用神经网络在维纳混沌展开中求解随机偏微分方程
Solving stochastic partial differential equations using neural networks in the Wiener chaos expansion
- 将神经网络嵌入截断的维纳混沌展开，逼近随机偏微分方程解
- 给出加性与乘性噪声下解的近似误差率
- 在热方程、HJM和Zakai方程上验证方法有效性
本文通过在截断的维纳混沌展开中使用（可能含随机性的）神经网络，数值求解随机偏微分方程（SPDEs）。同时，提供了对具有加性或乘性噪声的SPDE解的近似速率分析。最后，将该方法应用于三个具体实例：随机热方程、Heath-Jarrow-Morton方程和Zakai方程，以近似其解。
原文摘要 · Abstract (English)
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.
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