A brief review of the Deep BSDE method for solving high-dimensional partial differential equations
Jiequn Han, Arnulf Jentzen, Weinan E
TL;DR
The reviewed work addresses the challenge of solving high-dimensional PDEs by leveraging the Deep BSDE method, which reformulates a semilinear parabolic PDE for $u(t,x)$ as a forward SDE for $X_t$ together with a backward component $(Y_t,Z_t)$ governed by a BSDE. By approximating $u(0,X_0)$ and $\nabla_x u$ with neural networks and discretizing time via an Euler scheme, the method minimizes the loss $\mathbb{E}[|g(X_T) - Y_T|^2]$ through SGD, enabling solutions in dimensions far beyond traditional approaches. The paper also surveys a spectrum of subsequent developments—BSDE-based variants, least-squares / PINN approaches, Ritz and Galerkin formulations, and related theory—outlining a broad, rapidly evolving landscape for high-dimensional PDEs. These advances hold potential for substantial cross-disciplinary impact, including optimal control, probabilistic modeling, quantum mechanics, and plasma physics, while highlighting fundamental open questions in convergence and generalization theory for deep-learning PDE solvers.
Abstract
High-dimensional partial differential equations (PDEs) pose significant challenges for numerical computation due to the curse of dimensionality, which limits the applicability of traditional mesh-based methods. Since 2017, the Deep BSDE method has introduced deep learning techniques that enable the effective solution of nonlinear PDEs in very high dimensions. This innovation has sparked considerable interest in using neural networks for high-dimensional PDEs, making it an active area of research. In this short review, we briefly sketch the Deep BSDE method, its subsequent developments, and future directions for the field.