The approximation of solutions to second order Hamilton--Jacobi--Bellman (HJB) equations by deep neural networks is investigated. It is shown that for HJB equations that arise in the context of the optimal control of certain Markov processes the solution can be approximated by deep neural networks without incurring the curse of dimension. The dynamics is assumed to depend affinely on the controls and the cost depends quadratically on the controls. The admissible controls take values in a bounded set.
翻译:调查了深神经网络对二级汉密尔顿-Jacobi-Bellman(HJB)等式(HJB)的近似解决办法,发现对于某些Markov进程的最佳控制背景下产生的HJB等式而言,深神经网络可以近似解决办法,而不会引起尺寸的诅咒,假设动态会密切依赖控制,成本则取决于控制。可接受控制措施的数值是捆绑的。