论文标题
基于深度学习的基于随机部分微分方程和高维非线性滤波问题的数值近似算法
Deep learning based numerical approximation algorithms for stochastic partial differential equations and high-dimensional nonlinear filtering problems
论文作者
论文摘要
在本文中，我们介绍并研究了一种基于深度学习的近似算法，用于随机部分微分方程（SPDE）的解决方案。在提出的近似算法中，我们采用一个深层神经网络，每次实现SPDE的驱动噪声过程，以近似所考虑的SPDE的解决方案过程。在具有添加噪声的随机热方程，具有乘法噪声的随机热方程，具有乘法噪声的随机黑色 - choles方程以及来自非线性过滤的Zakai方程的情况下，我们测试了提出的近似算法的性能。在这些SPDE中，提出的近似算法在多达50个空间尺寸的短时间内产生准确的结果。
In this article we introduce and study a deep learning based approximation algorithm for solutions of stochastic partial differential equations (SPDEs). In the proposed approximation algorithm we employ a deep neural network for every realization of the driving noise process of the SPDE to approximate the solution process of the SPDE under consideration. We test the performance of the proposed approximation algorithm in the case of stochastic heat equations with additive noise, stochastic heat equations with multiplicative noise, stochastic Black--Scholes equations with multiplicative noise, and Zakai equations from nonlinear filtering. In each of these SPDEs the proposed approximation algorithm produces accurate results with short run times in up to 50 space dimensions.
