Semilinear parabolic partial differential equations (PDEs) are fundamental to modeling complex dynamical systems across scientific domains. The Deep Backward Stochastic Differential Equation (BSDE) method is a promising approach for high-dimensional PDEs; however, existing convergence results apply only to globally Lipschitz generators, excluding important cases such as Allen--Cahn and Hamilton--Jacobi--Bellman (HJB) equations. This paper presents both a theoretical and a computational advance for Deep BSDE methods. Theoretically, we establish the convergence theory for non--Lipschitz generators--covering Allen--Cahn equations with cubic nonlinearity and HJB equations with quadratic gradient growth--based on a bounded double--well lemma and a truncated-BSDE analysis within the Bouchard--Touzi--Zhang theory. Computationally, we instantiate the framework with XNet, a shallow architecture with $\mathcal O(L)$ parameters that preserves strong approximation while substantially reducing optimization and computational cost. Numerical experiments on 100--dimensional PDEs corroborate the predicted convergence behavior and demonstrate significant efficiency gains over standard feedforward implementations.

翻译：半线性抛物型偏微分方程是跨科学领域建模复杂动力系统的基础。深度倒向随机微分方程方法为高维偏微分方程提供了有前景的求解途径；然而，现有的收敛性结果仅适用于全局Lipschitz生成元，排除了Allen-Cahn方程和Hamilton-Jacobi-Bellman方程等重要情形。本文在理论计算两方面推进了深度BSDE方法的发展。理论上，基于有界双阱引理和Bouchard-Touzi-Zhang理论框架下的截断BSDE分析，我们建立了涵盖三次非线性Allen-Cahn方程和二次梯度增长HJB方程等非Lipschitz生成元的收敛理论。计算上，我们采用XNet实例化该框架——这是一种具有$\mathcal O(L)$个参数的浅层架构，在保持强逼近能力的同时显著降低了优化与计算成本。针对100维偏微分方程的数值实验验证了预期的收敛行为，并表明相较于标准前馈实现方法具有显著的效率优势。