Adam Bayesian Gaussian Process Regression with Combined Kernel-Function-Based Monte Carlo Reliability Analysis of Non-Circular Deep Soft Rock Tunnel
Abstract
:1. Introduction
2. Adam–Bayesian–GPR-Based MC Reliability Calculation Procedure Framework
2.1. Performance Function of Surrounding Rock Primary-Support Structure System
2.2. Gaussian Process Regression
2.3. Bayesian Inference
2.4. Bayesian Update of Geo-Mechanical Parameters
2.5. GPR-Based Relationship between Displacement and Geo-Mechanical Parameters
2.6. Combined Kernel Function Construction and Selection
2.7. Adam–Bayesian–GPR-Based MCS Reliability Analysis
3. Case Study
3.1. Problem Description
3.2. Performance Function and Random Variables
3.3. Kernel Function Selection
- (1)
- Data characteristics: The selection of kernel functions should take into account the characteristics of the data, such as dimensionality, linear separability, and the presence of noise.
- (2)
- Model performance: The selection of kernel functions should also be based on the performance requirements of the model. Evaluate the performance of different kernel functions through cross-validation and other methods, and select the kernel function that can provide the best predictive performance. For example, by comparing the training error and testing error of the model, a kernel function that can fit the data well while avoiding overfitting can be selected.
- (3)
- Computational resources: When choosing a kernel function, it is necessary to balance the complexity of the model and computational feasibility.
- (4)
- Parameter adjustments: Kernel functions typically have parameters that need to be adjusted through optimization processes to maximize model performance. Therefore, when selecting kernel functions, optimization algorithms for model hyperparameters should also be considered.
3.4. Results and Discussion
3.4.1. ABGPR–MCS Reliability Analysis Results
3.4.2. Comparison of Numerical Calculation and Surrogate Model Calculation Based on ABGPR, Kriging, and XGBoost for Monitoring Point Displacement
3.4.3. Comparison of Tunnel Reliability Based on ABGPR, Kriging, and XGBoost
- (1)
- ABGPR parameter values: constant kernel function (C) = 30.52, length scaling factor (l) = 0.0882.
- (2)
- The DACE [33] (Design and Analysis o I Computer Experiments) toolbox function [dmodel. perf] = dacefit (S, Y, regr, corr, theta) in MATLAB is used to construct Kriging surrogate model. Kriging parameter values: theta = 1.
- (3)
- XGBoost parameter values: max_depth = 1, learning_rate = 0.5, n_estimators = 4000, gamma = 0.
3.4.4. Comparison with the Polynomial Response Surface Method and ABGPR-MCS
3.4.5. Comprehensive System Reliability Index of Tunnel Structure
4. Conclusions
- A comparison among the prediction abilities of RBF, K1.5, K2.5, RBFK1.5, RBFK2.5, and CRBF indicates that the prediction ability of CRBF is the strongest. It is denoted that the prediction ability of the ABGP model can be greatly improved through the design of the combined kernel functions.
- A comparison between the reliability indices obtained from the proposed ABGPR-MCS method and the PRSM indicates that ABGPR-MCS is an efficient method for the probabilistic analyses of deep rock tunnel engineering. The most distinct advantage of the proposed method is that it can establish an optimized surrogate model for the numerical simulation calculation and obtain the mean value, variance, and confidence interval of geo-mechanical parameters.
- For deep rock tunnel structure systems with fewer failure elements, using the proposed ABGPR-MCS reliability analysis method and the system reliability theory can estimate its failure probability range and determine its comprehensive reliability index. The proposed method can quickly assess the chances of large deformation disasters in deep soft rock non-circular tunnels.
- From the system reliability calculation results obtained by using the proposed reliability method, it can be denoted that the support structure of the tunnel in this study is safe, which is consistent with the actual situation. For zones of the initial support structure of the tunnel with low reliability, the reliability and safety of the tunnel can be improved by strengthening the initial support measures.
- Using the ABGPR method to establish a surrogate model of the nonlinear mapping relationship between geo-mechanical parameters and their responses can significantly reduce the time needed for numerical simulation calculations. There are obvious advantages regarding the prediction error and the prediction ability of the proposed method (ABGPR) compared with Kriging and XGBoost.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
- Pan, Q.J.; Dias, D. An efficient reliability method combining adaptive support vector machine and Monte Carlo Simulation. Struct. Saf. 2017, 67, 85–95. [Google Scholar] [CrossRef]
- Feng, L.Y.; Zhang, L.M. Reliability–based multi–objective optimization in tunneling alignment under uncertainty. Struct. Multidisc. Optim. 2021, 63, 3007–3025. [Google Scholar] [CrossRef]
- Liu, W.; Chen, E.; Yao, E.; Wang, Y.; Chen, Y. Reliability analysis of face stability for tunnel excavation in a dependent system. Reliab. Eng. Syst. Saf. 2021, 206, 107306. [Google Scholar] [CrossRef]
- Xu, Z.Y.; Saleh, J.H. Machine learning for reliability engineering and safety applications: Review of current status and future opportunities. Reliab. Eng. Syst. Saf. 2021, 211, 107530. [Google Scholar] [CrossRef]
- Wang, Z.L.; Zhao, D.Y.; Guan, Y. Flexible–constrained time–variant hybrid reliability–based design optimization. Struct. Multidisc. Optim. 2023, 66, 89. [Google Scholar] [CrossRef]
- Wang, Q.; Fang, H.B. Reliability analysis of tunnels using an adaptive RBF and a first–order reliability method. Comput. Geotech. 2018, 98, 144–152. [Google Scholar] [CrossRef]
- Chakraborty, S.; Majumder, D. Hybrid reliability analysis framework for reliability analysis of tunnels. J. Comput. Civ. Eng. 2018, 32, 04018018. [Google Scholar] [CrossRef]
- da Silva, C.C.C.; Bernaud, D.; Real, M.D.; Maghous, S. Reliability analysis of bolt-supported tunnels regarded as homogenized structures. Int. J. Numer. Anal. Methods Geomech. 2023, 47, 2531–2561. [Google Scholar] [CrossRef]
- Do, D.P.; Vu, M.N.; Tran, N.T.; Armand, G. Closed-form solution and reliability analysis of deep tunnel supported by a concrete liner and a covered compressible layer within the viscoelastic Burger rock. Rock Mech. Rock Eng. 2021, 54, 2311–2334. [Google Scholar] [CrossRef]
- Wang, Q.; Huang, T.C. Time-variant reliability analysis of underground structures against delayed failure based on the probability density evolution method. Eng. Fail. Anal. 2022, 142, 106695. [Google Scholar] [CrossRef]
- Xu, J.C.; Sun, C.; Rui, G.R. NSGA–III–XGBoost–based stochastic reliability analysis of deep soft rock tunnel. Appl. Sci. 2024, 14, 2127. [Google Scholar] [CrossRef]
- Lü, Q.; Chan, C.L.; Low, B.K. System reliability assessment for a rock tunnel with multiple failure modes. Rock Mech. Rock Eng. 2013, 46, 821–833. [Google Scholar] [CrossRef]
- Zhao, H.B. High dimension model representation–based response surface for reliability analysis of tunnel. Math. Probl. Eng. 2018, 2018, 8049139. [Google Scholar] [CrossRef]
- Zhao, H.B.; Ru, Z.L.; Chang, X.; Yin, S.D.; Li, S.J. Reliability analysis of tunnel using least square support vector machine. Tunn. Undergr. Space Technol. 2014, 41, 14–23. [Google Scholar] [CrossRef]
- Zhou, S.Y.; Guo, X.F.; Zhang, Q.; Dias, D.; Pan, Q.J. Influence of a weak layer on the tunnel face stability—Reliability and sensitivity analysis. Comput. Geotech. 2020, 122, 103507. [Google Scholar] [CrossRef]
- Wang, Q.; Fang, H.B.; Shen, L. Reliability analysis of tunnels using a metamodeling technique based on augmented radial basis functions. Tunn. Undergr. Space Technol. 2016, 56, 45–53. [Google Scholar] [CrossRef]
- Feng, L.Y.; Zhang, L.M. Assessment of tunnel face stability subjected to an adjacent tunnel. Reliab. Eng. Syst. Saf. 2021, 205, 107228. [Google Scholar] [CrossRef]
- Xu, J.C.; Jiang, Y.; Yang, C.B. Landslide displacement prediction during the sliding process using XGBoost, SVR and RNNs. Appl. Sci. 2022, 12, 6056. [Google Scholar] [CrossRef]
- Yan, Y.H.; Li, X.R.; Sun, W.J.; Fang, X.; He, F.J.; Tu, J.Y. Semi–surrogate modelling of droplets evaporation process via XGBoost integrated CFD simulations. Sci. Total Environ. 2023, 895, 164968. [Google Scholar] [CrossRef]
- Wang, Y.; Gao, Y.; Liu, Y. Bayesian–entropy gaussian process for constrained metamodeling. Reliab. Eng. Syst. Saf. 2021, 214, 107762. [Google Scholar] [CrossRef]
- Li, T.Z.; Dias, D. Tunnel face reliability analysis using active learning Kriging model–Case of a two–layer soils. J. Cent. South Univ. 2019, 26, 1735–1746. [Google Scholar] [CrossRef]
- Langford, J.C.; Diederichs, M.S. Reliability based approach to tunnel lining design using a modified point estimate method. Int. J. Rock Mech. Min. 2013, 60, 263–276. [Google Scholar] [CrossRef]
- Jerez, D.J.; Jensen, H.A.; Beer, M. An effective implementation of reliability methods for Bayesian model updating of structural dynamic models with multiple uncertain parameters. Reliab. Eng. Syst. Saf. 2022, 225, 108634. [Google Scholar] [CrossRef]
- Liu, H.; Low, B.K. System reliability analysis of tunnels reinforced by rockbolts. Tunn. Undergr. Space Technol. 2017, 65, 155–166. [Google Scholar] [CrossRef]
- Ji, J.; Zhang, Z.; Wu, Z. An efficient probabilistic design approach for tunnel face stability by inverse reliability analysis. Geosci. Front. 2021, 12, 380–389. [Google Scholar] [CrossRef]
- Hamdia, K.M.; Ghasemi, H. Reliability analysis of the stress intensity factor using multilevel Monte Carlo methods. Probabilistic Eng. Mech. 2023, 74, 103497. [Google Scholar] [CrossRef]
- Sadr, M.; Torrilhon, M.; Gorji, M.H. Gaussian process regression for maximum entropy distribution. J. Comput. Phys. 2020, 418, 109644. [Google Scholar] [CrossRef]
- Kingma, D.P.; Ba, J.L. Adam: A method for stochastic optimization. In Proceedings of the 3rd International Conference on Learning Representations, San Diego, CA, USA, 7–9 May 2015. [Google Scholar]
- Contreras, L.F.; Brown, E.T. Slope reliability and back analysis of failure with geotechnical parameters estimated using Bayesian inference. J. Rock Mech. Geotech. Eng. 2019, 11, 628–643. [Google Scholar] [CrossRef]
- Rasmussen, C.E.; Williams, C.K.I. Gaussian Processes for Machine Learning; MIT Press: Cambridge, MA, USA, 2006. [Google Scholar]
- Fitzgerald, D.L. Statistical aspects of Tricomi’s function and modified Bessel functions of the second kind. Stoch. Environ. Res. Risk Assess. 2000, 14, 139–158. [Google Scholar] [CrossRef]
- Griffiths, D.; Fenton, G. Risk Assessment in Geotechnical Engineering; John Wiley & Sons: Hoboken, NJ, USA, 2008. [Google Scholar]
- Lophaven, S.N.; Nielsen, H.B.; Sondergaard, J. DACE: A MATLAB Kriging Toolbox, Tech. Rep. IMM-TR-2002-12; Technical University of Denmark: Kongens Lyngby, Denmark, 2002. [Google Scholar]
- Gunst, R. Response surface methodology: Process and product optimization using designed experiments. Technometrics 2008, 38, 284–286. [Google Scholar] [CrossRef]
Measuring Points/Lines | A | H | AB | AC | BC | AD | AE | DE | FG |
---|---|---|---|---|---|---|---|---|---|
Monitoring displacement/cm | –9.74 | 8.43 | –6.44 | –6.44 | –9.99 | –9.95 | –9.96 | –12.59 | –10.43 |
Random Variables | Distribution | COV | ||
---|---|---|---|---|
Normal | 0.33 | 0.033 | 0.1 | |
/° | Normal | 33 | 4.95 | 0.15 |
/MPa | Normal | 350 | 50.5 | 0.144 |
/kPa | Normal | 210 | 23 | 0.11 |
Parameters | /GPa | /MPa | /° | |
---|---|---|---|---|
Mean | 0.3306 | 0.338 | 0.1896 | 32.795 |
Standard deviation | 0.001875 | 0.001 | 0.001775 | 0.17575 |
Confidence interval of 95% | [0.3269, 0.3344] | [0.336, 0.340] | [0.1860, 0.1931] | [32.444, 33.147] |
Iteration Number | /GPa | /MPa | |||
---|---|---|---|---|---|
0 | 0.3306 | 0.338 | 0.1896 | 32.795 | |
1 | 0.3271 | 0.338 | 0.1874 | 32.483 | 2.87 |
2 | 0.3288 | 0.338 | 0.1864 | 32.424 | 2.94 |
3 | 0.3281 | 0.338 | 0.1864 | 32.507 | 2.77 |
4 | 0.3291 | 0.338 | 0.1860 | 32.484 | 2.80 |
5 | 0.3287 | 0.338 | 0.1863 | 32.491 | 2.77 |
6 | 0.3288 | 0.338 | 0.1863 | 32.514 | 2.64 |
7 | 0.3284 | 0.338 | 0.1867 | 32.560 | 2.40 |
8 | 0.3278 | 0.338 | 0.1873 | 32.480 | 2.68 |
9 | 0.3288 | 0.338 | 0.1864 | 32.484 | 2.71 |
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |
© 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Xu, J.; Yan, Z.; Wang, Y. Adam Bayesian Gaussian Process Regression with Combined Kernel-Function-Based Monte Carlo Reliability Analysis of Non-Circular Deep Soft Rock Tunnel. Appl. Sci. 2024, 14, 7886. https://doi.org/10.3390/app14177886
Xu J, Yan Z, Wang Y. Adam Bayesian Gaussian Process Regression with Combined Kernel-Function-Based Monte Carlo Reliability Analysis of Non-Circular Deep Soft Rock Tunnel. Applied Sciences. 2024; 14(17):7886. https://doi.org/10.3390/app14177886
Chicago/Turabian StyleXu, Jiancong, Ziteng Yan, and Yongshuai Wang. 2024. "Adam Bayesian Gaussian Process Regression with Combined Kernel-Function-Based Monte Carlo Reliability Analysis of Non-Circular Deep Soft Rock Tunnel" Applied Sciences 14, no. 17: 7886. https://doi.org/10.3390/app14177886