Seminar| Institute of Mathematical Sciences
Time: Tuesday, November 11th, 2025,16:00-17:00
Location: Online talk (Tencent Meeting Number: 198 379 314)
Speaker: Shuixin Fang, Chinese Academy of Sciences
Abstract: Solving high-dimensional parabolic partial differential equations (PDEs) with deep learning methods is often computation 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 computation. 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 [Cai et al., 2025, SIAM J. Sci. Comput.], without using stochastic calculus. The proposed method offers two main advantages: it avoids the need for computing derivatives in PDEs and enables parallel computation of the loss function in both time and space. Moreover, rigorous error estimates for the DRDM are proven for the linear drift-diffusion equation, which shows a first order accuracy in $h$, the time step used in the discretization of the paths of the SDEs 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 100,000 and 10,000, respectively.