Gaussian Process-Based Response Surface Method for Slope Reliability Analysis

School of Resource and Environmental Engineering, Wuhan University of Science and Technology, P.O. Box 430081, Wuhan, China Key Laboratory of Disaster Prevention and Structural Safety of Ministry of Education, School of Civil and Architecture Engineering, Guangxi University, Nanning, Guangxi 530004, China Key Laboratory of Ministry of Education on Safe Mining of Deep Metal Mines, Northeastern University, Shenyang 110819, China


Introduction
Slope stability analysis is an important issue in geotechnical engineering.e application of probabilistic concepts to the slope stability analysis has been pursued over the past decades.On real slope engineering, slope reliability can be obtained via an implicit limit state function with a numerical procedure employing the limit equilibrium methods [1,2] or finite element method [3,4].e classical reliability analysis methods including first-order reliability method (FORM) and secondorder reliability method (SORM) [5][6][7][8] that require limit state function gradients with respect to the basic variables may encounter great difficulties when direct or analytical differential is not possible.e Monte Carlo simulation (MCS) can handle the implicit performance function.However, the MCS was notorious for its unendurable computational cost.
e response surface method (RSM), which usually employs a polynomial function to approximate the unknown implicit performance function, has been used to the slope reliability analysis with the implicit performance function [4,[9][10][11].However, the RSM is a rigidly nonadaptive regression technique in the statistical learning perspective [12].Consequently, the RSM will become computationally impractical for problems involving many random variables, particularly when the involved random variables are mixed or statistically dependent.In addition, there is no guarantee that the fitted surface is a sufficiently close fit in all regions of interest.
In recent decades, machine learning methods such as artificial neural networks (ANN) [13][14][15][16][17], radial basis function network (RBFN) [18], and support vector machine (SVM) [19,20] have been widely used for the reliability analysis.ese methods can describe nonlinear and complex interactions among variables in a system and are therefore more suitable for solving reliability problems with highly nonlinear implicit performance.However, when these machine learning methods are used, some difficulties will be encountered.For ANN, it is always difficult to obtain an appropriate network topology and the optimum hyperparameters.In addition, the ANN has some inherent drawbacks such as limitation in solving the small sample problem [21].e SVM cannot avoid the blindness which is the common phenomenon in the artificial choice of the kernel function and its hyperparameter [21].Hence, it is desirable to develop an efficient framework for slope reliability analysis.
e Gaussian process (GP), based on statistical learning theory and Bayesian theory, is a newly developed machine learning technology [22].In recent years, the GP has gained considerable attention in the machine learning community and has been successfully applied in solving nonlinear, small samples, and high-dimensional problems [23][24][25][26][27][28].e GP has many advantages, such as it does not require a predefined structure, can approximate arbitrary function landscapes including discontinuities and multimodality, has meaningful hyperparameters, and includes a theoretical framework for obtaining the optimum hyperparameter selfadaptively.Furthermore, GP provides an uncertainty measure in the form of a standard deviation for predicted function values.
In the present study, a new GP-based RSM for slope reliability analysis was put forward by combining the GP with the FORM. is paper is constructed as follows.e used machine learning technique, namely, Gaussian process regression, is briefly described in Section 2. e GP-based RSM method is presented in detail in Section 3. Finally, four case studies are used to verify the flexibility and efficiency of the proposed method.

Gaussian Process
A Gaussian process is a collection of random variables, any finite set of which has a joint Gaussian distribution.A Gaussian process can be defined by its mean function m(x) and the covariance function k(x, x ′ ) as (1) , where x is an input vector with n dimensions and y is a scalar output or target.By using Bayesian forecasting, the distribution p(y * | x * , D) of output y * given a test input x * and a set of training points D can be calculated.Using the Bayesian rule, the posterior distribution for the Gaussian process outputs y * can be obtained.By conditioning on the observed targets in the training set, the predictive distribution is Gaussian: where the mean and variance are defined by where the compact forms of the notation setting for the matrix of the covariance functions are k * � K(X, x * ), K � K(X, X), σ 2 n is the unknown variance of the Gaussian noise.
Assuming α � (K + σ 2 n I) −1 y, equation (3) can be seen as a linear combination of m covariance functions, each one centered on a training point, by writing e Gaussian process procedure can handle interesting models by simply using a covariance function with an exponential term: where l denotes the characteristic length scale, σ 2 f denotes the signal variance, and δ pq denotes a Kronecker delta. is function expresses the idea the cases with nearby inputs will have highly correlated outputs.
e GP employs a set of hyperparameters θ including the length scale l, the signal variance σ 2 f , and the noise variance σ 2 n .e hyperparameters θ can be optimized based on the log-likelihood framework: e log-likelihood and its derivative with respect to θ can be described as where C � K + σ 2 n I. Hyperparameters θ are initialized to random values in a reasonable range and then optimized using an iterative, such as the conjugate gradient.
More detailed theory of the GP can be found in the literature [24].

Explicit Formation of Reliability Index Using GP.
e performance function of a slope can be built as follows: where X denotes the random variables with mean μ x and standard deviation σ x .g(X) > 0 indicates a stable slope, while g(X) < 0 indicates a failed slope.g(X) � 0 indicates that the slope is hovering between stable and unstable.F s is the safety factor.
Assuming that there is a point x * � (x * 1 , x * 2 , . . ., x * n ) T located on the limit state surface, which is called as the design point.Here, n is the dimension of x * .g(x) is generally a nonlinear function.However, it can be linearized at x * by neglecting the second-or upper-order term: 2 Advances in Civil Engineering Assuming x is statistically uncorrelated, the mean value and standard deviation of Z Q are defined by Reliability index can be calculated by According to the FORM, a tangent hyperplane is used to fit the limit state surface at the design point.erefore, the most important step in the method is to obtain the design point.Reliability index β is defined as the distance from the origin to the design point.Hence, the coordinate of the design point x * is denoted as where cos According to equation ( 5), the performance function can be approximated by the GP model as e first-order partial derivatives of the approximate function can be given by where Substituting equations ( 15) and ( 16) into ( 17), the reliability index β is given as According to equations ( 15) and ( 16), ( 14) can be rewritten as cos Substituting equations ( 17) and ( 18) into ( 13), one can obtain the coordinate of design point x * using GP approximation.
e failure probability can be calculated as where Φ(•) denotes the cumulative distribution function of the standard normal variable.

Procedure of GP-Based RSM.
e main difference between the GP-based RSM and the traditional RSM is that the former employs the GP to approximate the performance function and its first-order partial derivatives simultaneously.Moreover, from the viewpoint of machine learning theory, more knowledge means a better effect on the result of training or prediction of the machine learning model.us, the training samples, namely, knowledge set in the machine learning domain, is generated as a dynamic updating one to improve the performance of the GP model in the procedure of GP-based RSM.
As shown in Figure 1, reliability analysis using the GPbased RSM is described in detail as follows: Step 1. Assume initial values of the design point x * � (x 1 , x 2 , . . ., x n ).Usually, mean values of the random variables can be selected as the coordinates of the design point x * .
Step 2. According to the experimental plan of the RSM developed by Bucher and Bourgund [29]; the training samples are generated according to the intersection of the axis x(μ 1 , μ 2 , . . ., μ n ) and coordinates of x i � μ i and μ i ± fσ i , where μ i and σ i are the mean and standard deviation of the random variable x i .f, which is an integer, ranges from 1 to 4 and is usually set to 1. en, values of the performance function y(x 1 , x 2 , . . ., x n ) containing y(μ 1 , μ 2 , . . ., μ i ± fσ i , . . ., σ n ) and y(μ 1 , μ 2 , . . ., σ n ) are obtained using the slope analysis code.us, the training samples D � (x i , y i ) are established, and the number of training samples is 2n × s + 1. s is the number of selected f.Step 3. If the training target y has a nonzero mean, to improve the efficiency of the training process of the model, y is adjusted by the mean μ y of y Step 4. Train the GP model by learning the training samples D � (x i , en, we make predictions of performance function on the design point x * and obtain their predictions y(x * ) according to equation (5).Recall that the training targets were centered, thus we must adjust the predictions by offset:

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Step 5. Compute the reliability index of the k th iteration step β (k) using equation (17).
Step 6. Compute the values of the new design point according to equation ( 21).
Step 7. Check the convergence criterion for Step 8. Calculate the probability of failure p f using equation (19).
To apply the GP-Based RSM to slope reliability analysis, a general program package is developed using MATLAB.

Case 1: A Highly Nonlinear Performance Function.
e performance function of the rst example is de ned as [30] where where the covariance function k(x i , x j ) denotes the column of matrix K and the values of vector α and matrix K based on the initial training samples are as follows: Table 2 lists both calculation processes of the design point using the FORM and the proposed method.It clearly illustrates that the real performance and its rst derivatives are both tted very well by the GP model.
e failure probability and reliability index obtained by various methods in preexisting studies are listed in Table 3.
e exact reliability index is 2.9044 obtained by the direct MCS [30].e reliability index yielded by the FORM, RSM, and GP-based RSM are very close to that by the MCS.e GP-based RSM needs 7 function evaluations, while the RSM and direct MCS require 19 and 10 6 function evaluations, respectively.
Figure 2 shows the curves of approximated limit state function from the GP based on nal training samples composed of 5 initial samples and 2 new samples and that of real limit state function.As can be seen from Figure 3, the GP can capture the whole shape of the real limit state function, and the design point is found quickly by the use of the GP-based RSM. is shows that the dynamic update of the training samples can signi cantly improve the e ciency of searching the design point in the proposed method.

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Case 2:
A Two-Layered Slope.As illustrated in Figure 3, the second example considers a two-layered slope which has been studied by Xu and Low [4]; Cho [14]; Hassan and Wol [31]; Bhattacharya et al. [32]; and Chowdhury and Rao [33].Property parameters of the soil related to the slope, including the friction angle and cohesion, are taken as random variables, as listed in Table 4. ese random variables are assumed to obey normal distribution variables in the numerical analysis.Unit weight of the soil is assumed to be 19 kN/m 3 .By numerical analysis using SLIDE [24], the factor of safety F(x) of circular slip surface was calculated using the Bishop and Spencer methods, respectively.ere are 7 training samples (f 1) established for training the GP model as shown in Table 5. e safety factors of the other 10 samples created randomly were predicted using the trained GP model.e safety factors calculated by the Bishop code and Spencer method and values predicted by the GP are shown in Figure 4. e GP model can re ect the relationship between the random variables and the safety factor of the slope.en, the performance functions in the explicit form using the GP model were established to approximate the implicit real performance functions.

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Table 6 listed the results of the probabilistic analyses obtained from the previous literature for the same problem, while Tables 7 and 8 present the results obtained by the GPbased RSM.It can be observed from Tables 7 and 8, the result of reliability index by the early study shows that the exact solution is about 2.2 [33], and the reliability index estimation by the GP-based RSM with 7 initial samples is 2.249 from the Bishop method and 2.269 from the Spencer method, respectively.e results are quite in agreement with each other.
Furthermore, Tables 7 and 8 indicate that the change of the numbers of initial training samples slightly in uences the accuracy of estimated values of probability of failure.It is worth noting that a larger number of initial training samples may mean higher robust and computation cost of the proposed method at the same time.
e proposed method needs 12 function evaluations, while the ANN [14], MCS, rst-order HDMR-based response surface method, and second-order HDMR-based response surface method [33] requires 25, 10 6 , 13, and 61 number of original stability analysis, respectively.e larger the number of the function evaluations, the more the time needed to nish the computation is, the less e cient the used method is.Consequently, the GP-based RSM method is relatively e cient.
e slip surfaces from the Bishop method and Spencer method at the mean point and at the design point are shown in Figures 5 and 6, respectively.e slip surfaces at the mean point are very di erent from that at the design point.e values of the factor of safety F s at the design point are both very close to 1; in another words, g(X) ≈ 0, which mean the slope just reaches the limited equilibrium status.

Case 3:
A ree-Layered Slope.A complex slope with three di erent soil layers is studied.e cross section of the nonhomogeneous slope with a height of 10 m and an inclination of 2 : 1 is presented in Figure 7. e unit weight     9.
e calculations of the safety factor of the slope are conducted using the Bishop method and the Spencer method by the SLIDE code [35].11 initial training samples (f 1) are listed in Table 10 while the safety factor of the circular slip surface was calculated using the Bishop and Spencer methods, respectively.Advances in Civil Engineering e results of the slope reliability analysis using di erent methods are listed in Table 11.A sampling size of 10 6 is considered in the direct MCS by numerical analysis using SLIDE [35] to estimate the failure probability P f .e failure probability estimation by the GP-based RSM is in good agreement with that of the MCS.
e proposed method needs 12 function evaluations, while the number of function evaluations for the SVM-based RSM [20] and MCS are 19 and 10 6 , respectively.is indicates that the proposed GPbased RSM is much more e cient to achieve reasonable accuracy for slope reliability analysis than the SVM-based RSM [20].
In addition, the slip surfaces from the Bishop method and Spencer method at the design point obtained using the proposed method are both presented in Figure 8. and Xu [36] is considered.e soil pro le is presented in Figure 9. Two sets of circular slip surfaces are considered.
e rst set consists of potential failure surface tangential to the lower boundary of the Clay 2 layer (failure mode I), while the second considers slip surfaces tangential to the lower boundary of Clay 3 (failure mode II).
e property parameters of soil are listed in Table 12 and are assumed to obey the Gaussian distribution in the numerical analysis.e Bishop method is used to estimate the safety factor from the

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Advances in Civil Engineering numerical analysis using SLIDE [35].For evaluating the failure probability P f , f 1 is selected, which results in 7 initial training samples (Table 13).Without any iteration, totally 7 actual stability analyses using the Bishop method are needed for the GP-based RSM.It can be observed from Table 14 that the failure probability estimated by the GPbased RSM agrees well with that reported in the literature [33].However, the total numbers of slope stability analysis for the proposed method is 12, while that for the MCS, rstorder HDMR, and second-order HDMR are 5000, 13, and 61, respectively [33] is indicates that the proposed method is very e cient to achieve reasonable accuracy.e slip surface of failure mode I and failure mode II at the design point obtained using the proposed method are both presented in Figure 10.

Conclusions
A GP-based RSM was developed to predict failure probability of slope.In this method, the training samples for   Advances in Civil Engineering establishing the GP model are rst generated by the design method of the classical RSM method.e limit state function and its rst-order partial derivative are then approximated by the trained GP model.Finally, failure probability is estimated using the FORM.In addition, the viewpoint of more knowledge means the better e ect on the result of training or prediction of the machine learning model, and the use of the renewed continuously training samples may improve the performance of the GP model for approximating the limit state function around the design point.us, the GP signi cantly reduces the number of required training samples.Otherwise, GP also shows good performance to approximate the limit state function and then provides accurate estimation of the failure probability when connected with the FORM.Four numerical examples including both slope and nonslope problems illustrate the exibility and e ciency of the proposed method.Compared with the traditional RSM, the GP-based RSM is much more e cient to achieve reasonable accuracy for slope reliability analysis.Moreover, the present method can directly take advantage of existing slope software without modi cation and thus are convenient to be used for practitioner engineers.However, it should be noted that the proposed algorithm is intended as a possible complement rather than a replacement for existing classical methods.
Data Availability e data used to support the ndings of this study are included within the article.Advances in Civil Engineering

Figure 2 :
Figure 2: Compare the curve of limit state function from GP based on the nal training samples with that from the real limit state function, where the boxes refer to the design points at iterative step in GP-based RSM: (b) enlarged gure; (a) in the range of −0.001 ≤ x 1 ≤ 0.004, 2.990 ≤ x 2 ≤ 3.020.

Figure 4 :
Figure 4: Comparison between safety factors calculated by (a) Bishop method and GP prediction and (b) Spencer method and GP prediction.

Figure 5 :Figure 6 :
Figure 5: Example 2: results of the stability analysis by the Bishop method.Slip surface at (a) the mean point (F s 1.644) and (b) the design point (F s 1.008).

4. 4 .
Example 4: Congress St. Cut Model.In this example, the confessed Congress St. Cut model studied by Chowdhury

Figure 7 :
Figure 7: Geometry of the slope of Example 3.

Figure 8 :
Figure 8: Example 3: results of the stability analysis.Slip surface at the design point by (a) the Bishop method (F s 0.992) and (b) the Spencer method (F s 0.997).

Figure 10 :
Figure 10: Example 4: results of the stability analysis by Bishop method.Slip surface at the design point: (a) failure model I, F s 1.000 and (b) failure model II, F s 1.000.

Table 1 :
Training samples and the predicted values using GP for Example 1.

Table 2 :
Calculation process of the design point using FORM and the proposed method for Example 1.

Table 3 :
Results of Example 1 using di erent methods.

Table 5 :
e initial learning samples of Example 2.

Table 4 :
Properties of stochastic variables of Example 2.
c 19.5 kN•m −3 .e corresponding property parameters of the soil related to the slope, including the cohesion c and friction angle φ, are considered as random variables.ese random variables obey the normal distribution.e values of the mean and standard deviation of the random variables are listed in Table

Table 6 :
Results of probabilistic analysis from di erent methods for Example 2.

Table 7 :
Results of probabilistic analysis by the Bishop method for Example 2.

Table 8 :
Results of probabilistic analysis by the Spencer method for Example 2.

Table 9 :
Properties of stochastic variables of Example 3.

Table 10 :
e initial learning samples of Example 3.

Table 11 :
Results comparison of Example 3.

Table 12 :
Material properties of the soil of Example 4.

Table 13 :
e initial learning samples of Example 4.

Table 14 :
Results of probabilistic analyses using di erent methods for Example 4.