论文标题
基于FBSDE的神经网络算法，用于高维质抛物线PDES
FBSDE based Neural Network Algorithms for High-Dimensional Quasilinear Parabolic PDEs
论文作者
论文摘要
在本文中，我们提出了前向后和向后的随机微分方程（FBSDE）的深神经网络（DNN）学习算法，用于解决高维抛物性抛物线偏微分方程（PDE）的解决方案，这些方程（PDES）与Pardoux-Peng理论与FBSDES有关。该算法依靠学习过程来最大程度地减少两个离散随机过程之间的路径差异，该过程分别由FBSDE的时间离散化和PDE解决方案的DNN表示定义。所提出的算法被证明可以生成100维黑色 - choles-barenblatt方程的DNN溶液，在溶液空间中的有限区域中准确，并且具有类似于FBSDES使用的Euler--Maruyama ivepation的收敛速率。结果，可以使用Richardson外推技术来提高DNN溶液的准确性。对于时间振荡解决方案，显示多尺度DNN可改善FBSDE DNN的性能。
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.
