A neural network based policy iteration algorithm with global H²-superlinear convergence for stochastic games on domains

Author: 

REISINGER, C
Zhang, Y
Ito, K

Publication Date: 

18 May 2020

Journal: 

Foundations of Computational Mathematics

Last Updated: 

2021-07-26T14:42:55.34+01:00

Volume: 

N/A

DOI: 

10.1007/s10208-020-09460-1

page: 

N/A-N/A

abstract: 

In this work, we propose a class of numerical schemes for solving semilinear Hamilton–Jacobi–Bellman–Isaacs (HJBI) boundary value problems which arise naturally from exit time problems of diffusion processes with controlled drift. We exploit policy iteration to reduce the semilinear problem into a sequence of linear Dirichlet problems, which are subsequently approximated by a multilayer feedforward neural network ansatz. We establish that the numerical solutions converge globally in the 𝐻2-norm and further demonstrate that this convergence is superlinear, by interpreting the algorithm as an inexact Newton iteration for the HJBI equation. Moreover, we construct the optimal feedback controls from the numerical value functions and deduce convergence. The numerical schemes and convergence results are then extended to oblique derivative boundary conditions. Numerical experiments on the stochastic Zermelo navigation problem are presented to illustrate the theoretical results and to demonstrate the effectiveness of the method.

Symplectic id: 

1097619

Submitted to ORA: 

Submitted

Publication Type: 

Journal Article