中国物理B ›› 2024, Vol. 33 ›› Issue (1): 10202-10202.doi: 10.1088/1674-1056/ad071b

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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)   

  1. School of Mechanical Engineering, University of Shanghai for Science and Technology, Shanghai 200093, China
  • 收稿日期:2023-08-31 修回日期:2023-10-21 接受日期:2023-10-26 出版日期:2023-12-13 发布日期:2024-01-03
  • 通讯作者: Shenlong Wang E-mail:shenlongwang@usst.edu.cn
  • 基金资助:
    Project supported by the National Natural Science Foundation of China (Grant No. 12172226).

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)   

  1. School of Mechanical Engineering, University of Shanghai for Science and Technology, Shanghai 200093, China
  • Received:2023-08-31 Revised:2023-10-21 Accepted:2023-10-26 Online:2023-12-13 Published:2024-01-03
  • Contact: Shenlong Wang E-mail:shenlongwang@usst.edu.cn
  • Supported by:
    Project supported by the National Natural Science Foundation of China (Grant No. 12172226).

摘要: 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.

关键词: deep learning, prior knowledge, FPK equation, probability density function

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.

Key words: deep learning, prior knowledge, FPK equation, probability density function

中图分类号:  (Stochastic analysis methods)

  • 05.10.Gg
07.05.Mh (Neural networks, fuzzy logic, artificial intelligence) 05.10.Ln (Monte Carlo methods)