In this article, we introduce and analyze a deep learning based approximation algorithm for SPDEs. Our approach employs neural networks to approximate the solutions of SPDEs along given realizations of the driving noise process. If applied to a set of simulated noise trajectories, it yields empirical distributions of SPDE solutions, from which functionals like the mean and variance can be estimated. We test the performance of the method on stochastic heat equations with additive and multiplicative noise as well as stochastic Black-Scholes equations with multiplicative noise and Zakai equations from nonlinear filtering theory. In all cases, the proposed algorithm yields accurate results with short runtimes in up to 100 space dimensions.

翻译：本文提出并分析了一种基于深度学习的随机偏微分方程逼近算法。该方法利用神经网络沿驱动噪声过程的给定实现来逼近随机偏微分方程的解。若将其应用于一组模拟的噪声轨迹，则可得到随机偏微分方程解的经验分布，进而可估计均值、方差等泛函。我们在加性噪声与乘性噪声的随机热方程、乘性噪声的随机Black-Scholes方程以及非线性滤波理论中的Zakai方程上测试了该方法的性能。在所有案例中，所提算法在高达100个空间维度下均能以较短运行时间获得精确结果。