High-dimensional parabolic partial differential equations (PDEs) often involve large-scale Hessian matrices, which are computationally expensive for deep learning methods relying on automatic differentiation to compute derivatives. This work aims to address this issue. In the proposed method, the PDE is reformulated into a martingale formulation, which allows the computation of loss functions to be derivative-free and parallelized in time-space domain. Then, the martingale formulation is enforced using a Galerkin method via adversarial learning techniques, which eliminate the need of computing conditional expectations in the margtingale property. This method is further extended to solve Hamilton-Jacobi-Bellman (HJB) equations and the associated Stochastic optimal control problems, enabling the simultaneous solution of the value function and optimal feedback control in a derivative-free manner. Numerical results demonstrate the effectiveness and efficiency of the proposed method, capable of solving HJB equations accurately with dimensionality up to 10,000.

翻译：高维抛物型偏微分方程常涉及大规模Hessian矩阵，这对依赖自动微分计算导数的深度学习方法而言计算代价高昂。本研究旨在解决此问题。在所提方法中，偏微分方程被重构为鞅形式，这使得损失函数的计算无需导数且可在时空域并行化。随后，通过对抗学习技术采用伽辽金方法强制实施该鞅形式，从而避免了鞅性质中条件期望的计算。该方法进一步扩展至求解Hamilton-Jacobi-Bellman方程及相关随机最优控制问题，能够以无导数方式同步求解值函数与最优反馈控制。数值结果验证了所提方法的有效性与高效性，可精确求解维度高达10,000的HJB方程。