Three-dimensional slope stability predictions using artificial neural networks

To enable assess slope stability problems efficiently, various machine learning algorithms have been proposed recently. However, these developments are restricted to two-dimensional slope stability analyses (plane strain assumption), although the two-dimensional results can be very conservative. In this study, artificial neural networks are adopted and trained to predict three-dimensional slope stability and a program, SlopeLab has been developed with a graphical user interface. To reduce the number of variables, groups of dimensionless parameters to express stability of slopes in classic stability charts are adopted to construct the neural network architecture. The model has been trained with a dataset from slope stability charts for fully cohesive and cohesive-frictional soils. Furthermore, the impact of concave plan curvature on slope stability that is usually found by excavation in practice is investigated by introducing a dimensionless parameter, relative curvature radius. Slope stability analyses have been conducted with numerical calculations and the artificial neural networks are trained with dimensionless data. The performance of the trained artificial neural networks has been evaluated with the correlation coefficient ( R ) and root mean square error ( RMSE ). High accuracy has been found in all the trained models in which R > 0.999 and RMSE < 0.15. Most importantly, the proposed program can help engineers to estimate 3D effects of a slope quickly from the ratio of the factors of safety, FS3D/FS2D . When FS3D/FS2D is large (such as larger than 1.2), a 3D numerical modelling on slope stability analyses that can consider complex 3D geometry and boundary condition is advised. extended by a distance in the y- direction to simulate the 3D boundary conditions. The shear strength of soils is described on the basis of the Mohr-Coulomb failure criterion. The purely cohesive slopes are studied first before the slopes in cohesive-frictional soils. At last, the three-dimensional stability assessment of concave slopes in plan view is conducted. The ANN models are trained with dimensionless data based on stability charts.

slopes is quantified with the factor of safety (FS). In order to simplify the calculation process for FS, a vast majority of slope stability analyses in engineering practice are performed in two-dimensions (2D) under plane strain assumption, in which the slope is assumed to be infinitely wide such that three-dimensional (3D) end effects are negligible. However, numerous natural and constructed practical slopes exhibit a complex geometric configuration and a 3D state. [1][2][3][4] From reported slopes in the literature, 2,5,6 relative differences between the two-dimensional factor of safety (FS2D) and three-dimensional factor of safety (FS3D) are found greater than 50%. Therefore, the prediction based on 2D analysis can be excessively conservative and economically unjustified. 3D slope stability analysis should be used for accurate slope stability prediction, especially for the slopes with evident 3D end effects or complex slope geometries.
Limit equilibrium method (LEM) [7][8][9] is perhaps the oldest calculation method to predict the stability of slopes. The common limit equilibrium techniques are the ordinary method of slices, 10 Bishop simplified, 11 Spencer, 12 Janbu 13 and Morgenstern-Price. 14 In LEMs, assumptions on the shape and location of the failure surface need to be made. In contrast, the failure mechanism at the critical equilibrium state can be obtained automatically from finite element numerical results with the strength reduction method (SRM). [15][16][17] The factor of safety is found with the ratio of the soils' actual shear strength to the reduced strength at failure. Owing to its flexibility and robust performance, its application has been increasingly popular. An alternative to the strength reduction method is the limit analysis method, in which the soils are treated as a perfectly plastic material obeying an associated flow rule. 18 Its accuracy and powerful performance have been demonstrated in analysing the stability of various slopes, such as the seismic rock slope, 19 the two-layered cohesive soil slope 20 and the slope with cracks. 21 The advanced numerical methods can provide an accurate FS for slopes under complex conditions, and much detailed information is available from the numerical results. However, the calculation normally takes much longer time. Therefore, machine learning aided slope stability methods may be considered as an alternative when an FS is merely required for a slope.
In recent years, machine learning has received increasingly more attention in slope stability analyses. The accurate results have been demonstrated with various techniques such as extreme gradient boosting method, 22,23 Gaussian process regression, 24 relevance vector machine, 25 artificial neural networks, 26,27 extreme learning machine 28,29 and hybrid machine learning techniques. [30][31][32][33][34] Most recently, hybrid intelligent methods for predicting FS of slopes under seismic conditions are employed in. 35 An artificial neural network to investigate inhomogeneous soil slope stability under undrained conditions is developed in. 36 Machine learning aided probabilistic stability analysis of slopes in spatially variable soils has been conducted in. 37,38 However, as far as the authors know, the application of machine learning algorithms on slope stability analysis is restricted to two-dimensions. To better predict the stability of slopes, machine learning aided slope stability analysis algorithms need to be extended to three-dimensions.
In this study, artificial neural networks are adopted and trained for assessing the stability of 3D slopes. The straight slopes in both purely cohesive and cohesive-frictional soils are considered first before concave slopes in plan view. The effect of plan curvature of slopes is measured with a dimensionless parameter, the relative curvature radius of the slope and the stability analyses are conducted for a series of slopes using the strength reduction method. To reduce the number of variables in machine learning models, groups of dimensionless parameters based on slope stability charts 39-42 are used. As a result, a very good predictive performance can be achieved even with a relatively small training dataset. In addition, a program, SlopeLab, is developed for easy use by wrapping the trained neural networks with a graphical user interface. A ratio of the safety factor, FS3D/FS2D, can also be obtained from the program, providing a quick reference on 3D effects of a slope. The program's predictions have been validated with a series of examples.

ARTIFICIAL NEURAL NETWORKS WITH DIMENSIONLESS VARIABLES
Artificial neural networks (ANN) are a computing model inspired by the biological neural networks in animal brains. It has been used in the geotechnical engineering community for various challenges. [43][44][45][46][47][48][49][50] The architecture of ANN consists of the input layer, the output layer and several hidden layers, as demonstrated in Figure 1. The data is processed and transmitted between neurons via connection links. 51 During the learning process, neurons' weight is adjusted. Generally, more difficult problems require more neurons, and even more layers. Given very limited dimensionless parameters used in this work, one-hidden-layer feed forward neural networks are examined firstly. The number of neurons is increased until a satisfying result is reached. For a conventional ANN aided slope stability prediction approach, FS is treated as the output. The input is the slope factors that play a role in determining FS, commonly involving soils' shear strength (e.g., cohesion and internal friction angle), soils' unit weight and slope geometry, commonly described by slope height, slope width, slope depth and the slope F I G U R E 1 An ANN architecture with two hidden layers angle. Therefore, around seven independent variables need to be used as the input, resulting in high complexity of learning and computing.
Slope stability charts have been used in practice as a quick reference before personal computers become widely available. [52][53][54] While computational methods have made most stability charts obsolete, dimensionless slope parameters are formulated with a clear relationship in the charts. The charts have been redeveloped with more results obtained from advanced numerical methods. 20,55,56 In this study, dimensionless parameters based on stability charts are used, leading to a reduced number of independent variables and less complexity of the ANN model, as demonstrated in the next sections.
A dataset for the ANN is required to be randomly split into two independent subsets: the training dataset and the testing dataset. The ANN is trained with the training data, while the testing dataset is used to verify its predictive performance. In this study, 80% of the whole data is used for the training dataset while 20% is used for the testing dataset. The Levenberg-Marquardt algorithm 57 is employed to train the neural networks and the mean square error is minimised in the training process. The performance of the trained ANN is evaluated with the correlation coefficient R and root mean square error RMSE: where C is calculated solutions, P is predicted results,̄and̄are the mean value and n is the size of the dataset. R is a measure of the linear relationship between two variables. A value of 1 indicates a perfect positive correlation while 0 represents that there is no linear correlation between the variables. RMSE is a measure of absolute errors of predicted valuables. A lower RMSE represents a better performance of trained models and a value of 0 indicates a perfect fit.

THREE-DIMENSIONAL SLOPE STABILITY ASSESSMENT USING ARTIFICIAL NEURAL NETWORKS
The slope geometry for straight slopes is shown in Figure 2, in which the x-z dimension can be extended by a distance in the y-direction to simulate the 3D boundary conditions. The shear strength of soils is described on the basis of the Mohr-Coulomb failure criterion. The purely cohesive slopes are studied first before the slopes in cohesive-frictional soils. At last, the three-dimensional stability assessment of concave slopes in plan view is conducted. The ANN models are trained with dimensionless data based on stability charts.

H B
β D x y z F I G U R E 2 3D slope model geometry

Purely cohesive slopes
The model geometry of purely cohesive slopes is shown in Figure 2 involving the slope height H, the slope width B, the slope angle β and the slope depth D.
To present the results in a dimensionless manner, following, 52 a stability number is given in the dimensionless form: where N s is the stability number, c u is the undrained shear strength, γ is the unit weight, and H is the slope height. In the literature, 5 a series of stability charts have been developed with a wide range of slope angles β, the width ratio (B/H) and the depth ratio (D/H). B/H is increased from 1, 2, 3, 5 to ∞ (2D plane strain assumption). For example, dark lines in Figure 3 show three-and two-dimensional chart solutions for slopes with B/H = 1 and ∞, respectively.
To examine the chart solutions in, 5 the strength reduction method with a finite element software, Plaxis 3D 58 is employed. A slope is used as the first example with the following parameters: β = 45 • , B = 4.5 m, H = 4.5 m, D = 6 m, γ = 16 kN/m 3 and c u = 30 kPa. For the second slope, the slope angle β is set to 30 • and the slope depth D is changed to 9 m, and the rest of the parameters remain the same. From the charts, ∕ of the two slopes reads 0.11 and 0.09, respectively. Therefore, FS3D is calculated as 3.79 and 4.58, respectively. The numerical results of finite element strength reduction for two slopes show that FS3D = 3.62 and 4.62, respectively. The comparison of the two methods is plotted in Figure 3a, indicating a good agreement is achieved. Furthermore, a conventional 2D slope stability analysis is performed using the limit equilibrium method with the program, SLOPE/W. 59 The factor of safety of the two slopes above is FS2D = 2.41 and 2.42, respectively. The results in comparison with the chart solutions are shown in Figure 3b, indicating a good accuracy of the chart. In addition, the ratio of the safety factors of the two slopes, FS3D/FS2D, is calculated as 1.50 and 1.91, respectively, showing that the 2D analysis underestimates the stability of the two slopes greatly.
As the stability charts are verified, a total of 168 3D slope cases and 42 2D slope cases are collected and used for training the artificial neural networks. Thanks to the adopted dimensionless variables, the number of inputs of the ANN analysis is reduced to three: D/H, β and B/H. The dimensionless stability number ∕ is considered as the output. Figure 4 shows results of the trained ANN indicating that the predicted results agree with the calculated data very well with correlation coefficient R approximately equalling to 1. Meanwhile, all data lie in a straight line with the slope of 1.

Cohesive-frictional soil slopes
The geometry model as shown in Figure 2 is still used in this section. Since the slip surface of cohesive-frictional slopes is relatively shallow, commonly above the slope toe, D/H may be not required to be treated as an independent variable. Stability charts for cohesive-frictional soil slopes regard a dimensionless parameter FS/tanφ as a function of a dimensionless group ∕ tan for a variety of β and B/H, 39,53 where c is the cohesion and φ is the frictional angle. The stability charts presented in 55 are developed based on numerical results of limit analysis. A wide range of dimensionless parameters are used in the charts and β is increased from 15˚to 90˚with an interval of 15˚. For example, dark lines in Figure 5 illustrate the stability chart for the slope with a slope angle β = 30˚.  Figure 5 showing a satisfying agreement between the chart solutions and the numerical results.
After the charts are validated, a total of 1254 cohesive-frictional slope cases are taken from the charts. Independent variables ∕ tan , β and B/H are used as inputs of ANN, and a dimensionless number FS/tanφ is considered as the output. After the ANN is trained, its performance is evaluated, as demonstrated in Figure 6. A great performance of the trained neural networks is concluded with the correlation coefficient R being very close to 1.

Concave slopes in plan view
The design of stable concave slopes with plan curvature such as open-pit mine slopes is of significance in practical engineering. However, to date, few systematic studies have been conducted on stability prediction of concave slopes with a wide range of slope parameters. In this work, the shear strength reduction method with Plaxis 3D 58 and the limit equilibrium ethod with SLOPE/W 59 are employed for three-and two-dimensional slope stability analyses. Figure 7 shows 3D slope geometry for the defined problem, in which the radius of the plan curvature is denoted by r. The typical boundary condition for the slope analysis is adopted. The top surface of the slope is set free, the bottom and side surfaces are fully fixed, and the symmetric surface is fixed in the normal direction. It is necessary to note that for steep slopes that have non-circular slip surfaces, the two-dimensional stability analysis is conducted using the finite element strength reduction method.
Based on the numerical results, the stability charts are presented in Figure 8 for homogeneous concave slopes with plan curvature, in which a wide range of slope angles β, relative curvature radius of slopes (r/H), and the dimensionless parameter ∕ tan is utilized. The results show that an increase of the radius of the plan curvature reduces slope stability. When slope angles increase from 15˚to 90˚, ∕ tan decreases continuously. In addition, unit weight and slope height have a negative effect on slope stability.
A total of 234 slope cases from numerical results are collected to train the ANN. The input includes β, r/H and ∕ tan , while a dimensionless number FS/tanφ is used as the output. The performance of the trained ANN is shown  Figure 9. The results indicate that a very good performance of the trained ANN is achieved as R is greater than 0.999 and calculated data and predicated results lie closely in a straight line with the slope of 1.

APPLICATION EXAMPLES
Based on the trained ANN, a program, SlopeLab is built with a user-friendly graphical user interface. As shown in Figure 10, FS2D and FS3D of the slope can be easily obtained after setting a slope's parameters. Furthermore, a ratio of the factor of safety FS3D/ FS2D is displayed, which can be used to find if 3D end effects are significant or not. For example, if FS3D/ FS2D > 1.5, sophisticated 3D slope stability analyses may be conducted to account for 3D end effects. Comparing to the classic stability charts, selection of the chart and troublesome manual estimation are prevented. A series of examples have been employed to validate the proposed program and to demonstrate its easy use.

An undrained slope
The first example is taken from, 17 which is a slope in 'undrained clay' with γ = 16 kN/m 3 , c u = 14.4 kPa and β = 26.57˚(1:2 slope) whose geometry parameters include B/H = 4 and D/H = 1.5. A stability analysis using the finite element strength reduction method has been conducted for this slope yielding FS3D = 1.45.

F I G U R E 8 Stability charts for concave slopes with plan curvature
Stability charts can provide a solution but a cumbersome manual procedure is required. Since the stability charts 5 are presented based on slopes of B/H = 1, 2, 3, 5 and β = 7.5˚, 15˚, 22.5˚, 30˚, 45˚, 60˚and 75˚, the slope with B/H = 4 and β = 26.57˚cannot be found straightforwardly. Instead, four most relevant cases are determined, and their ∕ and FS are resolved in sequence as listed in Table 1. It turns out that FS of the slope is between 1.43 and 1.67 from the results. Finally, linear interpolation of the four slopes' FS gives an estimated value FS = 1.54.
In contrast, it is very simple to use the proposed program. For undrained slopes, the value of cohesion is equal to the undrained shear strength and a zero frictional angle is used. The default value for the plan curvature radius r is infinite, representing straight slope geometry. After setting the parameters on the left side, two-and three-dimensional factors of safety of the slope are obtained immediately, as shown in Figure 10.

A cohesive-frictional soil slope
The  After setting the slope's parameters in the program, Figure 11 shows the program's results: FS3D = 1.69 and FS2D = 1.53, which is in a very good agreement with the numerical results and chart solutions.

A concave slope in plan view
The third example involves a homogenous concave slope with γ = 18 kN/m 3 , c = 22.5 kPa, φ = 32˚, β = 50˚, H = 4 m and r = 10 m. Table 3 shows the manual procedures dedicated to estimating FS of the slope with r/H = 2.5 and β = 50˚. The chart solutions give FS3D = 3.31. Figure 12 shows results from the proposed program: FS3D = 3.15 and FS2D = 2.91. In addition, the slope's stability analyses are also performed using numerical methods. The calculation results yield factors of safety: FS3D = 3.28 and FS2D = 2.90. A satisfying agreement has been found between the program's results, chart solutions and numerical results, indicating the high accuracy of the trained ANN.

CONCLUSION
Artificial neural networks (ANN) are trained for three-dimensional slope stability predictions in this study. The training dataset is constructed with dimensionless parameters based on stability charts so that the number of variables is reduced and a wide range of slope parameters are guaranteed. Furthermore, for ease of use, a program, SlopeLab, is developed by wrapping the trained ANN with a graphical user interface. After setting the slope parameters, three-and two-dimensional factors of safety, FS3D and FS2D, can be obtained immediately. In addition, the three-dimensional end effects of slopes can be evaluated in the result, FS3D/FS2D. This would help engineers to decide if a 2D stability analysis solution is justifiable for a slope or not. The approach applies to slopes in purely cohesive soils and cohesive-frictional soils and concave slopes with plan curvature. A remarkable performance of the trained ANN is demonstrated with the measured correlation coefficient being very close to 1. The program has been adopted for a series of numerical examples, and numerical results and chart solutions have been derived for the purpose of validation. The results show that the proposed program can predict the stability of slopes with high accuracy.
The pore-water pressure conditions are not considered in this work. For future studies, this factor can be taken into account with an additional factor, the pore-water pressure coefficient representing a simplified average pore-water condition in a slope stability analysis. The slopes in this study are assumed to be homogenous and constant Mohr-Coulomb shear strength parameters are used.

A C K N O W L E D G E M E N T S
The authors wish to acknowledge the financial support from the BIG project of the Swedish Transport Administration (Trafikverket BIG project number: A2018-03). We thank the reference group, Dr. Kenneth Viking and Mr. Niklas Dannewitz from Swedish Transportation Administration (Trafikverket) and Prof. Göran Sällfors and Prof. Minna Karstunen from Chalmers University of Technology, for the helpful discussions.

D ATA AVA I L A B I L I T Y S TAT E M E N T
The data used to support the findings of this study are available from the corresponding author upon request.