Neural Network Approximation of Optimal Controls for Stochastic Reaction-Diffusion Equations

TL;DR

This paper introduces a neural network-based numerical algorithm for approximating optimal controls in stochastic reaction-diffusion equations, reducing computational complexity and applicable to high-dimensional stochastic PDEs.

Contribution

The paper develops a novel neural network approach for control approximation in stochastic PDEs, providing error rates and demonstrating efficiency improvements.

Findings

Algorithm reduces computational complexity significantly.

Neural networks effectively approximate feedback controls.

Method applicable to high-dimensional stochastic PDEs.

Abstract

We present a numerical algorithm that allows the approximation of optimal controls for stochastic reaction-diffusion equations with additive noise by first reducing the problem to controls of feedback form and then approximating the feedback function using finitely based approximations. Using structural assumptions on the finitely based approximations, rates for the approximation error of the cost can be obtained. Our algorithm significantly reduces the computational complexity of finding controls with asymptotically optimal cost. Numerical experiments using artificial neural networks as well as radial basis function networks illustrate the performance of our algorithm. Our approach can also be applied to stochastic control problems for high dimensional stochastic differential equations and more general stochastic partial differential equations.