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Modelling the mechanical behaviour of soils using machine learning algorithms with explicit formulations

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Abstract

This study systematically presents the application of machine learning (ML) algorithms for constructing a constitutive model for soils. A genetic algorithm is integrated with ML algorithms to determine the global optimum model, and the k-fold cross-validation method is used to enhance the models’ robustness. Three typical ML algorithms with formulations explicitly expressed [i.e., back-propagation neural network (BPNN), extreme learning machine (ELM) and evolutionary polynomial regression (EPR)], and two modelling strategies (i.e. total or incremental stress–strain strategies) are used. A synthetic database is first generated based on a simple constitutive model to objectively evaluate the performance of three ML algorithms and two modelling strategies. Next, the optimum ML algorithm and the well evaluated modelling strategy are applied to experimental tests for examining its robustness. All results indicate that a BPNN-based constitutive model using the incremental stress–strain strategy performs best in modelling the mechanical behaviour of soils in terms of interpolation and extrapolation abilities, followed by ELM and then EPR.

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Acknowledgements

This research was financially supported by the Research Grants Council (RGC) of Hong Kong Special Administrative Region Government (HKSARG) of China (Grant No.: 15209119, R5037-18F); Joint research project between SiChuan Province and National Universities funded by Science & Technology Department of Sichuan Province (No.2019YFSY0015), and Open research grant of MOE Key Laboratory of High-speed Railway Engineering.

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  1. Department of Civil and Environmental Engineering, The Hong Kong Polytechnic University, Hung Hom, Hong Kong, Kowloon, China

    Pin Zhang, Zhen-Yu Yin & Yin-Fu Jin

  2. Key Laboratory of High-Speed Railway Engineering of Ministry of Education, School of Civil Engineering, Southwest Jiaotong University, Chengdu, 610031, China

    Xian-Feng Liu

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Appendix A. Formulations of BPNN-based Kaolinite constitutive models

Appendix A. Formulations of BPNN-based Kaolinite constitutive models

$${\mathbf{H}} = f\left( {{\mathbf{WX}} + {\varvec\theta }} \right)$$
(1A)
$${\mathbf{O}} = g\left( {{\mathbf{VH}} + {\varvec\theta }_{{\text{o}}} } \right)$$
(2A)

where, X = [p, q, e, ε1, Δε1], matrix of input variables; H = matrix of the hidden layer output; O = [q, e], matrix of output variables; f = tansig formulation; g = purlin formulation. Herein,

$${\mathbf{W}} = \left[ \begin{gathered} - 1.35315 \, 1.907756 \, 0.915302 \, - 2.02378 \, - 2.16087 \hfill \\ 2.595596 \, - 0.32869 \, - 2.48042 \, 2.352501 \, - 2.32131 \hfill \\ - 1.36652 \, 0.169634 \, 3.622397 \, - 2.44086 \, - 3.21463 \hfill \\ - 0.72626 \, - 0.01880 \, - 0.22374 \, 0.908566 \, 1.41438 \hfill \\ - 0.02852 \, 0.013743 \, 0.634650 \, - 0.57525 \, - 0.11921 \hfill \\ - 0.13418 \, 0.036866 \, 0.933846 \, - 1.36281 \, - 1.07577 \hfill \\ - 0.19825 \, - 0.51040 \, - 6.04189 \, - 2.26296 \, - 1.74243 \hfill \\ - 0.19073 \, 0.055012 \, 1.177738 \, - 0.69086 \, - 2.03872 \hfill \\ \end{gathered} \right]$$
$${\varvec\theta } = \left[ {2.077058; \, 0.827601; \, 0.480095; \, - 0.95727; \, - 0.88088; \, - 1.21919; \, 0.862033; \, - 2.24696} \right]$$
$${\mathbf{V}} = \left[ \begin{gathered} - 0.1350 \, - 0.04501 \, - 2.23939 \, - 1.11558 \, 1.894278 \, - 1.86286 \, - 2.12292 \, 1.08516 \hfill \\ 0.065208 \, - 0.02499 \, - 0.12495{ 0}{\text{.603675 0}}{.853076 } - 0.0923 \, - 0.06472 \, - 0.63716 \hfill \\ \end{gathered} \right]$$
$${\varvec\theta }_{{\text{o}}} = \left[ {0.31; \, 0.113215} \right]$$

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Zhang, P., Yin, ZY., Jin, YF. et al. Modelling the mechanical behaviour of soils using machine learning algorithms with explicit formulations. Acta Geotech. 17, 1403–1422 (2022). https://doi.org/10.1007/s11440-021-01170-4

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