Application of GA-BP Neural Network Optimized by Grey Verhulst Model around Settlement Prediction of Foundation Pit

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Introduction
In recent years, China's engineering construction has developed rapidly, especially for deep foundation pit projects, caused by the need of large-scale public facilities and lots of exploitation of underground spaces [1][2][3][4][5][6][7][8][9]. However, the settlement of the foundation pit is affected by many factors, such as the excavation rate, the real-time excavation depth, changes in groundwater level, internal friction angle, soil weight, and number of supports [10][11][12][13][14]. Thus, the prediction of the foundation pit settlements and deformation is difficult and inaccurate [15]. Contributing to this situation are engineering accidents constantly occurring in many fields, such as building collapse, road or bridge cracks, excessive settlement of deep foundation pits, and pipeline bursts, accom-panied by huge casualties and economic losses [16][17][18][19][20][21][22]. So, it is of great significance to accurately predict settlement values of foundation pits [23]. Compared to the settlement of the top of foundation pit, the settlement around the foundation pit is more complex and difficult to accurately predict, while traditional finite element methods, such as Midas, Plaxis, and Flac, have difficulty in achieving good results for the complexity of foundation pits [24][25][26]. With the rapid development of urbanization, more and more foundation pits are in the interior of the city, which means that the influence of the foundation pit settlement is increased, such as road cracking, uneven settlement of pile foundation of highway, and collapse of buildings.
Nowadays, lots of methods to forecast the foundation pit settlement have been established. In 2011, a new neural network model was proposed by Ismail and Jeng, whose the features of SPT data along the excavation depth direction of the pile are set as the input samples to calculate the loadsettlement curve for predicting the subsequent settlement [27]. Ghorbani and Niavol developed a new model to predict the settlement of the foundation pit under the circumstance of dynamic-static [28]. Lv et al. proposed a new model, which is based on grey theory and BP neural network to calculate the settlement around a foundation pit [29]. The results showed that two models have good application in engineering project, while the error of the models is still high and time factor is still the only input factor to be considered. Eid and Shehada proposed a method to the initial elastic settlement for the rock foundations [30]. Xu et al. presented the hybrid GA/SIMPLS to study the deformation law of the foundation pit [31]. Guo et al. conducted a new multivariable grey selfmemory coupled prediction model, with high resolution prediction results of deep foundation pit [32]. Shahin established a model, which is based on the recurrent neural networks, to simulate the settlement response for bored piles under axial loading [33]. Doherty et al. studied an international project, which evaluated the responsiveness of geotechnical engineering, to analyze prediction of foundation settlement under the load of the undrained system [34]. Nejad and Jaksa established the ANN and CPT data to simulate the load settlement, while whole load-settlement relationship is obtained [35]. Cao et al. proposed a new neural network, which is an ensemble-based parameter sensitivity analysis paradigm, to study the impact of different parameters on the settlement. The result shows that the settlement is affected by many factors [36]. Su et al. put forward a settlement monitoring method on the basis of the Kalman filter, and the settlement is studied by forward modeling. The results show that it can predict the deformation of the following stage by analyzing the data of the prior stage [37]. Dai et al. filtered the observed that the noise and unmonitored data of the space and time domain are interpolated. The deformation of the dam was predicted through a Kalman filter recursive algorithm. The results demonstrate that the spatiotemporal noise of deformation can be effectively filtered out, and the deformation of the dam can be predicted well [38]. The wavelet packet transform and least-square support vector machines are combined, which are proposed by Zhang et al., to increase the accuracy and application in estimation of the ground subsidence under tunnel project [39]. Zhang et al. developed an optimized grey discrete Verhulst model-BP neural network to forecast the settlement of foundation pits [40].
Previous studies demonstrated that the relationship between the settlement and excavation time is characterized as an "S" curve, when the conditions are satisfied with linear loading [41,42]. Meanwhile, the grey Verhulst model is commonly adopted to predict the settlement caused by the "S" characteristics [43][44][45]. In fact, the grey Verhulst model is more suitable for the prediction of the settlement of foundation pits where the amount of monitoring data is lacking and there are small settlement fluctuations in the short term [32,40,46]. It is mainly because the grey Verhulst model lacked the ability of self-learning and correcting the error [47,48]. Nowadays, ANN has been used in various fields of engineer-ing and plays an important role in predicting and distinguishing, while the BP neural network is one of the most widely used ANN in engineering fields for its strong ability of self-learning, information processing, nonlinear mapping, error feedback adjustment, and fault tolerance [49,50]. Though the BP neural network has such advantages in predicting foundation pit settlements, it still has limitation in optimizing weights and thresholds, for easily falling into the local optimum [51,52], while the genetic algorithm, which can be obtained by the near-optimal solutions in every search space, can well solve these problems [53,54]. Therefore, a genetic algorithm (GA) is adopted to optimize the weights and thresholds of the BP neural network. On the contrary, the training process needs high-quantity and representative data [55]. In fact, it is hard to obtain accuracy and enough data in an actual engineering project, caused by the complicated influencing factors. Thus, the error will extremely lack an adequate training process. As for the grey Verhulst model, it can conduct a forecast for the data sequence in nonlinear and uncertain systems with insufficient data [32,47,56]. Therefore, the GA-BP neural network optimized by the grey Verhulst model is used to predict the settlement around foundation pits.
Meanwhile, in the preview study, the amount of training data is extremely insufficient, in which the amount of training data is often less than 20 sets, and the amount of prediction data is usually less than 10 sets [12,40,43,57,58]. It is mainly because the units of training data and prediction data are often set as month and week [40]. Not only that, time is often set as the only input parameter [12,57,58]. However, the settlement of the foundation pit is influenced by many factors, such as the internal friction angle, cohesion, bulk density, Poisson's ratio, void ratio, changes of real-time water level, permeability coefficient, the number of supports, and real-time excavation depth. Cause of the settlement of foundation pits is influenced by such many parameters; thus, it is inaccurate and meaningless to only study the influence of excavation time on the settlement of foundation pits. In addition, compared to the monthly or weekly accumulated settlement, the prediction of settlement that can be accurate to a certain day or a certain excavation depth has higher engineering significance. In this paper, the internal friction angle, cohesion, bulk density, Poisson's ratio of different soils, void ratio, changes of water level, permeability coefficient, number of supports, and real-time excavation depth are set as the input factors to predict the settlement around the foundation pits, which is nearby a pile foundation of a highway and larger settlement than other settlement monitoring points.

Methodology
The grey Verhulst model was proposed by Verhulst and Malthus to predict the procedure of featured saturation [59]. It is assumed that the x ð0Þ ðiÞ is the settlement value for the i-th monitoring, while the x ð1Þ ðiÞ is the accumulated generating operation of x ð0Þ ðiÞ. The x ð0Þ ðiÞ and x ð1Þ ðiÞ are shown as follows: 2 Geofluids where x ð1Þ ðkÞ = ∑ k i=1 x ð0Þ ðkÞ, k = 1, 2, 3, ⋯, n, while the Z ð1Þ = fz ð1Þ ð1Þ, z ð1Þ ð2Þ,⋯,z ð1Þ ðnÞg is the mean sequence of x ð1Þ ðkÞ, where z ð1Þ ðkÞ = 1/2ðx ð1Þ ðkÞ + x ð1Þ ðk − 1Þ, k = 2, 3, ⋯, n. The grey Verhulst model is shown as [60] x 0 The whitenization differential Equation (6) of the grey Verhulst model, which is the first-order differential equation, can be obtained from the x ð1Þ ðkÞ The resolution of the above Equation (6) is shown as where k = 1, 2, 3, ⋯, n − 1.
As mentioned before, the grey Verhulst model lacks the ability to self-learn and correct the error. Due to the high ability of the BP neural network in information process self-learning, nonlinear mapping, and so on, BP neural networks are adopted in this paper.
While the BP neural network still has limitations in optimizing thresholds and weights, the GA is taken in this paper. The GA is adopted to acquire the near-optimal solutions.
Generally, the GA starts with an initial population using binary bits, such as 1 and 0, string generated through random ways. All the potential solutions, the integers, and the real numbers are encoded. The fitness is regarded as the key factor to evaluate the quality of each string in the problem's domain. Then, a better population will be created through genetic operators. And the BP neural network is optimized by the GA. The flow chart of the GA-BP neural network optimized by the grey Verhulst model is shown in Figure 1.
It can be seen in Figure 1 that the weight and thresholds of the BP neural network are encoded, when the topological structure is determined. The training process is determined by the thresholds and weights. In the genetic algorithm part (within the red rectangle), the crossover, fitness value, selection, and mutation are calculated. It decides if the new group is satisfactory; if not, the weights and thresholds are changed till the requirement is satisfied. As for the genetic algorithm (GA), the near-optimal solutions are obtained. Commonly, the GA commonly starts with the initial population using binary bits, such as 1 and 0, strings generated through random ways. All the integers, potential solutions, and real numbers are encoded by binary strings. And these are taken from search space, including with all the potential solutions. Then, strings are decoded into the search space, while the performance of these strings is evaluated by computing the fitness value for the objective function. In particular, the fitness is the key factor of the quality of each string in the problem's domain. After the strings are evaluated, a better population will be created through the genetic operators. In the end, the optimized weights and thresholds are obtained. While in the grey Verhulst model part, the grey Verhulst model is determined after the original measured value is inputted. Then, the predicted values are determined through performed simulation. Moving forward, the grey prediction values are selected. As per the analysis of the flow chart, the training processes are conducted and the settlements are predicted based on the input parameters.
Most of the previous studies focused on the influence of time factor on the settlement prediction. Not only that, the time factor is often simplified to the unit of week and month. Meanwhile, the prediction of settlement is often the unit of the week or month, which is lacking in engineering guiding significance. Since the settlement is the result of multiple factors, it is inaccurate and meaningless to consider excavation time only.
In this paper, many factors that affect settlement are taken into account to predict the settlement. In the excavation of foundation pits, the soil mechanical parameters have great influence on the settlement, having small settlement values under good geological conditions, while having huge settlement values under bad geological conditions [36]. Previous studies rarely consider this factor, mainly because it is difficult to determine the type of soil at a certain day of excavation. Meanwhile, it is impractical to record the depth of 3 Geofluids excavation for each day, due to the complexity of excavation. To solve this issue, this paper proposed a method as follows. Firstly, supporting time and the position of different supports are precisely recorded. Then, the real-time excavation depth in the support position can be obtained, because these supports are set immediately when the foundation pit is excavated to the support position. After that, the excavation depth of the foundation pit is equally divided by the length of the excavation time between two adjacent support positions. Finally, the relationship between time and excavation depth can be established, which means the real-time excavation soil style can be confirmed, according to the drilling data. As for the internal support, the number of internal supports is increasing when the excavation depth is increasing, which can effectively decrease the deformation and settlement, while the number of the supports is adopted as the input parameters.

Application of Different Models in Settlement Prediction
The deep foundation pit project is located in Foshan City, Guangdong Province. This project consists of the receiving well, the jacking well, and the pipe jacking tunnel. Compared with the receiving well, the jacking well is taken as the research object due to its more complicated geological conditions, deeper excavation depth, and proximity to a bridge pier, to which the settlement around the foundation pit can induce adverse effects. In the process of foundation pit excavation, the settlement must be monitored during the whole excavation procedure. Different forms of monitoring points are set in Figure 2.
Determining the topologicaal structure of BPNN

Geofluids
The settlement monitoring points around foundation pits are set as in Figure 2, which consist of D1-D8. Compared with other settlement monitoring points, the settlement of the D4 point (approximately 20 m southeast of the foundation pit) is the largest. Not only that, the D4 point is also close to the bridge pier of the highway. It means that the settlement of D4 may have a bad impact to the bridge pier, which is part of the highway. Therefore, the settlement of D4 is studied in this paper. Meanwhile, the project is located in Foshan city, Guangdong Province, where rainfall is heavy and concentrated. Thus, the water level around this foundation pit should also be considered. The SW1, SW2, SW3, and SW4 are water level monitoring points. In this paper, the water level changes are set as the input factor for training and prediction.
The characteristics and distribution of the rock-soil mass are obtained by geological data and drilling result. In particular, specimens of rock-soil mass are precisely obtained along different depths of drilling holes to obtain the rock-soil mass properties. Thus, the properties of the rock-soil mass are shown in Table 1.
The relationship between the real-time excavation time and the excavation soil type is demonstrated in detail above in this paper. Meanwhile, the water level changes are precisely match with the excavation time, while different permeability coefficients of the rock-soil mass are also considered in this paper to improve the accuracy of settlement prediction. The foundation pit project started on February 2, 2019, and the excavation to the bottom was on August 8, 2019 (188 days in total). The part monitoring data about the foundation pit settlement and related soil physical parameters are shown in Table 2 (for the detailed data, please refer to the supplemental files of Table 2).
As shown in Table 2, a day is 1 monitoring period, and settlement data sets of 188 days are selected, which is the whole process of foundation pit excavation. When the foundation pit was excavated to the depth of -10 meters (150 th point) above sea level, the settlement of D4 was -31.52 mm. Up to 150 days, there was still 13 m deep of soil that needed to be excavated. In order to prevent the settlement of D4 from being too large, it is extremely important to predict the settlement of D4 continuing the existing construction conditions. Thus, the first 150 data sets are used to establish the model, and the last 38 data sets are taken to verify the accuracy of the trained model. In order to verify the accuracy and application of the GA-BP neural network optimized grey Verhulst model, four other models are compared, which consists of the grey Verhulst model, BP neural network, BP neural network optimized by grey Verhulst model, and BP neural network optimized by genetic algorithm (GA-BP neural network).
The grey Verhulst model is obtained through the first 150 actual settlement measured values (shown as the following equation): Then, predicted results of the grey Verhulst model are inputted to the BP neural network, while the original data are set as the target value of the input vector to the BP neural The structure of the foundation pit and the distribution along the depth direction of soils are precisely matched as shown in Figure 3.

Guiding wall Crown beam
The first concrete support The second concrete support The third concrete support The fourth concrete support The fifth concrete support The sixth concrete support        11 Geofluids network. Then, the weights and thresholds are optimized by the genetic algorithm. The training max epochs are 50000; the learning rate is 3e -3 . The target error of the training is set as 1E-10, while the highest failure time is selected as 6. The gradient descent method is used in this study. In MATLAB, the input layer uses the function "tansig," the hidden layer uses the function "logsig," and the "purelin" are selected in the output layer, while the best hidden neurons are 6, which is obtained after the training and testing processes. As for the best neurons of the GA-BP neural network and BP neural network, the best hidden neurons are both 6. The other parameters are same as the GA-BP neural network for the purpose of comparison. The comparison between measured values and different models, which consist of the BP neural network, the GA-BP neural network, the BP neural network optimized by grey Verhulst model, and the GA-BP neural network optimized by grey Verhulst model, is shown in Figure 4.

Geofluids
As shown in Figure 4, the predicted values of the BP neural network are compared with the measured values. It shows that the accuracy of the predicted values and the consistently measured values is relatively low. In particular, since day 167, the error between measured values and the predicted value has largely increased. The fluctuation of settlement values with time cannot be reflected well by the BP neural network, while the consistency between measured values and predicted values can be basically accepted before day 167, and the consistency is low after day 167. At day 188, the error between measured values and the predicted value is 3.502, while the relative error at day 188 is 8.42%. The changes of error show an increasing trend with the increase of time. And it is basically fitted with the feature of the prediction result, in which the prediction resolution decreased with the increase of the predicted data sets.
Due to the "S" characteristics of foundation pit settlements, the grey Verhulst model is often adopted to predict settlement due to its own characteristics. Then, the BP neural network is optimized by the grey Verhulst model to forecast the settlement. The comparison between predicted values and measured values of the BP neural network optimized by the grey Verhulst model is shown in Figure 5.
As shown in Figure 5, it can be easily found that the consistency is relatively high. It is mainly because the prediction process is optimized by the grey Verhulst model, which is very suitable for the prediction of the settlement. The blue oval ( Figure 5) marks the peaks of the predicted polyline and the measured polyline with high consistency and resolution, while the green rectangle ( Figure 5) marks the part polylines that also show a high consistency. Although the consistency between predicted values and measured values is high, it is worth noting that the error between measured values and predicted values, which is before day 170, is too large. It is mainly because the BP neural network easily falls into a local optimum, which is limited in optimizing weights and thresholds. As for the predicted values after day 170, the resolution of predicted results highly contributes to the correction of the grey Verhulst model. Due to the lack of predicted value accuracy before day 170, the BP neural network is optimized by the GA, which has the ability to optimize the weights and thresholds. The comparison between measured values and the predicted value of the GA-BP neural network are shown in Figure 6.
As shown in Figure 6, it can be obtained that the predicted values of the GA-BP neural network have high accuracy, in which the polyline of predicted values is almost in the internal position of the polyline of the measured values.    12 Geofluids Comparing the Figures 4 and 6, it can be easily found that the prediction accuracy of the GA-BP neural network has largely improved compared to the prediction of the BP neural network, while compared to the BP neural network optimized by the grey Verhulst model, it is worth noting that the GA-BP neural network has low reflection to the fluctuation of the measured values. This conclusion also proved that the grey Verhulst model has good application to predict the settlement around the foundation pits on the other side. Through the above analysis, the GA-BP neural network optimized by the grey Verhulst model is used to predict the settlement of foundation pits, which has taken the advantage of the genetic algorithm and the grey Verhulst model. The comparison between measured values and predicted values of the GA-BP neural network optimized by the grey Verhulst model are shown in Figure 7. As shown in Figure 7, the predicted values of the GA-BP neural network optimized by the grey Verhulst model have shown extremely high resolution and accuracy with the measured values. The polyline of predicted values almost coincides with the polyline of the measured values, which means extremely high consistency between predicted values and measured values. Comparing Figures 5 and 7, it can be seen that the accuracy of predicted values of the BP neural network optimized by the grey Verhulst model is greatly improved by the genetic algorithm, which means the genetic algorithm has good application in optimizing the BP neural network, which has been already optimized by the grey Verhulst model. Similarly, the reflection to the fluctuation of the GA-BP neural network is greatly improved through the optimization by the grey Verhulst model. In general, the GA-BP neural network optimized by the grey Verhulst model has high application and accuracy in the prediction of the settlement.
Usually, mean absolute error (MAE), coefficient of correlation (R 2 ), mean absolute percentage error (MAPE), mean square error (MSE), and root mean square error (RMSE) are used to evaluate the prediction resolution. The model is considered as excellent, when the errors in terms of MAPE, MAE, RMSE, and MSE are close to 0 and the R 2 is close to 1. In this paper, these evaluated parameters are used to further compare the predicted values of the BP neural network, BP neural network optimized by grey Verhulst model, GA-BP neural network, and GA-BP neural network optimized by grey Verhulst model with measured values. The comparison of these models is shown in Table 3.
As listed in Table 3, it can be obtained that the R 2 of the GA-BP neural network optimized by the grey Verhulst model is the closest to 1, while the RMSE, MSE, MAE, and MAPE are all the lowest compared with three other models. It means that the GA-BP neural network is superior to the three models in the prediction of settlement. In particular, though the consistency has greatly increased from the BP neural network to the BP neural network optimized by the grey Verhulst model, these evaluated parameters of the BP neural network optimized by the grey Verhulst are not greatly improved compared with the BP neural network, such as the RMSE has improved from 1.801 to 1.523, while compared to the GA-BP neural network and the BP neural network, the MSE has decreased from 3.245 to 0.942. In particular, the MAPE has decreased from 6.67% to 2.02%. But the analysis above demonstrates that the GA-BP neural network has low reflection to the fluctuation. So, the GA-BP neural network optimized by the grey Verhulst model, which has taken the two advantages of GA and the grey Verhulst model, should have higher resolution than the three other models in theory. The predicted result has well proven this assumption.

Summary and Conclusions
Because of the influence of many factors, settlement around the foundation pit is hard to predict. In this paper, the settlement of D4 is studied due to a large settlement value and being close to the bridge pier of a highway. The internal friction angle, cohesion, bulk density, Poisson's ratio, void ratio, water level changes, permeability coefficient, number of supports, and excavation depth, which influence the settlement around foundation pits, are adopted as the input parameters of the prediction model. Since the supporting time is precisely recorded, the correspondence between the real-time excavation depth and the excavation time can be obtained. Then, the first 150 data sets are used to establish the model, and the last 38 data sets are taken to verify the accuracy of the established model. To obtain a suitable model, the BP neural network, GA-BP neural network, BP neural network optimized by grey Verhulst model, and GA-BP neural network optimized by grey Verhulst model are used to predict the settlement of foundation pits, and the comparisons are detailed analysed. The following conclusions can be advanced from this paper: (1) Due to insufficient consideration of influencing factors in previous studies, the internal friction angle, cohesion, bulk density, Poisson's ratio, void ratio, water level changes, permeability coefficient, number of supports, and excavation depth are taken into consideration in this study. Through the analysis of the prediction results, the selection of these input parameters has high guiding significance for the prediction settlement around foundation pits (2) This paper proposed a new model, which is combined with the BP neural, genetic algorithm, and grey Verhulst models, to predict the settlement of a certain day or excavation to a certain depth in a long period of time in the future, which has guiding significance for engineering construction

Data Availability
The experimental data used to support the findings of this study are included within the article.