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A deep learning method based on prior knowledge with dual training for solving FPK equation |
| Denghui Peng(彭登辉), Shenlong Wang(王神龙)†, and Yuanchen Huang(黄元辰) |
| School of Mechanical Engineering, University of Shanghai for Science and Technology, Shanghai 200093, China |
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Abstract The evolution of the probability density function of a stochastic dynamical system over time can be described by a Fokker—Planck—Kolmogorov (FPK) equation, the solution of which determines the distribution of macroscopic variables in the stochastic dynamic system. Traditional methods for solving these equations often struggle with computational efficiency and scalability, particularly in high-dimensional contexts. To address these challenges, this paper proposes a novel deep learning method based on prior knowledge with dual training to solve the stationary FPK equations. Initially, the neural network is pre-trained through the prior knowledge obtained by Monte Carlo simulation (MCS). Subsequently, the second training phase incorporates the FPK differential operator into the loss function, while a supervisory term consisting of local maximum points is specifically included to mitigate the generation of zero solutions. This dual-training strategy not only expedites convergence but also enhances computational efficiency, making the method well-suited for high-dimensional systems. Numerical examples, including two different two-dimensional (2D), six-dimensional (6D), and eight-dimensional (8D) systems, are conducted to assess the efficacy of the proposed method. The results demonstrate robust performance in terms of both computational speed and accuracy for solving FPK equations in the first three systems. While the method is also applicable to high-dimensional systems, such as 8D, it should be noted that computational efficiency may be marginally compromised due to data volume constraints.
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Received: 31 August 2023
Revised: 21 October 2023
Accepted manuscript online: 26 October 2023
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PACS:
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05.10.Gg
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(Stochastic analysis methods)
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07.05.Mh
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(Neural networks, fuzzy logic, artificial intelligence)
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05.10.Ln
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(Monte Carlo methods)
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| Fund: Project supported by the National Natural Science Foundation of China (Grant No. 12172226). |
Corresponding Authors:
Shenlong Wang
E-mail: shenlongwang@usst.edu.cn
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Cite this article:
Denghui Peng(彭登辉), Shenlong Wang(王神龙), and Yuanchen Huang(黄元辰) A deep learning method based on prior knowledge with dual training for solving FPK equation 2024 Chin. Phys. B 33 010202
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[1] Weigel A V, Fox P D, Akin E J, Ecklund K H, Tamkun M M and Krapf D 2012 Biophysical Journal 103 1727
[2] Nadler W and Schulten K 1986 J. Chem. Phys. 84 4015
[3] Belomestny D, Hübner T and Krätschmer V 2022 Finance and Stochastics 26 461
[4] Han B T and Jiang D Q 2022 Chaos, Solitons & Fractals 162 112458
[5] Fa K S 2017 Journal of Statistical Mechanics:Theory and Experiment 2017 033207
[6] Chakraborty I and Balachandran B 2011 Nonlinear Dyn. 66 427
[7] Galán R F, Ermentrout G B and Urban N N 2007 Phys. Rev. E 76 056110
[8] Sepehrian B and Karimi Radpoor M 2020 Computational Methods for Differential Equations 8 493
[9] Xu Y, Zan W R, Jia W T and Kurths J 2019 J. Comput. Phys. 394 41
[10] Biazar J, Gholamin P and Hosseini K 2010 Journal of the Franklin Institute 347 1137
[11] Hirvijoki E, Kurki-Suonio T, Äkäslompolo S, Varje J, Koskela T and Miettunen J 2015 J. Plasma Phys. 81 435810301
[12] Chen N and Majda A J 2018 J. Comput. Phys. 354 242
[13] Han J Q, Jentzen A and E W N 2018 Proc. Natl. Acad. Sci. USA 115 8505
[14] Raissi M 2023 Peter Carr Gedenkschrift:Research Advances in Mathematical Finance 18 637
[15] Raissi M, Perdikaris P and Karniadakis G E 2019 J. Comput. Phys. 378 686
[16] Berg J and Nyström K 2018 Neurocomputing 317 28
[17] Sirignano J and Spiliopoulos K 2018 J. Comput. Phys. 375 1339
[18] Xu Y, Zhang H, Li Y G, Zhou K, Liu Q and Kurths J 2020 Chaos:An Interdisciplinary Journal of Nonlinear Science 30 013133
[19] Zhang H, Xu Y, Liu Q, Wang X L and Li Y G 2022 Nonlinear Dyn. 108 4029
[20] Zhang H, Xu Y, Liu Q and Li Y G 2023 Engineering Applications of Artificial Intelligence 121 106036
[21] Wang X, Jiang J, Hong L, Chen L C and Sun J Q 2023 Probabilistic Engineering Mechanics 73 103470
[22] Chen X L, Yang L, Duan J Q and Karniadakis G E 2021 SIAM Journal on Scientific Computing 43 B811
[23] Zhai J Y, Dobson M and Li Y 2021 Proceedings of the 2nd Mathematical and Scientific Machine Learning Conference, August 16-19, 2021, Virtual Conference, p. 568
[24] Lin B, Li Q X and Ren W Q 2022 Journal of Scientific Computing 91 77
[25] Zhang Y and Yuen K V 2022 International Journal of Nonlinear Mechanics 147 104202
[26] Tang K J, Wan X L and Liao Q F 2022 J. Comput. Phys. 457 111080
[27] Zhang H, Xu Y, Li Y G and Kurths J 2020 International Journal of Dynamics and Control 8 1129
[28] Lepage G P 2021 J. Comput. Phys. 439 110386
[29] Newby J M 2012 Physical Biology 9 026002 |
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