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 quasilinear parabolic partial differential equations (PDEs), which are related to the FBSDEs by the Pardoux-Peng theory. The algorithms rely on a learning process by minimizing the pathwise difference between two discrete stochastic processes, defined by the time discretization of the FBSDEs and the DNN representation of the PDE solutions, respectively. The proposed algorithms are shown to generate DNN solutions for a 100-dimensional Black--Scholes--Barenblatt equation, accurate in a finite region in the solution space, and has a convergence rate similar to that of the Euler--Maruyama discretization used for the FBSDEs. As a result, a Richardson extrapolation technique over time discretizations can be used to enhance the accuracy of the DNN solutions. For time oscillatory solutions, a multiscale DNN is shown to improve the performance of the FBSDE DNN for high frequencies.
翻译:在本文中,我们提出前方和后方随机差异方程式(FBSDEs)基于深神经网络(DNN)的深神经网络(DNN)的学习算法(DDE),用于解决高维准线性抛光性部分差异方程式(PDE),这些算法与Pardou-Peng理论的FBSDEs相关,与FBSDEs使用的Euler-Maruyama离散率相似。结果之一是,可以使用由FBSDEs的时间离散和PDE解决方案的DNN代表分别界定的两个离散的随机异方程之间的路径差异最小化方法来提高DNN的准确性。关于时间分解解决方案的时间分解度,拟议的算法显示为100维的黑-Scholes-Barenblat方程式生成DNNN(DNNN)的DNN(D)解决方案,在解决方案的有限区域中准确,并且具有类似于FBS-DNEDDD的高级频率性。