Approximation of optimal feedback controls for stochastic reaction-diffusion equations

In this paper we present a method to approximate optimal feedback controls for stochastic reaction-diffusion equations. We derive two approximation results providing the theoretical foundation of our approach and allowing for explicit error estimates. The approximation of optimal feedback controls by neural networks is discussed as an explicit application of our method. We illustrate our findings in the case of a linear quadratic control problem with a numerical example.