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

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 explicit Hessian computation. When incorporated into a Galerkin framework, the DRDM eliminates the need for pointwise evaluation of expectations, resulting in efficient implementation. We further extend the approach to Hamilton-Jacobi-Bellman (HJB) equations. Notably, the DRDM recovers existing martingale deep learning methods for PDEs (Cai et al., 2024, arXiv:2405.03169), without using the tools of stochastic calculus. The proposed method offers two main advantages: it removes the dependence on AD for PDE derivatives and enables parallel computation of the loss function in both time and space. We provide rigorous error estimates for the DRDM in the linear case, which shows a first order accuracy in $\Delta t$ used in the sampling of the 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 $10^4$ and $10^5$, respectively.