Martingale deep learning for very high dimensional quasi-linear partial differential equations and stochastic optimal controls

TL;DR

This paper introduces a parallel, derivative-free martingale neural network approach for solving high-dimensional quasi-linear PDEs and stochastic control problems, avoiding gradient computations and enabling efficient, accurate solutions in very high dimensions.

Contribution

The paper presents a novel martingale neural network method that reformulates PDEs into a martingale problem, eliminating the need for gradient or Hessian calculations and allowing highly parallelized solutions.

Findings

Successfully solves PDEs in up to 10,000 dimensions.

Achieves accurate solutions efficiently without gradient computations.

Demonstrates effectiveness for stochastic optimal control problems.

Abstract

In this paper, a highly parallel and derivative-free martingale neural network learning method is proposed to solve Hamilton-Jacobi-Bellman (HJB) equations arising from stochastic optimal control problems (SOCPs), as well as general quasilinear parabolic partial differential equations (PDEs). In both cases, the PDEs are reformulated into a martingale formulation such that loss functions will not require the computation of the gradient or Hessian matrix of the PDE solution, while its implementation can be parallelized in both time and spatial domains. Moreover, the martingale conditions for the PDEs are enforced using a Galerkin method in conjunction with adversarial learning techniques, eliminating the need for direct computation of the conditional expectations associated with the martingale property. For SOCPs, a derivative-free implementation of the maximum principle for optimal controls is also introduced. The numerical results demonstrate the effectiveness and efficiency of the proposed method, which is capable of solving HJB and quasilinear parabolic PDEs accurately in dimensions as high as 10,000.