Deep random difference method for high-dimensional quasilinear parabolic partial differential equations

TL;DR

This paper introduces a deep random difference method (DRDM) for efficiently solving high-dimensional quasilinear parabolic PDEs and HJB equations by avoiding derivative computations and enabling parallel processing, with proven error bounds and strong numerical results.

Contribution

The paper proposes a novel DRDM that approximates PDE operators using first-order differences, extending to HJB equations without stochastic calculus, and provides rigorous error estimates.

Findings

Efficiently solves PDEs in dimensions up to 10^5.

Achieves first-order accuracy in time step h.

Demonstrates high accuracy and computational efficiency in numerical experiments.

Abstract

Solving high-dimensional parabolic partial differential equations (PDEs) with deep learning methods is often computationally and memory intensive, primarily due to the need for automatic differentiation (AD) to compute large Hessian matrices in the PDE. In this work, we propose a deep random difference method (DRDM) that addresses these issues by approximating the convection-diffusion operator using only first-order differences and the solution by deep neural networks, thus avoiding Hessian and other derivative computations. The DRDM is implemented within a Galerkin framework to reduce sampling variance, and the solution space is explored using stochastic differential equations (SDEs) to capture the dynamics of the convection-diffusion operator. The approach is then extended to solve Hamilton-Jacobi-Bellman (HJB) equations, which recovers existing martingale deep learning methods for PDEs [{\it SIAM J. Sci. Comput.}, 47 (2025), pp. C795-C819], without using stochastic calculus. The proposed method offers two main advantages: it avoids the need to compute derivatives in PDEs and enables parallel computation of the loss function in both time and space. Moreover, a rigorous error estimate is proven for the quasi-linear parabolic equation, showing first-order accuracy in $h$, the time step used in the discretization of the SDE paths by the Euler-Maruyama scheme. Numerical experiments demonstrate that the method can efficiently and accurately solve quasilinear parabolic PDEs and HJB equations in dimensions up to $10^5$ and $10^4$, respectively.