AI 中文总结
研究针对高维基于鞅的PDE，提出iSMART迭代采样-回归技术，通过停止梯度等方法规避对抗优化，引入冻结-补偿技术优化HJB方程收敛性，经多类高维数值实验验证其准确性、效率与鲁棒性。
AI 中文摘要
本文提出了用于求解高维基于鞅的偏微分方程（PDE）的迭代采样-回归技术（iSMART）。利用条件期望的L²投影性质并采用停止梯度技术，iSMART将从PDE导出的连续鞅条件重新表述为迭代框架内一系列易处理的采样-回归问题。该方法仅依赖标准随机微分方程（SDE）路径模拟和普通平方误差损失最小化，完全规避了以往方法中对抗优化或嵌套期望估计的需求。iSMART可在统一迭代流程中处理线性、半线性及完全非线性的基于鞅的PDE。特别地，针对完全非线性的哈密尔顿-雅可比-贝尔曼（HJB）方程，引入冻结-补偿技术以将部分非线性项策略性转移至SDE漂移项，从而提升迭代的收敛性能。针对具有陡峭梯度的线性反应-扩散方程、半线性伯格斯型方程及完全非线性HJB方程开展的大量数值实验，验证了所提方法在不同高维场景下的准确性、效率与鲁棒性。
英文摘要
We propose the {\bf i}terative {\bf S}a{\bf M}pling-{\bf A}nd-{\bf R}egression {\bf T}echnique (iSMART) for high-dimensional martingale-based partial differential equations (PDEs) in this paper. By leveraging the $L^2$-projection property of conditional expectation and adopting the stop-gradient technique, iSMART reformulates the continuous martingale condition derived from PDEs into a sequence of tractable sampling-regression problems within an iterative framework. This approach relies solely on standard SDE path simulation and plain squared-error loss minimization, completely bypassing the need for adversarial optimization or nested expectation estimation in previous methods. iSMART accommodates linear, semi-linear, and fully nonlinear martingale-based PDEs within a unified iterative procedure. In particular, for fully nonlinear Hamilton-Jacobi-Bellman (HJB) equations, a freezing-and-compensating technique is introduced to strategically shift a portion of the nonlinearity into the SDE drift, thereby improving the convergence behavior of the iterations. Numerous numerical experiments on linear reaction-diffusion equations with sharp gradients, semilinear Burgers-type equations, and fully nonlinear HJB equations demonstrate the accuracy, efficiency, and robustness of the proposed approach in various high dimensions.
Comments32 pages, 9 figures, 4 tables
