikonal equations are presented to validate the effectiveness of our proposed method.
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