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.

翻译：在本篇文章中,我们引入并研究一个基于深层次学习的理论近似算法,用于解决随机部分差异方程式(SPDEs)的解决方案。在拟议的近似算法中,我们使用一个深层神经网络,用于实现SPDE的驱动噪音过程,以近似审议中的SPDE的解决方案过程。我们测试了在具有添加噪音的随机热方程式、具有多复制噪音的随机热方程式、具有多复制性噪音的随机黑色切换方程式和来自非线性过滤的Zakai方程式的情况下,拟议的近似算法的性能,这些近似方程式在50个空间尺寸的短期内产生准确的结果。