In this paper, we solve stochastic partial differential equations (SPDEs) numerically by using (possibly random) neural networks in the truncated Wiener chaos expansion of their corresponding solution. Moreover, we provide some approximation rates for learning the solution of SPDEs with additive and/or multiplicative noise. Finally, we apply our results in numerical examples to approximate the solution of three SPDEs: the stochastic heat equation, the Heath-Jarrow-Morton equation, and the Zakai equation.

翻译：本文通过使用（可能随机的）神经网络在截断Wiener混沌展开中数值求解随机偏微分方程。此外，我们为学习具有加性和/或乘性噪声的随机偏微分方程解提供了若干近似速率估计。最后，我们将所得结果应用于数值算例，对三类随机偏微分方程——随机热方程、Heath-Jarrow-Morton方程及Zakai方程——的解进行近似计算。