Application of ANN in Predicting the Cantilever Wall Deﬂection in Undrained Clay

: The main objective of this study is to propose an artiﬁcial neural network (ANN)-based tool for predicting the cantilever wall deﬂection in undrained clay. The excavation width, the excavation depth, the wall thickness, the at-rest lateral earth pressure coefﬁcient, the soil shear strength ratio at mid-depth of the wall, and the soil stiffness ratio at mid-depth of the wall were selected as the input parameters, whereas the cantilever wall deﬂection was selected as an output parameter. A set of veriﬁed numerical data were utilized to train, test, and validate the ANN models. Two commonly used performance indicators, namely, root mean square error (RMSE) and mean absolute error (MAE), were selected to evaluate the performance of the proposed model. The results indicated that the proposed model can reliably predict the cantilever wall deﬂection in undrained clay. Moreover, the sensitivity analysis showed that the excavation depth is the most important parameter. Finally, a graphical user interface (GUI) tool was developed based on the proposed ANN model, which is much easier and less expensive to be used in practice. The results of this study can help engineers to better understand and predict the cantilever wall deﬂection in undrained clay.


Introduction
In recent decades, embedded retaining structures have been increasingly used for excavations in urban areas around the world [1][2][3][4]. The cantilever wall is a conventional earth-retaining structure with a relatively simple construction process and is commonly used for excavations no more than 6 m deep. Generally, they provide open excavations and do not require bracing or anchoring. However, excavation work will inevitably cause cantilever wall deflection, i.e., the wall top displacement, and ground movement, ultimately posing a risk to adjacent structures. Therefore, the control of deformation is as important as the safety requirements against collapse in the design of such retaining walls [5,6]. In this regard, an accurate and practical tool for predicting the cantilever wall deflection should be of great interest to engineers and stakeholders.
Numerous studies have been conducted to analyze the cantilever wall deflection using various methods, such as field measurements [7][8][9], centrifuge modelling tests [10,11], numerical analyses [12][13][14][15][16], and analytical methods [17][18][19]. Zhou et al. [9] analyzed the monitoring data of diaphragm wall deflection using a Bayesian network. Kunasegaram et al. [11] developed a centrifuge modelling system to study the response of a cantilever retaining wall. Hashash et al. [12] conducted a series of numerical analyses to investigate the ground movement caused by deep excavations in soft clay. Sert et al. [13] studied the effect of soil spatial variability on the excavation-induced lateral wall deflection through a large number of numerical simulations. Zhang et al. [16] developed a simplified polynomial regression model to predict the maximum wall deflection. More recently, Qi et al. [19] developed an efficient probabilistic back-analysis method for braced excavations using wall from both horizontal and vertical movements, while the left-and right-side bounda were only restricted horizontally. Moreover, the soil mass and the retaining structure w modelled by about 1200 15-node triangular soil elements and 15 5-node plate eleme respectively.
It is worth noting that the finite element method (FEM) analysis was rigorously ified based on 14 centrifuge tests and 45 field cases [30]. Figure 2 presents the comparis between the cantilever wall deflections δFEM, i.e., the wall top displacements, calculated Plaxis 2D and the measured displacements δm. Since the results of FEM were quite c to the 45° trend line, the rationality of the analysis of FEM was demonstrated. Theref the use of database from the verified finite element model in ANN is appropriate reliable. More detailed information about the numerical models used and the valida process can be found in the authors' previous work [30]. As a result, a total of 191 num ical simulations of the soil-cantilever wall system were conducted, and the correspond numerical data of the cantilever wall deflection δ as well as other associated parame were obtained. It is worth noting that the finite element method (FEM) analysis was rigorously verified based on 14 centrifuge tests and 45 field cases [30]. Figure 2 presents the comparisons between the cantilever wall deflections δ FEM , i.e., the wall top displacements, calculated by Plaxis 2D and the measured displacements δ m . Since the results of FEM were quite close to the 45 • trend line, the rationality of the analysis of FEM was demonstrated. Therefore, the use of database from the verified finite element model in ANN is appropriate and reliable. More detailed information about the numerical models used and the validation process can be found in the authors' previous work [30]. As a result, a total of 191 numerical simulations of the soil-cantilever wall system were conducted, and the corresponding numerical data of the cantilever wall deflection δ as well as other associated parameters were obtained. From the above numerical results, it is noted that the cantilever wall deflection δ significantly influenced by the excavation width B, the excavation depth Hc, the wall thick ness t, the at-rest lateral earth pressure coefficient K0, the soil shear strength ratio at mid depth of the wall su/σv′, and the soil stiffness ratio at mid-depth of the wall Eur/su. There fore, these six important parameters were selected as the main influential factors used i the following ANN model. Finally, as described previously, a total of 191 results of finit element analyses of the cantilever wall deflection were used, and the statistical analysis o these data sets is presented in Table 1. In this database, the excavation width B is distrib uted between 10.00 and 150.00 m, the excavation depth Hc is distributed between 0.50 an 6.00 m, the wall thickness t is distributed between 0.60 and 1.60 m, the at-rest lateral eart pressure coefficient K0 is distributed between 0.50 and 1.10, the soil shear strength ratio a mid-depth of the wall su/σv′ is distributed between 0.30 and 0.70, and the soil stiffness rati at mid-depth of the wall Eur/su is distributed between 199.38 and 900.00.  Figure 3 presents the relationship between each input and output parameter. Th results in Figure 4 indicate that the correlation between all input parameters is weak, s the input datasets used in this study are appropriate. From the above numerical results, it is noted that the cantilever wall deflection δ is significantly influenced by the excavation width B, the excavation depth H c , the wall thickness t, the at-rest lateral earth pressure coefficient K 0 , the soil shear strength ratio at mid-depth of the wall s u /σ v , and the soil stiffness ratio at mid-depth of the wall E ur /s u . Therefore, these six important parameters were selected as the main influential factors used in the following ANN model. Finally, as described previously, a total of 191 results of finite element analyses of the cantilever wall deflection were used, and the statistical analysis of these data sets is presented in Table 1. In this database, the excavation width B is distributed between 10.00 and 150.00 m, the excavation depth H c is distributed between 0.50 and 6.00 m, the wall thickness t is distributed between 0.60 and 1.60 m, the at-rest lateral earth pressure coefficient K 0 is distributed between 0.50 and 1.10, the soil shear strength ratio at mid-depth of the wall s u /σ v is distributed between 0.30 and 0.70, and the soil stiffness ratio at mid-depth of the wall E ur /s u is distributed between 199.38 and 900.00.  Figure 3 presents the relationship between each input and output parameter. The results in Figure 4 indicate that the correlation between all input parameters is weak, so the input datasets used in this study are appropriate.

Introduction of ANN
The ANN, which works similarly to human cognition, is one of the best machine learning models to handle complex nonlinear problems, and it has been successfully applied to many geotechnical problems. Typically, an ANN model consists of an input layer, one or more hidden layers, and an output layer. The input layer receives the information from the input data and transforms it into the hidden layer. The hidden layer and the output layer perform the complex computations of the ANN model. In this work, the ANN model was constructed in the spirit of multilayer perceptron, typically trained using the feed-forward back-propagation algorithm [31]. For simplicity, only one hidden layer was used, as previous research [32,33] has reported that the ANN model with one hidden layer can perform well in handling similar problems. Hence, the final ANN model consisted of a single input, hidden, and output layer.

Introduction of ANN
The ANN, which works similarly to human cognition, is one of the best machine learning models to handle complex nonlinear problems, and it has been successfully applied to many geotechnical problems. Typically, an ANN model consists of an input layer, one or more hidden layers, and an output layer. The input layer receives the information from the input data and transforms it into the hidden layer. The hidden layer and the output layer perform the complex computations of the ANN model. In this work, the ANN model was constructed in the spirit of multilayer perceptron, typically trained using the feed-forward back-propagation algorithm [31]. For simplicity, only one hidden layer was used, as previous research [32,33] has reported that the ANN model with one hidden layer can perform well in handling similar problems. Hence, the final ANN model consisted of a single input, hidden, and output layer.
The so-called weights, biases, and activation functions are used to connect to the neurons of different layers, and based on them, the whole ANN model can be represented in a mathematical form, as shown below: The so-called weights, biases, and activation functions are used to connect to the neurons of different layers, and based on them, the whole ANN model can be represented in a mathematical form, as shown below: where Y is the final output for the ANN model; X is the matrix of input parameters; f 1 , B 1 , W 1 are the activation function, the weight matrix, and the biases vector of the hidden layer, respectively; and f 2 , B 2 , W 2 are the activation function, the weight matrix, and the biases vector of the output layer, respectively. For the activation functions, this study adopted the commonly used LOGSIG function and PURELIN function in the hidden and output layers, respectively. Their functions are presented in Figure 5.
where Y is the final output for the ANN model; X is the matrix of input parameters; f1, b1, W1 are the activation function, the weight matrix, and the biases vector of the hidden layer, respectively; and f2, b2, W2 are the activation function, the weight matrix, and the biases vector of the output layer, respectively. For the activation functions, this study adopted the commonly used LOGSIG function and PURELIN function in the hidden and output layers, respectively. Their functions are presented in Figure 5. In the training process of the ANN model, the connecting weights and biases are adjusted to minimize the error of the ANN. A term named the mean square error (MSE) is widely utilized to terminate the training of the model. The MSE can be calculated by Equation (3) below: where N represents the number of input datasets; ei is the difference between the predicted and actual results.

Architecture of the Developed ANN Model for Cantilever Wall Deflection
To select the appropriate hidden neurons to obtain the best ANN geometry, a trialand-error procedure was performed. First, the collected datasets were normalized in a range of −1 and 1 based on Equation (4) to achieve dimensional consistency for all the parameters and avoid potential overfitting issues.
where Zn represents the normalized data sample; Zi represents the data sample; Zmax and Zmin stand for the maximum and minimum values of the data for the interested parameter, respectively. Second, the collected database was randomly spilt into training, testing, and validation sets, respectively. The training ratio was set as 0.70, while the testing and validation ratios were set as equal, i.e., 0.15, respectively. The training set was utilized to compute the associated weights matrix and the biases vector in Equation (2), and the testing set was used to evaluate the performance of the ANN model. Furthermore, the validation set was used to monitor ANN model overfitting. The error of validation data will increase accordingly once overfitting occurs. In the training process of the ANN model, the connecting weights and biases are adjusted to minimize the error of the ANN. A term named the mean square error (MSE) is widely utilized to terminate the training of the model. The MSE can be calculated by Equation (3) below: where N represents the number of input datasets; ei is the difference between the predicted and actual results.

Architecture of the Developed ANN Model for Cantilever Wall Deflection
To select the appropriate hidden neurons to obtain the best ANN geometry, a trialand-error procedure was performed. First, the collected datasets were normalized in a range of −1 and 1 based on Equation (4) to achieve dimensional consistency for all the parameters and avoid potential overfitting issues.
where Z n represents the normalized data sample; Z i represents the data sample; Z max and Z min stand for the maximum and minimum values of the data for the interested parameter, respectively. Second, the collected database was randomly spilt into training, testing, and validation sets, respectively. The training ratio was set as 0.70, while the testing and validation ratios were set as equal, i.e., 0.15, respectively. The training set was utilized to compute the associated weights matrix and the biases vector in Equation (2), and the testing set was used to evaluate the performance of the ANN model. Furthermore, the validation set was used to monitor ANN model overfitting. The error of validation data will increase accordingly once overfitting occurs.
Finally, the number of neurons in the hidden layer varied from 1 to 12, and accordingly, a total of 12 ANN models were trained. Figure 6 presents the computed MSE of the training, testing, and validation sets for different numbers of neuron. It can be found that the case of 6 neurons in the hidden layer results in the best performance. The obtained best ANN architecture is schematically shown in Figure 7. It shows 6 neurons in the input layer, 6 neurons in the hidden layer and 1 neuron in the output layer.
ingly, a total of 12 ANN models were trained. Figure 6 presents the c training, testing, and validation sets for different numbers of neuron the case of 6 neurons in the hidden layer results in the best perform best ANN architecture is schematically shown in Figure 7. It shows 6 layer, 6 neurons in the hidden layer and 1 neuron in the output layer : Connecting weight between each layer  Finally, the number of neurons in the hidden layer varied from 1 to 12, and accordingly, a total of 12 ANN models were trained. Figure 6 presents the computed MSE of the training, testing, and validation sets for different numbers of neuron. It can be found that the case of 6 neurons in the hidden layer results in the best performance. The obtained best ANN architecture is schematically shown in Figure 7. It shows 6 neurons in the input layer, 6 neurons in the hidden layer and 1 neuron in the output layer.        Figure 9a compares the cantilever wall deflection derived from the proposed ANN model with the actual numerical data in the training process. In general, there is a fairl good agreement between the two datasets. Figure 9b shows the evolution of the error wit the change in the number of samples. The histogram of the error values with the mea and standard deviation (StD) during the training is also shown in Figure 9c. It can be see that most of the errors during the training process are close to 0. A similar phenomeno was also observed when comparing the results of the ANN model and the correspondin numerical data for the testing and validation, as well as for all data (see Figures 10-12 The above results confirm the effectiveness of the developed ANN model and show tha it is reliable in predicting the cantilever wall deflection in undrained clay.  Figure 9a compares the cantilever wall deflection derived from the proposed ANN model with the actual numerical data in the training process. In general, there is a fairly good agreement between the two datasets. Figure 9b shows the evolution of the error with the change in the number of samples. The histogram of the error values with the mean and standard deviation (StD) during the training is also shown in Figure 9c. It can be seen that most of the errors during the training process are close to 0. A similar phenomenon was also observed when comparing the results of the ANN model and the corresponding numerical data for the testing and validation, as well as for all data (see Figures 10-12). The above results confirm the effectiveness of the developed ANN model and show that it is reliable in predicting the cantilever wall deflection in undrained clay.

Performance of the Proposed ANN Model
Furthermore, the actual values of the cantilever wall deflection δ were compared with the values predicted by the ANN model using the regression analysis, as shown in Figure 13. It can be observed that the values of R 2 for the training, testing, validation, and all data are 0.9823, 0.9135, 0.9915, and 0.9811, respectively. They are all quite close to 1, which again indicates the acceptable performance of the developed ANN model.  Furthermore, the actual values of the cantilever wall deflection δ with the values predicted by the ANN model using the regression analysis, as shown in Figure 13. It can be observed that the values of R 2 for the training, testing, validation, and Two commonly used performance indicators, root mean square error (RMSE) and mean absolute error (MAE), were selected to evaluate the network performance. The expression for each indicator is given in Equations (5) and (6). The value of the MAE represents the absolute differences between the predicted and actual values, while the RMSE represents the existing deviation between the predicted and actual values. The results of two indicators for training, testing, validation, and all data are shown in Figure 14. This again demonstrates that the predicted cantilever wall deflection using the developed ANN model were quite similar to the actual values. These results confirm that the developed ANN model performs well and could be successfully used to coherently predict the cantilever wall deflection in undrained clay. Equations (5) and (6) are calculated as follows: Two commonly used performance indicators, root mean square error (RMSE) and mean absolute error (MAE), were selected to evaluate the network performance. The expression for each indicator is given in Equations (5) and (6). The value of the MAE represents the absolute differences between the predicted and actual values, while the RMSE represents the existing deviation between the predicted and actual values. The results of two indicators for training, testing, validation, and all data are shown in Figure 14. This again demonstrates that the predicted cantilever wall deflection using the developed ANN model were quite similar to the actual values. These results confirm that the developed ANN model performs well and could be successfully used to coherently predict the cantilever wall deflection in undrained clay. Equations (5) and (6) are calculated as follows: where N represents the input datasets number; e i is the difference between the predicted and actual results.
N (6) where N represents the input datasets number; ei is the difference between the predicted and actual results.

Parametric Study
A parametric study was further conducted to evaluate the influences of different input parameters on the cantilever wall deflection. For this purpose, in the following analyses, each input parameter was changed from the minimum to the maximum values according to the database in Table 1, while the other parameters corresponded to the mean values. However, it is noted that the influences of multiple variations of different input parameters on the cantilever wall deflection were not studied in this section. Figure 15a-f present the influence of six input parameters on the cantilever wall deflection. It is obvious that the cantilever wall deflection is increased as the parameters of the excavation width B, and the excavation depth Hc changed from the minimum and the maximum values, as shown in Figure 15a,b. For the parameter of the at-rest lateral earth pressure coefficient K0, the cantilever wall deflection was found to increase slightly as K0 varied from the minimum and the mean values, while it was decreased significantly as K0 increased from the mean to the maximum values. For the other three parameters of the wall thickness t, the soil shear strength ratio at mid-depth of the wall su/σv′ and the soil stiffness ratio at mid-depth of the wall Eur/su, the cantilever wall deflection was decreased as they increased from the minimum and the maximum values, as presented in Figure  15c,e,f. The above observations highlighted that all the considered input parameters had moderate effects on the cantilever wall deflection.

Parametric Study
A parametric study was further conducted to evaluate the influences of different input parameters on the cantilever wall deflection. For this purpose, in the following analyses, each input parameter was changed from the minimum to the maximum values according to the database in Table 1, while the other parameters corresponded to the mean values. However, it is noted that the influences of multiple variations of different input parameters on the cantilever wall deflection were not studied in this section. Figure 15a-f present the influence of six input parameters on the cantilever wall deflection. It is obvious that the cantilever wall deflection is increased as the parameters of the excavation width B, and the excavation depth H c changed from the minimum and the maximum values, as shown in Figure 15a,b. For the parameter of the at-rest lateral earth pressure coefficient K 0 , the cantilever wall deflection was found to increase slightly as K 0 varied from the minimum and the mean values, while it was decreased significantly as K 0 increased from the mean to the maximum values. For the other three parameters of the wall thickness t, the soil shear strength ratio at mid-depth of the wall s u /σ v and the soil stiffness ratio at mid-depth of the wall E ur /s u , the cantilever wall deflection was decreased as they increased from the minimum and the maximum values, as presented in Figure 15c,e,f. The above observations highlighted that all the considered input parameters had moderate effects on the cantilever wall deflection.

Sensitivity Analysis
Since the proposed ANN model in this study has only one hidden layer, the Garson factor [34] can be used to evaluate the relative importance of various input parameters to the output. The importance factor of each input parameter is computed using the following equation: where E ∑ r=1 w rh is the weights summation within the hidden neuron h and the input neuron E; and v ho is the connection weight for the hidden neuron h and the output neuron o.

Sensitivity Analysis
Since the proposed ANN model in this study has only one hidden layer, the Garson factor [34] can be used to evaluate the relative importance of various input parameters to the output. The importance factor of each input parameter is computed using the following equation:  Figure 16 shows the relative importance of the different input parameters through sensitivity analysis. In general, it can be seen that these input parameters all have a significant effect on the cantilever wall deflection δ. In particular, the excavation depth H c is the most important parameter, followed by the soil stiffness ratio at mid-depth of the wall E ur /s u , while the at-rest lateral earth pressure coefficient K 0 has the least importance. Obviously, the parameter of H c is demonstrated to be the most sensitive parameter, which means that it plays a key role in predicting the cantilever wall deflection. ron E; and vho is the connection weight for the hidden neuron h a Figure 16 shows the relative importance of the different i sensitivity analysis. In general, it can be seen that these input pa icant effect on the cantilever wall deflection δ. In particular, the most important parameter, followed by the soil stiffness ratio Eur/su, while the at-rest lateral earth pressure coefficient K0 has th ously, the parameter of Hc is demonstrated to be the most se means that it plays a key role in predicting the cantilever wall d Figure 16. Sensitivity analysis of input parameters.

ANN-Based Formulation
For the application of the developed ANN model to practi form of ANN can be converted into an explicit mathematical eq cedures discussed in the above sections, the formula for calcu deflection δ can be obtained directly from the proposed AN weights, biases, activation functions, and normalization factors ver wall deflection δN is a function of six considered input param 7. The above procedures are further illustrated in the Equations where T = 6 is the number of the hidden neurons. Table 2 shows p0 to pn and Ci0 to C i6, for the formulation of the cantilever wall d

ANN-Based Formulation
For the application of the developed ANN model to practical problems, the current form of ANN can be converted into an explicit mathematical equation. Based on the procedures discussed in the above sections, the formula for calculating the cantilever wall deflection δ can be obtained directly from the proposed ANN model by utilizing its weights, biases, activation functions, and normalization factors. The normalized cantilever wall deflection δ N is a function of six considered input parameters presented in Figure 7. The above procedures are further illustrated in the Equations (8) and (9) as below: where t = 6 is the number of the hidden neurons. Table 2 shows the other coefficients, i.e., p 0 to p n and C i0 to C i6 , for the formulation of the cantilever wall deflection. It is noted that the output in Equation (8) of the cantilever wall deflection δ N is a normalized value ranged between −1 and 1. Hence, it needs to be transformed into the real value, and the cantilever wall deflection δ (m) can then be further calculated as: where δ N and δ stand for the normalized and real value of the cantilever wall deflection, respectively.

GUI Tool
In recent years, geotechnical engineers have paid more attention to the development of user-friendly and practical software, because the complex mathematical equations are usually challenging for practitioners. Therefore, a graphical user interface (GUI) tool was established by MATLAB [31] to predict the cantilever wall deflection δ in undrained clay, as shown in Figure 17. In accordance with the proposed ANN model, six considered input parameters, i.e., from X1 to X6, are presented in this tool. People can input the values for the excavation width B, the excavation depth H c , the wall thickness t, the at-rest lateral earth pressure coefficient K 0 , the soil shear strength ratio at mid-depth of the wall s u /σ v , and the soil stiffness ratio at mid-depth of the wall E ur /s u . After entering the input parameters, the cantilever wall deflection δ is immediately displayed when you click on the Start Predict button. In addition, it should be noted that the scope of the developed ANN and this GUI tool is limited to the lower and upper bound of these six input parameters because the ANN model is generally not accurate for extrapolation [25].

Conclusions
In this study, an application of ANN to predict the cantilever wall deflection in undrained clay based on a set of numerical data from verified finite element models was presented. The proposed ANN model was well trained, tested, and validated, indicating its reliable performance. In addition, a series of parametric studies were conducted to evaluate the influences of six input parameters on the cantilever wall deflection. Based on the analysis, the following conclusions were drawn:


The proposed ANN model can predict the cantilever wall deflection in undrained clay with acceptable performance.  The most influential parameter in determining the cantilever wall deflection in un-  Figure 17. ANN interactive graphical user interface.

Conclusions
In this study, an application of ANN to predict the cantilever wall deflection in undrained clay based on a set of numerical data from verified finite element models was presented. The proposed ANN model was well trained, tested, and validated, indicating its reliable performance. In addition, a series of parametric studies were conducted to evaluate the influences of six input parameters on the cantilever wall deflection. Based on the analysis, the following conclusions were drawn: • The proposed ANN model can predict the cantilever wall deflection in undrained clay with acceptable performance.

•
The most influential parameter in determining the cantilever wall deflection in undrained clay is the excavation depth, followed by the soil stiffness ratio at middepth of the wall, while the at-rest lateral earth pressure coefficient presents least influence among the considered parameters. • An ANN model-based formula, which included all input parameters, was developed to calculate the cantilever wall deflection. • A new GUI was developed to easily predict the cantilever wall deflection in undrained clay. The developed GUI tool is much simpler and more cost-efficient and can be recommended for use in practical engineering in advance.
However, there are some limitations to the proposed ANN model. First, it should be noted that the database in this study is generally small, and more high-quality datasets urgently need to be generated for a future investigation. Second, commonly used K-fold cross-validation methods can be used to improve the performance and robustness of the proposed ANN model. Third, comparisons should be made between the proposed ANN model and other mathematical models to verify the performance and reliability of the proposed ANN model. Finally, to enhance the ability and robustness of the proposed ANN model in more complicate problems, extra work on the use of other strong machine learning algorithms, such as the modular neural network and radial basis function neural network, could be performed.  Data Availability Statement: All data generated or analyzed during this study are included in this published article.

Acknowledgments:
The author would like to thank the editors for their helpful and constructive comments that greatly contributed to improving the quality of the paper. The helpful comments from three anonymous reviewers are also greatly appreciated by the authors.

Conflicts of Interest:
The authors declare no conflict of interest.