Deep neural network approximation for high-dimensional parabolic Hamilton-Jacobi-Bellman equations

TL;DR

This paper demonstrates that deep neural networks can effectively approximate solutions to high-dimensional Hamilton-Jacobi-Bellman equations in optimal control problems, overcoming the curse of dimensionality.

Contribution

It provides a theoretical foundation showing neural networks can approximate HJB solutions in high dimensions without the curse of dimensionality, under specific control assumptions.

Findings

Neural networks can approximate HJB solutions in high dimensions.

The approach avoids the curse of dimensionality in certain control problems.

Applicable to HJB equations with affine dynamics and quadratic costs.

Abstract

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.