Separable Gaussian neural networks for high-dimensional nonlinear stochastic systems高维非线性随机系统的可分离 Gauss 神经网络
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Wang X, Xing SY, Jiang J, Hong L, Sun JQ, 2024. Separable Gaussian neural networks for high-dimensional nonlinear stochastic systems. Probabilistic Engineering Mechanics, 76: 103594.DOI: 10.1016/j.probengmech.2024.103594
摘要 | Abstract
本文将最近开发的可分离 Gauss 神经网络 (separable Gaussian neural networks, SGNN) 方法扩展到高维状态空间中的 Fokker-Planck-Kolmogorov (FPK) 方程的求解。解决了将可分离 Gauss 神经网络扩展到高维状态空间时的几个挑战，包括 Gauss 神经元的域定义和数据采样域定义、以及边缘概率密度函数的数值积分问题。通过可分离 Gauss 神经网络方法，对三个具有不同复杂性和维度的基准非线性动力系统进行了分析。特别地，使用可分离 Gauss 神经网络方法获得了响应稳态概率密度，并与大量蒙特卡罗模拟结果进行了比较。需指出，对于非线性动力系统高维 FPK 方程的某些解，除采用可分离 Gauss 神经网络外很难获得。关键词: 径向基函数神经网络设计,
非线性随机系统, 稳态概率密度函数, Fokker-Planck-Kolmogorov 方程This paper extends the recently developed method of separable Gaussian neural networks (SGNN) to obtain solutions of the Fokker-Planck-Kolmogorov (FPK) equation in high-dimensional state space. Several challenges when extending SGNN to high-dimensional state space are addressed including proper definition of domain for placing Gaussian neurons and region for data sampling, and numerical integration issue of evaluating marginal probability density functions. Three benchmark nonlinear dynamic systems with increasing complexity and dimension are examined with the SGNN method. In particular, the steady-state probability density of the response is obtained with the SGNN method and compared with the results of extensive Monte Carlo simulations. It should be pointed out that some solutions of high-dimensional FPK equations for nonlinear dynamic systems would be very difficult to obtain without SGNN.Keywords: Design of radial basis function neural networks; Nonlinear stochastic system; Stationary probability density function; Fokker-Planck-Kolmogorov equationFig. 1. Structure of RBFNN
图 2: 含多个隐藏层的可分离 Gauss 神经网络结构Fig. 2. Structure of an SGNN with d hidden layers
图 3: 可分离 Gauss 神经网络方法与蒙特卡罗模拟获得的 2 维 Van der Pol 系统概率密度函数解对比Fig. 3. Comparison of the PDF solution obtained by the SGNN method and MCS of the 2D Van der Pol system
图 4: 2 维 Van der Pol 系统的边缘概率密度函数Fig. 4. marginal PDFs of p (x_1) and p (x_2) of the 2D Van der Pol system
图 5: 2 维 Van der Pol 系统的边缘概率密度函数半对数图Fig. 5. Semi-logarithmic plots of marginal PDFs p (x_1) and p (x_2) of the 2D Van der Pol system
图 6: 2 维 Van der Pol 系统的损失函数Fig. 6. Loss function of the 2D Van der Pol system
图 7: 4 维非线性系统的边缘稳态联合概率密度函数Fig. 7. Marginal steady-state joint PDFs of system (43)
图 8: 可分离 Gauss 神经网络方法与蒙特卡罗模拟获得的 4 维非线性系统单变量边缘概率密度函数解对比Fig. 8. Comparison of the single-variant marginal PDF solutions obtained by the SGNN method and MCS of system (43)
图 9: 2 自由度非线性系统的边缘概率密度函数半对数图Fig. 9. Semi-logarithmic plots of marginal PDFs p (x_1) and p (x_2) of the 2-DOF nonlinear system
Fig. 10. Variation of the loss function of the 4D system (43) during training
图 11: 6 维非线性系统的边缘稳态联合概率密度函数Fig. 11. Marginal steady-state joint PDFs of system (45)
图 12: 可分离 Gauss 神经网络方法与蒙特卡罗模拟获得的 6 维非线性系统单变量边缘概率密度函数解对比Fig. 12. Comparison of the single-variant marginal PDF solutions obtained by the SGNN method and MCS of system (45)
图 13: 6 维非线性系统的边缘概率密度函数半对数图Fig. 13. Semi-logarithmic plots of marginal PDFs p (x_1) and p (x_2) of system (45)
Fig. 14. Variation of the loss function of the 6D system (45) during training
图 8: 6 维非线性系统的第一与第二维状态量之差的稳态概率密度函数Fig. 8. Steady-state PDF of system (45) on the x_1 - x_2 with x_3 = x_4 = x_5 = x_6 = 0
作者信息 | Authors
西安交通大学 (Xi’an Jiaotong University) 机械结构强度与振动国家重点实验室
美国加利福尼亚州立理工大学 (California Polytechnic State University) 机械工程系
西安交通大学 (Xi’an Jiaotong University) 机械结构强度与振动国家重点实验室
西安交通大学 (Xi’an Jiaotong University) 机械结构强度与振动国家重点实验室
孙建桥 Jian-Qiao Sun, 通讯作者 (Corresp.) 美国加利福尼亚大学默塞德分校 (University of California Merced) 工学院Email: jqsun@ucmerced.edu
律梦泽 M.Z. Lyu | 编辑 (Ed)
P.D. Spanos | 审校 (Rev)
陈建兵 J.B. Chen | 审校 (Rev)
彭勇波 Y.B. Peng | 审校 (Rev)
