A method for predicting random vibration response of train-track-bridge system based on GA-BP neural network

Abstract

To enhance the efficiency of stochastic vibration analysis for the Train-Track-Bridge (TTB) coupled system, this paper proposes a prediction method based on a Genetic Algorithm-optimized Backpropagation (GA-BP) neural network. First, initial track irregularity samples and random parameter sets of the Vehicle–Bridge System (VBS) are generated using the stochastic harmonic function method. Then, the stochastic dynamic responses corresponding to the sample sets are calculated using a developed stochastic vibration analysis model of the TTB system. The track irregularity data and vehicle–bridge random parameters are used as input variables, while the corresponding stochastic responses serve as output variables for training the BP neural network to construct the prediction model. Subsequently, the Genetic Algorithm (GA) is applied to optimize the BP neural network by considering the randomness in excitation and parameters of the TTB system, improving model accuracy. After optimization, the trained GA-BP model enables rapid and accurate prediction of vehicle–bridge responses. To validate the proposed method, predictions of vehicle–bridge responses under varying train speeds are compared with numerical simulation results. The findings demonstrate that the proposed method offers notable advantages in predicting the stochastic vibration response of high-speed railway TTB coupled systems.

Keywords

Train-track-bridge system / Genetic algorithm / BP neural network / Random response prediction / Random parameters

CRediT authorship contribution statement

Zhiwu Yu: Investigation. Mansoor Khan: Investigation. Li Zheng: Writing – review & editing, Writing – original draft, Software, Resources, Methodology, Data curation, Conceptualization. Yun Zhang: Investigation. Jianfeng Mao: Writing – review & editing, Writing – original draft, Resources, Methodology, Investigation, Funding acquisition, Formal analysis, Data curation, Conceptualization.

Declaration of Competing Interest

The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: Zhiwu Yu is currently employed by China Railway Group Limited.

Acknowledgements

This research is supported by the China State Railway Group Co., Ltd. Science and Technology Research and Development Program Project (Grant No. L2024G007); the Natural Science Foundation of Hunan Province (Grant No. 2024JJ5427); the National Natural Science Foundation of China (Grant No. 52478321, 52078485); the Science and Technology Research and Development Program Project of China Railway Group Limited (Grant No. 2021-Special-08, 2022-Key-06 & 2023-Key-22).

References

[1] N. Zhang, H. Xia, W.W. Guo, Vehicle-bridge interaction analysis under high-speed trains, J. Sound Vib. 309(3)(2007) 407-425.

[2] X.P. Cai, W.L. Liu, L. Gao, et al., Mechanical deformation properties of continuous welded rail on kilometer-span suspension bridge for high-speed railway, High-speed Railw. 1(2)(2023) 97-109.

[3] W.M. Zhai, Z.L. Han, Z.W. Chen, et al., Train-track-bridge dynamic interaction: A state-of-the-art review, Veh. Syst. Dyn. 57(7)(2019) 984-1027.

[4] H. Xia, N. Zhang, W.W. Guo, Analysis of resonance mechanism and conditions of train-bridge system, J. Sound Vib. 297(3-5)(2006) 810-822.

[5] Y.B. Yang, J. Li, 2.5D prediction of soil vibrations due to railway loads by the isogeometric analysis with scaled boundary, Eng. Anal. Bound. Elem. 134(2022) 341-359.

[6] B. Enright, E.J. O, Monte Carlo simulation of extreme traffic loading on short and medium span bridges, Struct. Infrastruct. Eng. 9(12)(2013) 1267-1282.

[7] F.TK. Au, J.J. Wang, Y.K. Cheung, Impact study of cable-stayed railway bridges with random rail irregularities, Eng. Struct. 24(5)(2002) 529-541.

[8] J.Q. Sun, Random vibration analysis of time-delayed dynamical systems, Probabilistic Eng. Mech. 29(2012) 1-6.

[9] T. Janesupasaeree, V. Boonyapinyo, Determination of flutter derivatives of bridge decks by covariance-driven stochastic subspace identification, Int. J. Struct. Stab. Dyn. 11(1)(2011) 73-99.

[10] Z.P. Zeng, X.F. He, Y.G. Zhao, et al., Random vibration analysis of train-slab track-bridge coupling system under earthquakes, Struct. Eng. Mech. 54(5)(2015) 1017-1044.

[11] Z.C. Zhang, J.H. Lin, Y.H. Zhang, et al., Non-stationary random vibration analysis of three-dimensional train-bridge systems, Veh. Syst. Dyn. 48(4)(2010) 457-480.

[12] Y. Cheng, G.S. Fang, L. Zhao, et al., Uncertainty propagation of flutter analysis for long-span bridges using probability density evolution method, Reliab. Eng. Syst. Saf. 251(2024) 110361.

[13] Z.W. Yu, J.F. Mao, F.Q. Guo, et al., Non-stationary random vibration analysis of a 3D train-bridge system using the probability density evolution method, J. Sound Vib. 366(2016) 173-189.

[14] J. Li, J.B. Chen, W.L. Sun, et al., Advances of the probability density evolution method for nonlinear stochastic systems, Probabilistic Eng. Mech. 28(2012) 132-142.

[15] J.F. Mao, Y.J. Xiao, Z.W. Yu, et al., Probabilistic model and analysis of coupled train-ballasted track-subgrade system with uncertain structural parameters, J. Cent. South Univ. 28(7)(2021) 2238-2256.

[16] L. Xin, X. Li, Y. Zhu, et al., Uncertainty and sensitivity analysis for train-ballasted track–bridge system, Veh. Syst. Dyn. 58(3)(2020) 453-471.

[17] X. Xiao, Y. Yan, B. Chen, Stochastic dynamic analysis for vehicle-track-bridge system based on probability density evolution method, Eng. Struct. 188(2019) 745-761.

[18] Z. Xun, W. Lidong, H. Yan, et al., An efficient method for predicting wheel-rail forces in coupled nonlinear train-track-bridge system using artificial neural networks, Adv. Struct. Eng. 26(7)(2023) 1228-1241.

[19] H.L. Li, T.Y. Wang, G. Wu, A Bayesian deep learning approach for random vibration analysis of bridges subjected to vehicle dynamic interaction, Mech. Syst. Signal Process., 170(2022), https://doi.org/10.1590/1679-78256238.

[20] R. Wu, M.R. Jahanshahi, Deep convolutional neural network for structural dynamic response estimation and system identification, J. Eng. Mech., 145(1)(2019).

[21] T. Kim, O. Kwon, J. Song, Response prediction of nonlinear hysteretic systems by deep neural networks, Neural Netw. 111(2018) 1-10.

[22] X. Han, H. Xiang, Y. Li, et al., Predictions of vertical train-bridge response using artificial neural network-based surrogate model, Adv. Struct. Eng. 22(12)(2019) 2712-2723.

[23] Z.J. Shao, P. Zhang, X.N. Xie, et al., A new three-dimensional model of train-track-bridge coupled system based on meshless method and its graph neural network-based surrogate model, Comput. Struct. 315(2025) 107786.

[24] H.L. Li, T.Y. Wang, G. Wu, Dynamic response prediction of vehicle-bridge interaction system using feedforward neural network and deep long short-term memory network, Structure. 34(2021) 2415-2431.

[25] X. Zhang, L.D. Wang, Y. Han, et al., An efficient method for predicting wheel-rail forces in coupled nonlinear train-track-bridge system using artificial neural networks, Adv. Struct. Eng. 26(7)(2023) 1228-1241.

[26] J. Mao, Z. Li, Z. Yu, et al., A novel hybrid approach combining PDEM and Bayesian optimization deep learning for stochastic vibration analysis in train-track-bridge coupled system, Reliab. Eng. Syst. Saf. 257(2025) 110827.

[27] T. Kim, O. Kwon, J. Song, Response prediction of nonlinear hysteretic systems by deep neural networks, Neural Netw. 111(2018) 1-10.

[28] Q. Zhang, X.P. Cai, Y.L. Zhong, et al., Dynamic response prediction of high-speed train on cable-stayed bridge based on genetic algorithm and fused neural networks, Eng. Struct. 306(2024) 117869.

[29] H.L. Li, G. Wu, M.D. Cui, A machine learning based approach for efficient safety evaluation of the high speed train and short span bridge system, Lat. Am. J. Solids Struct. 17(7)(2020), https://doi.org/10.1590/1679-78256238.

[30] S.S. Liu, Q.X. He, W.J. Fu, et al., A two-step data-driven method for predicting the wear of train wheel treads using GA-BPNN and LSTM algorithms, Int. J. Rail Transp. 13(1)(2025) 85-102.

[31] S.M. Motsa, G.E. Stavroulakis, G.A. Drosopoulos, A data-driven, machine learning scheme used to predict the structural response of masonry arches, Eng. Struct. 296(2023) 116912.

[32] H. Wang, S.H. Wang, R.G. Yang, et al., An optimized dynamic model improved deep discriminative transfer learning network for fault detection in rotation vector reducers, Reliab. Eng. Syst. Saf. 251(2024) 110331.

[33] L.V. Jospin, H. Laga, F. Boussaid, et al., Hands-on bayesian neural networks-a tutorial for deep learning users, IEEE Comput. Intell. Mag. 17(2)(2022) 29-48.

[34] Y. Wu, S. Lan, T.T. Liu, Research on engineering cost prediction based on GA-BP neural network, Int. J. Bus. Intell. Data Min. 26(3–4)(2025) 362-381.

[35] L.L. Pang, B. Xia, Z.F. Cheng, et al., Research on IoT network performance prediction model of power grid warehouse based on nonlinear GA-BP neural network, Nonlinear Eng. 14(1)(2025) 20240061.

[36] P.F. Jie, Y. Zhou, Z.J. Zhang, et al., Heating energy consumption prediction based on improved GA-BP neural network model, Energ. 328(2025) 136392.

[37] X. Yang, H. Ren, H. Zhao, Application of BP dual network model considering internal and external factors in short-term load forecasting, J. Phys.: Conf. Ser. 1631(2020) 012137.

[38] J.H. Holland, Building blocks, cohort genetic algorithms, and hyperplane-defined functions, Evolut. Comput. 8(4)(2000) 373-391.

[39] L. Wang, X.G. Wu, H.Y. Chen, et al., Multi-objective optimization of building energy consumption based on random forest and genetic algorithm. The 8th International Symposium on Project Management, China (ISPM2020), Beijing, 2020.

[40] B. Liu, Y. Wang, P. Hu, et al., Impact coefficient and reliability of mid-span continuous beam bridge under action of extra heavy vehicle with low speed, J. Cent. South Univ. 22(4)(2015) 1510-1520.

[41] J. Chen, F. Kong, Y. Peng, A stochastic harmonic function representation for non-stationary stochastic processes, Mech. Syst. Signal Process. 96(2017) 31-44.

[42] J. Li, J.B. Chen, Stochastic Dynamics of Structures, John Wiley & Sons, New York, 2009.

[43] Y. Liu, J. Dai, G. Chen, et al., Integrating signal pairing evaluation metrics with deep learning for wind power forecasting through coupled multiple modal decomposition and aggregation, Knowl. Based Syst. 317(2025) 113394.