Artificial neural network solver for time-dependent Fokker–Planck equations
Artificial neural network solver for time-dependent Fokker–Planck equations
复制标题
用于求解瞬态福克普朗克方程的人工神经网络求解器
DOI：
10.1016/j.amc.2023.128185
复制
发表时间：
2023
影响因子：
4
通讯作者：
Meredith, Caleb
中科院分区：
文献类型：
--
作者：
Li, Yao;Meredith, Caleb
随机微分方程（SDEs）在各种应用中扮演着至关重要的角色，无论是随机扰动或混沌动力学系统在更快的时间尺度建模。随机微分方程的概率分布的时间演化由Fokker-Planck方程描述，它是一个二阶抛物型偏微分方程（PDE）。以前的工作结合人工神经网络和蒙特卡罗数据来求解固定的福克-普朗克方程。本文将这种方法推广到含时的Fokker-Planck方程。主要的重点是研究训练具有多尺度损失函数的神经网络的算法。此外，提出了一种新的配置点采样方法。给出了几个一维和二维数值算例。
Stochastic differential equations (SDEs) play a crucial role in various applications for modeling systems that have either random perturbations or chaotic dynamics at faster time scales. The time evolution of the probability distribution of a stochastic differential equation is described by the Fokker–Planck equation, which is a second order parabolic partial differential equation (PDE). Previous work combined artificial neural networks and Monte Carlo data to solve stationary Fokker–Planck equations. This paper extends this approach to time dependent Fokker–Planck equations. The main focus is on the investigation of algorithms for training a neural network that has multi-scale loss functions. Additionally, a new approach for collocation point sampling is proposed. A few 1D and 2D numerical examples are demonstrated.