In science and engineering, the majority of non-linear stochastic systems can be represented as the quasi-Hamiltonian systems. Moreover, the corresponding Hamiltonian system offers two concepts of integrability and resonance that can fully describe the global relationship among the degrees-of-freedom (DOFs) of the system. This paper proposes an effective and promising approximate semi-analytical method for the stationary response of multi-dimensional quasi-Hamiltonian systems. To be specific, the trial solution of the reduced Fokker-Plank-Kolmogorov (FPK) equation constructs with radial basis function (RBF) neural networks. Then taking into account the residual generated by substituting the trial solution into the reduced FPK equation, a loss function is constructed by combining random sampling technique, and the unknown weight coefficients are optimized by minimizing the loss function through the Lagrange multiplier method. Moreover, an efficient sampling strategy is employed to promote the implementation of algorithms. Finally, two numerical examples are studied in detail, and all the semi-analytical solutions are in contrast with Monte Carlo simulations (MCS) results. The results indicate that the complex non-linear dynamic features of the system response can be captured through the proposed scheme accurately.

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Hysteresis widely exists in civil structures, and dissipates the mechanical energy of systems. Research on the random vibration of hysteretic systems, however, is still insufficient, particularly when the excitation is non-Gaussian. In this paper, the radial basis function (RBF) neural network (RBF-NN) method is adopted as a numerical method to investigate the random vibration of the Bouc-Wen hysteretic system under the Poisson white noise excitations. The solution to the reduced generalized Fokker-Planck-Kolmogorov (GFPK) equation is expressed in terms of the RBF-NNs with the Gaussian activation functions, whose weights are determined by minimizing the loss function of the reduced GFPK equation residual and constraint associated with the normalization condition. A steel fiber reinforced ceramsite concrete (SFRCC) column loaded by the Poisson white noise is studied as an example to illustrate the solution process. The effects of several important parameters of both the system and the excitation on the stochastic response are evaluated, and the obtained results are compared with those obtained by the Monte Carlo simulations (MCSs). The numerical results show that the RBF-NN method can accurately predict the stationary response with a considerable high computational efficiency.

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