Deep neural network approximation for high-dimensional parabolic Hamilton-Jacobi-Bellman equations
This addresses the challenge of solving high-dimensional HJB equations in optimal control, which is important for applications in finance and engineering, but the approach is incremental as it builds on existing neural network methods with specific assumptions.
The paper tackles the problem of approximating solutions to high-dimensional parabolic Hamilton-Jacobi-Bellman (HJB) equations using deep neural networks, showing that this can be done without the curse of dimension for certain optimal control problems with affine dynamics and quadratic costs.
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.