福克-普朗克方程
可解释性
离散化
随机微分方程
应用数学
概率密度函数
偏微分方程
高斯分布
数学
稳健性(进化)
计算机科学
动力系统理论
微分方程
数学优化
人工智能
数学分析
量子力学
生物化学
基因
统计
物理
化学
作者
Wenqing Sun,Jinqian Feng,Jin Su,Yunyun Liang
出处
期刊:Chaos
[American Institute of Physics]
日期:2022-03-01
卷期号:32 (3)
被引量:1
摘要
The Fokker-Planck (FP) equation provides a powerful tool for describing the state transition probability density function of complex dynamical systems governed by stochastic differential equations (SDEs). Unfortunately, the analytical solution of the FP equation can be found in very few special cases. Therefore, it has become an interest to find a numerical approximation method of the FP equation suitable for a wider range of nonlinear systems. In this paper, a machine learning method based on an adaptive Gaussian mixture model (AGMM) is proposed to deal with the general FP equations. Compared with previous numerical discretization methods, the proposed method seamlessly integrates data and mathematical models. The prior knowledge generated by the assumed mathematical model can improve the performance of the learning algorithm. Also, it yields more interpretability for machine learning methods. Numerical examples for one-dimensional and two-dimensional SDEs with one and/or two noises are given. The simulation results show the effectiveness and robustness of the AGMM technique for solving the FP equation. In addition, the computational complexity and the optimization algorithm of the model are also discussed.
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