We develop a generic convolutional neural network (CNN) based numerical scheme to simulate the 2-tuple adapted strong solution to a unified system of backward stochastic partial differential equations (B-SPDEs) driven by Brownian motions, which can be used to model many real-world system dynamics such as optimal control and differential game problems. The dynamics of the scheme is modeled by a CNN through conditional expectation projection. It consists of two convolution parts: W layers of backward networks and L layers of reinforcement iterations. Furthermore, it is a completely discrete and iterative algorithm in terms of both time and space with mean-square error estimation and almost sure (a.s.) convergence supported by both theoretical proof and numerical examples. In doing so, we need to prove the unique existence of the 2-tuple adapted strong solution to the system under both conventional and Malliavin derivatives with general local Lipschitz and linear growth conditions.

我们开发了一种基于通用卷积神经网络 (CNN) 的数值方案，以模拟由布朗运动驱动的后向随机偏微分方程 (B-SPDE) 统一系统的 2 元组适应强解，该系统可用于模拟许多真实的- 世界系统动力学，例如最优控制和微分博弈问题。该方案的动力学由 CNN 通过条件期望投影进行建模。它由两个卷积部分组成：W层后向网络和L加固迭代层。此外，它是一种在时间和空间上完全离散和迭代的算法，具有均方误差估计和几乎肯定的收敛性，得到理论证明和数值示例的支持。在这样做时，我们需要证明在具有一般局部 Lipschitz 和线性增长条件的常规和 Malliavin 导数下系统的 2 元组适应强解的独特存在。