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，在其对应解的截断Wiener混沌展开中，对随机偏微分方程（SPDEs）进行数值求解。此外，我们为学习具有加性和/或乘性噪声的SPDEs解提供了一些近似速率。最后，我们在数值算例中应用我们的结果来近似求解三个SPDE：随机热方程、Heath-Jarrow-Morton方程以及Zakai方程。