The probability density function of stochastic differential equations is governed by the Fokker-Planck (FP) equation. A novel machine learning method is developed to solve the general FP equations based on deep neural networks. The proposed algorithm does not require any interpolation and coordinate transformation, which is different from the traditional numerical methods. The main novelty of this paper is that penalty factors are introduced to overcome the local optimization for the deep learning approach, and the corresponding setting rules are given. Meanwhile, we consider a normalization condition as a supervision condition to effectively avoid that the trial solution is zero. Several numerical examples are presented to illustrate performances of the proposed algorithm, including one-, two-, and three-dimensional systems. All the results suggest that the deep learning is quite feasible and effective to calculate the FP equation. Furthermore, influences of the number of hidden layers, the penalty factors, and the optimization algorithm are discussed in detail. These results indicate that the performances of the machine learning technique can be improved through constructing the neural networks appropriately.
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January 2020
Research Article|
January 23 2020
Solving Fokker-Planck equation using deep learning
Yong Xu;
Yong Xu
a)
1
Department of Applied Mathematics, Northwestern Polytechnical University
, Xi’an 710072, China
2
MIIT Key Laboratory of Dynamics and Control of Complex Systems, Northwestern Polytechnical University
, Xi’an 710072, China
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Hao Zhang;
Hao Zhang
1
Department of Applied Mathematics, Northwestern Polytechnical University
, Xi’an 710072, China
3
Department of Engineering Mechanics, Northwestern Polytechnical University
, Xi’an 710072, China
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Yongge Li;
Yongge Li
4
Center for Mathematical Sciences and School of Mathematics and Statistics, Huazhong University of Science and Technology
, Wuhan 430074, China
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Kuang Zhou;
Kuang Zhou
1
Department of Applied Mathematics, Northwestern Polytechnical University
, Xi’an 710072, China
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Qi Liu;
Qi Liu
1
Department of Applied Mathematics, Northwestern Polytechnical University
, Xi’an 710072, China
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Jürgen Kurths
Jürgen Kurths
5
Potsdam Institute for Climate Impact Research
, Potsdam 14412, Germany
6
Department of Physics, Humboldt University Berlin
, Berlin 12489, Germany
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a)
Author to whom correspondence should be addressed: hsux3@nwpu.edu.cn
Note: This paper is part of the Focus Issue, “When Machine Learning Meets Complex Systems: Networks, Chaos and Nonlinear Dynamics.”
Chaos 30, 013133 (2020)
Article history
Received:
October 21 2019
Accepted:
December 23 2019
Citation
Yong Xu, Hao Zhang, Yongge Li, Kuang Zhou, Qi Liu, Jürgen Kurths; Solving Fokker-Planck equation using deep learning. Chaos 1 January 2020; 30 (1): 013133. https://doi.org/10.1063/1.5132840
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