论文标题
与时间相关的fokker-Planck方程的人工神经网络求解器
Artificial Neural Network Solver for Time-Dependent Fokker-Planck Equations
论文作者
论文摘要
当在更快的时间尺度上具有随机扰动或混乱动力学的系统建模时，随机微分方程在各种应用中起重要作用。 Fokker-Planck方程描述了随机微分方程的概率分布的时间演变，这是二阶抛物线偏差方程。先前的工作结合了人工神经网络和蒙特卡洛数据来求解固定的fokker-planck方程。本文将这种方法扩展到时间依赖的fokker-planck方程。重点是研究具有多尺度损失函数的神经网络的算法。此外，提出了一种新的搭配点采样方法。证明了一些1D和2D数值示例。
Stochastic differential equations play an important role in various applications when 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. Previous work combined artificial neural network and Monte Carlo data to solve stationary Fokker-Planck equations. This paper extends this approach to time dependent Fokker-Planck equations. The 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.
