Skip to main navigation menu Skip to main content Skip to site footer


Vol. 10 (2023)

Small Sample Data-Driven Method for Interval Prediction on Structural Responses

May 24, 2023


Abstract: This paper proposes an approach of intervals prediction on the structural responses under the limited samples by using Grey-BP neural networks (GBP) and genetic algorithm (GA). For the response prediction of complex structures, the presented method can ensure the necessary accuracy and can greatly reduce the cost of computation. Meanwhile, the method is different from the other traditional neural networks, which represents the superiority of dealing with the interval prediction under limited samples with help of the grey theory. In this paper, the Grey-BP neural networks are established by introducing the grey theory, which is used to achieve the mapping relationship between input and output of the system and accomplish the approximation of real mapping. The accuracy of built the mapping model can be tested by error analysis. Subsequently, the issue of interval prediction can be translated to optimal problem for extremum value. Although the explicit expression of the established mapping relationship is unknown, the fitness of every value includes the information of extremum. So, the best fitness and worst fitness can be searched in the given bound of uncertain variables based on genetic algorithm, and then achieve the upper bound and lower bound of the system response depending on the capability of global search. After proposed technologies are given in detail, one numerical example and one engineering example are presented and the results are discussed with common methods based on traditional BP neural network and Monte Carlo, which demonstrates the validity and reasonability of the developed methodology.


  1. Gameel, M.S. and K. El-Geziry, Predicting Financial Distress: Multi Scenarios Modeling Using Neural Network. International Journal of Economics & Finance, 2016. 8(11): p. 159.
  2. Xu, Q., et al., Nonparametric conditional autoregressive expectile model via neural network with applications to estimating financial risk. Applied Stochastic Models in Business & Industry, 2016. 32(6): p. 882-908.
  3. Baklacioglu, T., Modeling the fuel flow-rate of transport aircraft during flight phases using genetic algorithm-optimized neural networks. Aerospace Science & Technology, 2016. 49(3): p. 52-62.
  4. Hashemi, R.R., et al., A neural network for transportation safety modeling. Expert Systems with Applications, 1995. 9(3): p. 247-256.
  5. Vlahogianni, E.I. and M.G. Karlaftis, Testing and Comparing Neural Network and Statistical Approaches for Predicting Transportation Time Series. Transportation Research Record Journal of the Transportation Research Board, 2013. 2399(2399): p. 9-22.
  6. Smith, A.E., D.W. Coit, and Y.C. Liang, Neural Network Models to Anticipate Failures of Airport Ground Transportation Vehicle Doors. IEEE Transactions on Automation Science & Engineering, 2009. 7(1): p. 183-188.
  7. Elhewy, A.H., E. Mesbahi, and Y. Pu, Reliability Analysis of Structure Using Neural Network method. Probabilistic Engineering Mechanics, 2006. 21(1): p. 44-53.
  8. Gomes, H.M. and A.M. Awruch, Comparison of response surface and neural network with other methods for structural reliability analysis. Structural Safety, 2004. 26(1): p. 49-67.
  9. Kao, C.Y. and S.L. Hung, Detection of structural damage via free vibration responses generated by approximating artificial neural networks. Computers & Structures, 2003. 81(28): p. 2631-2644.
  10. Xuan, S.N., et al., Adaptive response surface method based on a double weighted regression technique. Probabilistic Engineering Mechanics, 2009. 24(2): p. 135-143.
  11. Gavin, H.P. and S.C. Yau, High-order limit state functions in the response surface method for structural reliability analysis. Structural Safety, 2008. 30(2): p. 162-179.
  12. Wong, S.M., R.E. Hobbs, and C. Onof, An adaptive response surface method for reliability analysis of structures with multiple loading sequences. Structural Safety, 2005. 27(4): p. 287-308.
  13. Alibrandi, U., N. Impollonia, and G. Ricciardi, Probabilistic eigenvalue buckling analysis solved through the ratio of polynomial response surface. Computer Methods in Applied Mechanics & Engineering, 2010. 199(9): p. 450-464.
  14. Zheng, Y. and P.K. Das, Improved response surface method and its application to stiffened plate reliability analysis. Engineering Structures, 2000. 22(5): p. 544-551.
  15. Allaix, D.L. and V.I. Carbone, An improvement of the response surface method. Structural Safety, 2011. 33(2): p. 165-172.
  16. Basaga, H.B., A. Bayraktar, and I. Kaymaz, An improved response surface method for reliability analysis of structures. Structural Engineering & Mechanics, 2012. 42(2): p. 175-189.
  17. Gupta, S. and C.S. Manohar, An improved response surface method for the determination of failure probability and importance measures. Structural Safety, 2004. 26(2): p. 123-139.
  18. Kim, C., Reliability-Based Design Optimization Using Response Surface Method With Prediction Interval Estimation. Journal of Mechanical Design, 2008. 130(12): p. 121401.
  19. Kilickap, E., Modeling and optimization of burr height in drilling of Al-7075 using Taguchi method and response surface methodology. International Journal of Advanced Manufacturing Technology, 2010. 49(9-12): p. 911-923.
  21. Zhao, W. and Z. Qiu, An efficient response surface method and its application to structural reliability and reliability-basedoptimization. Finite Elements in Analysis & Design, 2013. 67(5): p. 34-42.
  22. Goel, T., et al., Response surface approximation of Pareto optimal front in multi-objective optimization. Computer Methods in Applied Mechanics & Engineering, 2007. 196(4): p. 879-893.
  23. Lefik, M. and B.A. Schrefler, Artificial neural network as an incremental non-linear constitutive model for a finite element code. Computer Methods in Applied Mechanics & Engineering, 2003. 192(28): p. 3265-3283.
  24. Patnaik, S.N., J.D. Guptill, and D.A. Hopkins, Subproblem optimization with regression and neural network approximators. Computer Methods in Applied Mechanics & Engineering, 2003. 194(30): p. 3359-3373.
  25. Chakraverty, S., V.P. Singh, and R.K. Sharma, Regression based weight generation algorithm in neural network for estimation of frequencies of vibrating plates. Computer Methods in Applied Mechanics & Engineering, 2006. 195(33-36): p. 4194-4202.
  26. Chokshi, P., R. Dashwood, and D.J. Hughes, Artificial Neural Network (ANN) based microstructural prediction model for 22MnB5 boron steel during tailored hot stamping. Computers & Structures, 2017. 190: p. 162-172.
  27. Biswas, M.A.R., M.D. Robinson, and N. Fumo, Prediction of residential building energy consumption: A neural network approach. Energy, 2016. 117: p. 84-92.
  28. Li, P., et al., Neural network prediction of flow stress of Ti-15-3 alloy under hot compression. Journal of Materials Processing Tech, 2004. 148(2): p. 235-238.
  29. Fu, Z., et al., Using genetic algorithm-back propagation neural network prediction and finite-element model simulation to optimize the process of multiple-step incremental air-bending forming of sheet metal. Materials & Design, 2010. 31(1): p. 267-277.
  30. Escrivá-Escrivá, G., et al., New artificial neural network prediction method for electrical consumption forecasting based on building end-uses. Energy & Buildings, 2011. 43(11): p. 3112-3119.
  31. Mazloumi, E., et al., Prediction intervals to account for uncertainties in neural network predictions: Methodology and application in bus travel time prediction. Engineering Applications of Artificial Intelligence, 2011. 24(3): p. 534-542.
  32. Weerdt, E.D., Q.P. Chu, and J.A. Mulder, Neural Network Output Optimization Using Interval Analysis. IEEE Transactions on Neural Networks, 2009. 20(4): p. 638-53.
  33. Pierce, S.G., K. Worden, and A. Bezazi, Uncertainty analysis of a neural network used for fatigue lifetime prediction. Mechanical Systems & Signal Processing, 2008. 22(6): p. 1395-1411.
  34. Khosravi, A., et al., Lower upper bound estimation method for construction of neural network-based prediction intervals. IEEE Transactions on Neural Networks, 2011. 22(3): p. 337.
  35. Pai, T.Y., et al., Grey and neural network prediction of suspended solids and chemical oxygen demand in hospital wastewater treatment plant effluent. Computers & Chemical Engineering, 2007. 31(10): p. 1272-1281.
  36. Liu, C., et al., An improved grey neural network model for predicting transportation disruptions. Expert Systems with Applications An International Journal, 2016. 45(C): p. 331-340.
  37. Liua, X., B.M. Cuartas, and A.S.G. Muñiz, A Grey neural network and input-output combined forecasting model. Primary energy consumption forecasts in Spanish Economic sectors. Energy, 2016. 115: p. 1042-1054.
  38. Abdulshahed, A.M., et al., Thermal error modelling of a gantry-type 5-axis machine tool using a Grey Neural Network Model. Journal of Manufacturing Systems, 2016. 41: p. 130-142.
  39. Yang, Y., et al., Research of on-line noise source identification based on the Grey neural network. International Journal of Acoustics & Vibration, 2008. 13(4): p. 144-150.
  40. Ming, J. and B. Zuo, Prediction of mechanical properties of aging B.mori silk fabric based on grey neural network model. Fibers & Polymers, 2012. 13(5): p. 653-657.
  41. Deng, J.L., Control problems of grey systems. Systems & Control Letters, 1982. 1(5): p. 288-294.