Stochastic differential equations are commonly used to describe the evolution of stochastic processes. The uncertainty of such processes is best represented by the probability density function (PDF), whose evolution is governed by the Fokker-Planck partial differential equation (FP-PDE). However, it is generally infeasible to solve the FP-PDE in closed form. In this work, we show that physics-informed neural networks (PINNs) can be trained to approximate the solution PDF using existing methods. The main contribution is the analysis of the approximation error: we develop a theory to construct an arbitrary tight error bound with PINNs. In addition, we derive a practical error bound that can be efficiently constructed with existing training methods. Finally, we explain that this error-bound theory generalizes to approximate solutions of other linear PDEs. Several numerical experiments are conducted to demonstrate and validate the proposed methods.

翻译：随机微分方程常被用于描述随机过程的演化规律。此类过程的不确定性最宜用概率密度函数来表征，其演化受福克-普朗克偏微分方程支配。然而，通常难以以闭式解形式求解该方程。本研究表明，通过现有方法可训练物理信息神经网络来逼近概率密度函数解。主要贡献在于近似误差分析：我们建立了利用物理信息神经网络构建任意紧致误差界的理论框架。此外，推导出可通过现有训练方法高效构建的实用误差界。最后论证该误差界理论可推广至其他线性偏微分方程的近似求解。通过数值实验验证了所提方法的有效性。