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  Solving Fokker-Planck equation using deep learning

Xu, Y., Zhang, H., Li, Y., Zhou, K., Liu, Q., Kurths, J. (2020): Solving Fokker-Planck equation using deep learning. - Chaos, 30, 1, 013133.
https://doi.org/10.1063/1.5132840

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Xu, Yong1, Author
Zhang, Hao1, Author
Li, Yongge1, Author
Zhou, Kuang1, Author
Liu, Qi1, Author
Kurths, Jürgen2, Author              
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1External Organizations, ou_persistent22              
2Potsdam Institute for Climate Impact Research, ou_persistent13              

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 Abstract: 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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 Dates: 2020
 Publication Status: Finally published
 Pages: -
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 Rev. Type: Peer
 Identifiers: DOI: 10.1063/1.5132840
PIKDOMAIN: RD4 - Complexity Science
MDB-ID: No data to archive
Working Group: Network- and machine-learning-based prediction of extreme events
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Title: Chaos
Source Genre: Journal, SCI, Scopus, p3
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Pages: - Volume / Issue: 30 (1) Sequence Number: 013133 Start / End Page: - Identifier: CoNE: https://publications.pik-potsdam.de/cone/journals/resource/180808
Publisher: American Institute of Physics (AIP)