发表机构
Research Institute of Intelligent Complex Systems, Fudan University; Department of Mathematics, Friedrich-Alexander-Universität Erlangen–Nürnberg（复旦大学智能复杂系统研究院; 埃尔朗根-纽伦堡大学数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种深度截断FBSDE方法，通过梯度截断迭代解耦与虚构博弈平均，结合路径一致性项，实现对高维非线性PDE和全耦合FBSDE的稳定求解，并在数值实验中验证了其高精度与稳健性。
AI 中文摘要
本文提出了一种用于高维偏微分方程（PDE）的深度截断正倒向随机微分方程（FBSDE）方法。与现有的用于全耦合FBSDE的深度学习求解器相比，这些求解器中强耦合可能导致数值不稳定，我们的方法表现出改进的稳定性。所提出的方法将梯度截断迭代解耦与虚构博弈平均相结合，以在耦合框架中分离正向和反向过程。这保留了耦合动力学，同时减少了优化过程中由参数依赖的正向路径引起的不稳定反馈。此外，我们引入路径一致性项以创建显式局部梯度捷径，从而提供一种可能缓解梯度消失的结构性机制。我们还推导了基于残差的误差估计，并在适当条件下建立了全离散数值逼近的条件收敛性，其中路径一致性损失不是必需的。我们的方法在对流主导方程中特别有效，其中耦合公式为非线性输运提供了稳定的表示，而无需在BSDE中引入奇异项。数值实验表明，在低维和高维问题中，准确性和稳定性均有提高，并且对强耦合问题具有稳健性能。
英文摘要
In this paper, we introduce a deep truncated forward-backward stochastic differential equation (FBSDE) method for high-dimensional partial differential equations (PDEs). Compared with existing deep-learning solvers for fully coupled FBSDEs, where strong coupling may lead to numerical instability, our approach exhibits improved stability. The proposed method combines gradient-truncated iterative decoupling with fictitious-play averaging to separate the forward and backward processes in a coupled framework. This preserves the coupled dynamics while reducing the unstable feedback induced by parameter-dependent forward paths during optimization. Furthermore, we incorporate a pathwise consistency term to create explicit local gradient shortcuts, thereby providing a structural mechanism that may mitigate gradient vanishing. We also derive a residual-based error estimate and establish conditional convergence of the fully discrete numerical approximations under suitable conditions, in which the pathwise consistency loss is not required. Our approach is particularly effective for convection-dominated equations, where the coupled formulation provides a stable representation of nonlinear transport without introducing singular terms into the BSDE. Numerical experiments demonstrate improved accuracy and stability in both low- and high-dimensional problems and robust performance for strongly coupled problems.
Comments40 pages, 9 figures, 7 tables
