FBSDE based neural network algorithms for high-dimensional quasilinear parabolic PDEs
FBSDE based neural network algorithms for high-dimensional quasilinear parabolic PDEs
复制标题
基于 FBSDE 的高维拟线性抛物线偏微分方程神经网络算法
DOI：
10.1016/j.jcp.2022.111557
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发表时间：
2022
影响因子：
4.1
通讯作者：
Cai, Wei
中科院分区：
文献类型：
--
作者：
Zhang, Wenzhong;Cai, Wei
在本文中，我们提出了基于深度神经网络（DNN）的前向和后向随机微分方程（FBSDE）学习算法来求解高维准线性抛物型偏微分方程（PDE），这与Pardoux-Peng理论中的FBSDE相关。该算法依赖于学习过程，通过最小化两个离散随机过程之间的路径差异来实现，这两个随机过程分别由 FBSDE 的时间离散化和 PDE 解的 DNN 表示来定义。所提出的算法可以为 100 维 Black-Scholes-Barenblatt 方程生成 DNN 解，该解在解空间的有限区域内是准确的，并且收敛速度接近用于离散 FBSDE 的 Euler-Maruyama 方案。
In this paper, we propose forward and backward stochastic differential equations (FBSDEs) based deep neural network (DNN) learning algorithms for the solution of high dimensional quasi-linear parabolic partial differential equations (PDEs), which is related to the FBSDEs from the Pardoux-Peng theory. The algorithms rely on a learning process by minimizing the path-wise difference between two discrete stochastic processes, which are defined by the time discretization of the FBSDEs and the DNN representation of the PDE solution, respectively. The proposed algorithms are shown to generate DNN solution for a 100-dimensional Black–Scholes–Barenblatt equation, which is accurate in a finite region in the solution space, and has a convergence rate close to that of the Euler–Maruyama scheme used for discretizing the FBSDEs.