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We propose a novel numerical method for high dimensionalHamilton--Jacobi--Bellman (HJB) type elliptic partial differential equations(PDEs). The HJB PDEs, reformulated as optimal control problems, are tackled bythe actor-critic framework inspired by reinforcement learning, based on neuralnetwork parametrization of the value and control functions. Within theactor-critic framework, we employ a policy gradient approach to improve thecontrol, while for the value function, we derive a variance reducedleast-squares temporal difference method using stochastic calculus. Tonumerically discretize the stochastic control problem, we employ an adaptivestep size scheme to improve the accuracy near the domain boundary. Numericalexamples up to $20$ spatial dimensions including the linear quadraticregulators, the stochastic Van der Pol oscillators, the diffusive Eikonalequations, and fully nonlinear elliptic PDEs derived from a regulator problemare presented to validate the effectiveness of our proposed method.

Actor-Critic Method for High Dimensional Static Hamilton--Jacobi--Bellman Partial Differential Equations based on Neural Networks