Response of Gaussian white noise excited oscillators with inertia nonlinearity based on the RBFNN method基于径向基函数神经网络方法的 Gauss 白噪声激励下非线性惯性振子响应
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Hu YQ, Ge G, 2024. Response of Gaussian white noise excited oscillators with inertia nonlinearity based on the RBFNN method. Probabilistic Engineering Mechanics, 77: 103637.DOI: 10.1016/j.probengmech.2024.103637
尽管随机平均法对于求解宽带噪声激励下强非线性刚度振子响应非常有效，但其在处理强非线性惯性 (即坐标相关质量) 或多重势井振子时似乎无效。为解决这一局限性，本文采用径向基函数神经网络 (radial basis function neural network, RBFNN) 算法来预测同时具有强非线性惯性和多重势井的振子响应。模型选择常见的 Gauss 函数作为径向基函数。然后，将近似稳态概率密度函数 (probability density function, PDF) 表示为加权 Gauss 基函数 (Gaussian basis function, GBF) 之和。采用 Lagrange 乘子法最小化 Fokker-Plank-Kolmogorov (FPK) 方程近似解的平方误差，从而确定最佳权重系数。通过三个算例展示了非线性惯性和势井如何影响振子响应。提供了蒙特卡罗模拟 (Monte Carlo simulation, MCS) 与径向基函数神经网络预测的均方误差。结果表明，径向基函数神经网络的预测与蒙特卡罗模拟结果完美一致。关键词: 径向基函数神经网络, 非线性惯性, Gauss 白噪声Although stochastic averaging methods have proven effective in solving the responses of nonlinear oscillators with a strong stiffness term under broadband noise excitations, these methods appear to be ineffective when dealing with oscillators that have a strong inertial nonlinearity term (also known as coordinate-dependent mass) or multiple potential wells. To address this limitation, a radial basis function neural network (RBFNN) algorithm is applied to predict the responses of oscillators with both a strong inertia nonlinearity term and multiple potential wells. The well-known Gaussian functions are chosen as radial basis functions in the model. Then, the approximate stationary probability density function (PDF) is expressed as the sum of Gaussian basis functions (GBFs) with weights. The squared error of the approximate solution for the Fokker-Plank-Kolmogorov (FPK) function is minimized using the Lagrange multiplier method to determine optimal weight coefficients. Three examples are presented to demonstrate how inertia nonlinearity terms and potential wells affect the responses. The mean square errors between Monte Carlo simulations (MCS) and RBFNN predictions are provided. The results indicate that RBFNN predictions align perfectly with those obtained from MCS.
Keywords: Radial basis function neural network; Inertia nonlinearity; Gaussian white noise.![]()
Fig. 1. Structure of the RBFNN
Fig. 2. Diagrammatic sketch of approximate fitting
Fig. 3. Joint probability density p(x,y)
Fig. 4. Probability density p(x) and p(y)
Fig. 5. Phase portrait of the Hamiltonian function
Fig. 6. Triple-well potential energy
Fig. 7. Joint probability density p(x,y) of the triple-well system
Fig. 8. Probability density p(x) and p(y) of the triple well system
Fig. 9. Asymmetric double-well potential energy
Fig. 10. Joint probability density p(x,y) of the double welled system
Fig. 11. Probability density p(x) and p(y) of the double well system
作者信息 | Authors
天津工业大学 (Tiangong University) 机械学院
葛根 Gen Ge, 通讯作者 (Corresp.)天津工业大学 (Tiangong University) 航空航天学院Email: gegen@tiangong.edu.cn
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)
