加性/乘性白噪声下强非线性系统高维稳态 Fokker-Planck-Kolmogorov 方程求解的高效方法
提出了一种复杂非线性系统随机分析的高效神经网络格式
采用高效的径向基函数神经网络从先前数据中近似等价漂移和扩散系数 研究了四个不同复杂度和非线性程度的算例，以验证所提格式的有效性和优势
A highly efficient NN-based scheme for the stochastic analysis of complex nonlinear system is proposed.
The powerful RBFNN is employed to approximate the equivalent drift coefficients (EDCs) and diffusion coefficients (EDFs) from a few prior data. Four examples with various levels of complexity and nonlinearity are studied to verify the effectiveness and advantage of the proposed scheme.
Fig. 1. Connection of the regression function r(x_1) and conditional expectation E(Y | X_1 = x_1), where only one interested quantity is considered
Fig. 2. Scheme of RBFNN: (a) Architecture of NN; (b) Illustration of RBFs
Fig. 3. Flowchart of the RBFNN model to approximate EDCs and EDFs
Fig. 4. Flowchart of the PINN to solve the steady-state FPK equation
Fig. 5. MSE and R-squared vs. the number of samples in training RBFNN for EDC a_2^e(x_1,x_2)
Fig. 6. Exact EDC a_2^e(x_1,x_2) and its approximation by respectively the RBFNN and Lowess regressions with 300 samples
Fig. 7. Comparison of the steady-state marginal velocity PDF by the proposed method with the exact solution, 300 MCS, and GV-GDEE solution: (a) Velocity in linear coordinates; (b) Velocity in logarithmic coordinates
Fig. 8. Exact joint PDF p(x_1,x_2): (a) 3D view of the PDF; (b) Contour of the PDF
Fig. 9. MSE and R-squared vs. the number of samples in training RBFNN for a_2^e(x_1,x_2) and EDF b_22^e(x_1,x_2)
Fig. 10. Exact EDC a_2^e(x_1,x_2), and its approximation by respectively the RBFNN and Lowess with 400 samples: (a) EDC by RBFNN; (b) Comparison between the exact EDC and approximated EDC by RBFNN; (c) EDC by Lowess; (d) Comparison between the exact EDC and approximated EDC by Lowess
Fig. 11. Exact EDF b_22^e(x_1,x_2), and its approximation by respectively the RBFNN and Lowess with 400 samples: (a) EDF by RBFNN; (b) Comparison between the exact EDF and approximated EDF by Lowess; (c) EDF by Lowess; (d) Comparison between the exact EDF and approximated EDF by RBFNN
Fig. 12. Comparison of the steady-state marginal displacement and velocity PDF by the proposed method with the exact solution, 400 MCS, and GV-GDEE solution: (a) Displacement in linear coordinates; (b) Displacement in logarithmic coordinates; (c) Velocity in linear coordinates; (d) Velocity in logarithmic coordinates
Fig. 13. Exact joint PDF p(x_1,x_2): (a) 3D view of the PDF; (b) Contour of the PDF
Fig. 14. MSE and R-squared vs. the number of samples in training RBFNN for a_2^e(x_1,x_2) and EDF b_22^e(x_1,x_2): (a) MSE for EDC; (b) R-squared for EDC; (c) MSE for EDF; (d) R-squared for EDF
Fig. 15. Exact EDC a_2^e(x_1,x_2), and its approximation by respectively the RBFNN and Lowess with 400 samples: (a) EDC by RBFNN; (b) Comparison between the exact EDC and approximated EDC by RBFNN; (c) EDC by Lowess; (d) Comparison between the exact EDC and approximated EDC by Lowess
Fig. 16. Exact EDF b_22^e(x_1,x_2), and its approximation by respectively the RBFNN and Lowess with 400 samples: (a) EDF by RBFNN; (b) Comparison between the exact EDF and approximated EDF by Lowess; (c) EDF by Lowess; (d) Comparison between the exact EDF and approximated EDF by RBFNN
Fig. 17. Comparison of the steady-state marginal displacement x_1 and velocity x_2 PDF by the proposed method with the exact solution, 400 MCS, and GV-GDEE solution: (a) Displacement x_1 in linear coordinates; (b) Displacement x_1 in logarithmic coordinates; (c) Velocity x_2 in linear coordinates; (d) Velocity x_2 in logarithmic coordinates
Fig. 18. 10-story frame structure model
Fig. 19. MSE and R-squared vs. the number of samples in training RBFNN for EDC a_2^e(x_10,x_10): (a) MSE; (b) R-squared
Fig. 20. Approximation of a_2^e(x_10,x_10) by RBFNN and Lowess with 400 samples at the top floor: (a) EDC by RBFNN; (b) EDC by Lowess
Fig. 21. Joint PDF p(x_10,x_10) obtained by MCS: (a) 3D view of the PDF; (b) Contour of the PDF
Fig. 22. Comparison of the steady-state marginal displacement and velocity PDF at the top floor by the proposed method with the MCS and GV-GDEE solution: (a) Displacement x_10 in linear coordinates; (b) Displacement x_10 in logarithmic coordinates; (c) Velocity x_10 in linear coordinates; (d) Velocity x_10 in logarithmic coordinates
Fig. 23. Typical hysteretic restoring force curve for hysteretic system: (a) Bottom floor; (b) Top floor
Fig. 24. MSE and R-squared against number of samples in training RBFNN for EDCs: (a) MSE for EDC a_1^e(x_10,z_1); (b) R-squared for EDC a_1^e(x_10,z_1); (c) MSE for EDC a_2^e(x_10,z_1); (d) R-squared for EDC a_2^e(x_10,z_1)
Fig. 25. Approximation of EDC with 400 samples at the bottom floor: (a) Approximated EDC a_1^e(x_10,z_1) by RBFNN; (b) Approximated EDC a_1^e(x_10,z_1) by Lowess; (c) Approximated EDC a_2^e(x_10,z_1) by RBFNN; (d) Approximated EDC a_2^e(x_10,z_1) by Lowess
Fig. 26. Joint PDF p(x_10,z_1) obtained by MCS: (a) 3D view of the PDF; (b) Contour of the PDF
Fig. 27. Comparison of the steady-state marginal hysteretic displacement z_1 PDF at the top floor by the proposed method with the MCS and GV-GDEE solution: (a) Hysteretic displacement in linear coordinates; (b) Hysteretic displacement in logarithmic coordinates
作者信息 | Authors
哈尔滨工业大学 (深圳) (Harbin Institute of Technology Shenzhen) 土木与环境工程学院
华侨大学 (Huaqiao University) 土木工程学院
Email: duanzd@hit.edu.cn
美国加利福尼亚大学默塞德分校 (University of California Merced) 工学院
哈尔滨工业大学 (深圳) (Harbin Institute of Technology Shenzhen) 土木与环境工程学院
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)
