Deep-Control BSDE：用于高维半线性偏微分方程的分层布朗加权回归
Deep-Control BSDE: Layerwise Brownian-Weighted Regression for High-Dimensional Semilinear PDEs
AI总结：
针对高维半线性偏微分方程的维度灾难问题，提出Deep-Control BSDE方法，通过分层布朗加权回归实现价值与控制过程的可靠联合近似，在多基准测试中表现出良好平衡与稳定性。
AI中文摘要：
高维半线性抛物型偏微分方程出现于随机控制、金融工程与不确定性量化领域，但经典空间离散化方法会遭遇维度灾难。受去噪分数匹配中的高斯扰动与条件回归启发，我们提出Deep-Control BSDE（\textbf{DCBSDE}），这是一种针对马尔可夫型倒向随机微分方程的分层控制回归方法。从隐式欧拉格式出发，我们将离散理想控制$z_n^\boldsymbol{\u03C0}=\boldsymbol{\u004D}_n u_{n+1}^\boldsymbol{\u03C0}$识别为后继价值响应对一步布朗增量的条件投影系数。在每个时间层，该方法冻结后继价值函数，利用有限分支近似所得的布朗条件矩目标，回归控制，再通过隐式BSDE关系拟合价值。冻结线性响应基线与对偶配对可减少有限分支波动，同时保留条件目标，层内修正则协调价值与控制近似。我们通过布朗投影的压缩性质建立后向稳定性，并推导了当离散化、局部学习、数值及有限分支误差共同消失时的条件一致性。在六个基准测试上的实验表明，\textbf{DCBSDE}在价值精度、控制精度与动态一致性之间取得了良好的整体平衡，且在不同随机种子下表现普遍稳定。这些结果支持将\textbf{DCBSDE}应用于需要可靠联合近似价值与控制过程的场景。
英文摘要：
High-dimensional semilinear parabolic partial differential equations arise in stochastic control, financial engineering, and uncertainty quantification, but classical spatial discretizations suffer from the curse of dimensionality. Motivated by Gaussian perturbation and conditional regression in denoising score matching, we propose Deep-Control BSDE (\DCBSDE{}), a layerwise control-regression method for Markovian backward stochastic differential equations. From an implicit Euler scheme, we identify the discrete ideal control $z_n^π=\mathcal M_nu_{n+1}^π$ as the conditional projection coefficient of the successor value response onto the one-step Brownian increment. At each time level, the method freezes the successor value function, approximates the resulting Brownian conditional-moment target using finitely many branches, regresses the control, and then fits the value through the implicit BSDE relation. A frozen linear-response baseline and antithetic pairing reduce finite-branch fluctuations while preserving the conditional target, and a within-layer correction coordinates the value and control approximations. We establish backward stability through a contraction property of the Brownian projection and derive conditional consistency when discretization, local learning, numerical, and finite-branch errors vanish jointly. Experiments across six benchmarks demonstrate that \DCBSDE{} achieves a favorable overall balance among value accuracy, control accuracy, and dynamic consistency while exhibiting generally stable performance across random seeds. These results support the use of \DCBSDE{} in applications requiring reliable joint approximation of the value and control processes.
