Intelligent Prediction Model (IPM) of Foundation Pit Displacement Based on Extreme Learning Machine (ELM) and Its Application

Abstract

In order to effectively predict the dynamic displacement and disaster, according to the analysis of the influencing parameters affecting the deformation of a subway foundation pit supported by piles (walls), the rough set attribute reduction method (RSARM) and the average influence value algorithm (AIVA) are used to simplify the influencing factors of foundation pit deformation. Those simplified factors are taken as the input of the ELM, with the output being the displacement of the foundation pit. Finally, the IPM of foundation pit displacement derived from the ELM is obtained, which is finally used for engineering practice. The results show that it is feasible to simplify the influencing factors of the deformation of the foundation pit by RSARM and AIVA. The proposed IPM of foundation pit displacement has high accuracy and good generalization ability, which can be used for the deformation prediction.

1. Introduction

As the subway foundation pit develops toward greater depth and larger scale, monitoring during the construction period is important to guarantee the safety of surrounding workers, environment, and the foundation pit. However, it is difficult to accurately and rapidly predict the changes of the displacement and corresponding positions of the foundation pit, which mainly attributes to the restriction of such factors (the rigid parameters of the structure of the foundation pit, geological environment, rains, and construction conditions) and makes it difficult to be accurately predict the displacement [1,2]. So far, a variety of intelligent methods have been used for the prediction of displacement changes and corresponding position. For example, Chen et al. [3,4] used GM (1,1) to predict the deformation of foundation pits in time series. To address the problem of low prediction accuracy and limitation of a single model, Cui [5] optimized and improved the BP network model and GM (1,1) model by the PSO optimization algorithm, respectively, so as to establish the PSO-GM-BP model to predict the excavating deformation in the Changsha subway station. The results tell that the PSO-GM-BP model has high applicability and accuracy. Jia et al. [6,7,8] utilized Elman neural network to establish a model about the time series prediction to predict the foundation pit deformation. Wang et al. [9,10] used the ELM to predict the foundation pit deformation. Li et al. [11,12] established a model about the prediction of foundation pits based on the supporting vector machines. Zhao et al. [13] analyzed the applicability of the support vector machine model in deformation prediction of deep foundation pits according to the practical engineering, and the results indicated that the prediction results of the support vector machine model matched well with the measured data. Song et al. [14,15,16,17] established a deformation prediction analysis model according to the BP neural network (BPNN). In order to improve the precision of predicting the deformation of foundation pits, the genetic algorithm (GA) was used to optimize the BPNN [18,19] and a modified BPNN was proposed. The results showed that the modified BPNN optimized by genetic algorithm has high prediction ability on foundation pit deformation compared with the random forest model (RFM) and support vector regression model (SVRM). Feng [20] used an artificial bee colony algorithm to improve the BPNN and applied the modified method to the composite ground excavation deformation prediction. The result showed that the modified BPNN has higher predicting accuracy. To reduce the effect of disturbing parameters in pit deformation prediction, Liu [21] introduced the wavelet transforming method into the gray BPNN method and eventually formed the improved gray BPNN, which was later used for the prediction of the deformation of the deep foundation pits. The result showed that the gray BPNN improved by the wavelet transform method has high accuracy and strong reliability in pit deformation prediction. To solve the problem of high noise in pit deformation prediction, Cui [22] introduced the residual network (ResNet), combined with GA and back propagation (NNBP) algorithm, to create a new algorithm whose effectiveness was verified by actual engineering examples. In addition, in order to solve the difficult problem of the relation between real-time excavation depth and excavation time, Liu [23] proposed a new method for foundation pit settlement prediction by the gray VelHulst model and genetic algorithm. To improve the prediction precision of the deformation of the adjacent tunnels caused by pit excavation, the artificial bee colony algorithm was adopted in the random forest model [24], which was finally used for the prediction of the tunnel deformation caused by adjacent pit excavation. As a result, the new model has higher computational efficiency and accuracy. In addition, Jing [25] proposed a model to improve the predicting accuracy of pit deformation by pit deformation using the improved supply and demand exponential power product (ISDO-EPP) model. According to the comparison between the improved model and existing traditional models, it was concluded that the ISDO-EPP model was effective for pit deformation prediction. Moreover, Ma [26] combined NARX neural network and principal component analysis (PCA) to form a pit settlement prediction model, which was verified by actual engineering and proved to have good ability to predict pit displacements.

Currently, there are two ways to predict displacement based on neural networks. One method is to select a number of influencing factors as input variables based on experience to predict the displacement. Another one is to directly use the displacement time series to predict the displacement. However, some problems exist in the above two ways. The input variables of the first way are artificially selected, hence it is difficult to guarantee the correlation within the predicted displacements according to the selected factors, which easily leads to uncertainty in the accuracy. In addition, the second way has poor accuracy in the long term and suddenly changes in displacement prediction. Moreover, most prediction models need to adjust parameters, whose accuracy are greatly affected by the parameters. Hence, the displacement value at a fixed depth (position) is often obtained, which is difficult to be applied to the displacement analysis and safety warning of foundation pit engineering.

According to the analysis of the influencing factors of subway foundation pits, supported by bracing piles (walls), deformation, quantifying the expressions of the influencing factors to prepare data collection, factor selection, and model training is, firstly, concentrated on. Secondly, the filtering method of the factors affecting the displacement of the subway foundation pit supported by brace piles (walls) is studied to simplify the model input variables and improve prediction accuracy. Thirdly, the optimization of key parameters of the prediction model is determined, so as to solve the difficulty in determining key parameters of the model. Finally, based on the above study, a new prediction model is proposed to predict the deformation of the foundation pits and provide methods and data support for early warning of foundation pit construction.

2. Analysis and Quantification of Influencing Factors of Foundation Pit Deformation

With the improvement of the environmental protection requirements around foundation pits, current engineering designing of foundation pits is shifting from the designing mode of strength and stability controlling to deformation controlling. Hence, it is significant to control the deformation of foundation pits. However, the influencing factors of foundation pit deformation are complicated, and it is necessary to conduct analysis on the influencing factors before predicting the deformation by neural network.

2.1. Analysis of Influencing Factors of Foundation Pit Deformation

As the influencing factors of the deformation of foundation pits are complicated, they were determined to four parts, which are geology, designing, construction, and environment.

2.1.1. Influence of Geology

The influence of geology can be determined by the physical and mechanical properties of site rock and soil and groundwater level. The physical and mechanical properties of rock and soil are the most essential reason, which determines the pressure difference between the two sides of the retaining structure and the changes of the stress state of the soil, including the $\varphi$ (internal friction angle), $c$ (cohesion), $\gamma$ (bulk density), $E$ (elastic modulus), and $\mu$ (Poisson’s ratio). In addition, the hazards of groundwater to the pit included the effect of static water pressure on the stability of the foundation pit, quicksand, piping, subduction, and corrosion under hydrodynamic pressure. Therefore, the foundation pit is often used for reducing the groundwater level, to increase the strength of the soil, so that the hydraulic slope and the confined water can be effectively reduced. However, dewatering the foundation pit significantly reduces the pore water pressure and enhances the effective stress of the soil, which significantly causes the settlement of the soil around the pit and finally leads to the uneven settlement of surrounding pipelines, structures, and roads.

2.1.2. Influence of Designing

The designing factors affecting the foundation pit mainly include the rigidity of the supporting enclosure system, supporting spacing, insertion depth of the enclosure structure, and reinforcement of the passive zone. The greater rigidity of the supporting enclosure system can not only effectively resist the active earth pressure, but also reduce the displacement caused by soil deformation. In addition, the pit foundation with relatively large supporting rigidity and low rigidity of retaining structure has a soil arching effect that redistributes the active earth pressure behind the retaining structure [27,28,29], which can make full use of the soil resistance. The supporting spacing includes horizontal and vertical directions, which should meet the supporting strength and stability when ensuring the convenience and economy of construction. Moreover, it is necessary to appropriately reduce the excavation depth before the first support and reduce the distance between the bottom support and the bottom of the pit, since the former is very important for reducing the displacement of the enclosure structure, and the latter is significant for reducing the earth pressure in the passive zone, and a larger insertion depth of the enclosure structure can effectively control the pit-bottom uplift caused by the soil. According to [30,31], it is clear that the insertion depth of the enclosure structure generally reaches (0.6~0.8) H; therefore, when the deformation controlling is relatively high, it is suggested to increase the insertion depth. As the strength and rigidity of the soil in the passive zone are very important for controlling the uplift of the pit bottom and the bottom of the enclosure structure, it is necessary to determine whether to reinforce, or not, and the reinforcement parameters according to the actual situation.

2.1.3. Influence of Construction

According to Bian et al. [32], there are 342 accidents related to foundation pit construction, in which 270 accidents are caused by the fault of construction. It should be noted that the main construction factors affecting the deformation of foundation pits include the current excavation depth, the depth of the retaining structure buried below the excavation surface, over-excavation, the timeliness of support installation, and the axial force of each support, etc. More specifically, the current excavation depth and the depth of enclosure structure buried below the excavation surface can reflect the “promotion” and “inhibition” effects of the enclosure structure. When increasing the excavation depth, the deformation of the foundation pit tends to increase. As a result, the ability of the passive side restraining deformation is gradually reduced, which leads to an increase in deformation. In addition, over-excavation leads the passive soil to stay in a high stress state. When the structure insertion depth is small, over-excavation may cause the overall instability of the foundation pit.

2.1.4. Influence of Environment

The environmental factors affecting the deformation of foundation pit mainly contain the buildings (structures) around the foundation pit, pipelines, roads, various overloading effects, and possible pipeline leakage and rainwater immersion.

Table 1. Factors of the foundation pit deformation.

Number	Factor Category	Specific Factor
a1	Geology	Physical and mechanical properties of rock and soil
a2		The rise and fall of groundwater
b1	Construction	Current excavation depth
b2		The depth of enclosure structure inserted into the bottom of pit
b3		Airline cumulative time
b4		Whether the support is installed in time
b5		Over-excavation or not
b6		Supporting axial force
c1	Designing	The rigidity of the enclosure structure
c2		Supporting stiffness
d1	Environment	Daily rainfall
d2		Leakage of foundation pit
d3		Ground overload around pit
d4		Daily temperature

shows the influence of each factor on foundation pit deformation:

2.2. Quantitative Expression of Factors Affecting the Foundation Pit Deformation

According to Table 1, both qualitative and quantitative factors significantly affect the foundation pit deformation. As shown in

Table 2. Quantification of influencing factors.

Number	Factor	Category	Graded Value
			1	2	3	4
b4	Whether the support is installed in time	Qualitative factors	Very timely	Very timely	Not timely	Particularly untimely
b5	Over-excavation		None	There is a local over-excavation smaller than one layer and smaller than B/3	There is over-excavation greater than one layer or greater than B/3	There are more than two layers and more than B/3 over-excavation
d2	Leakage of foundation pit		None	Drip water seepage and very low water pressure	Strand-shaped water seeps and the seeping water pressure is small.	Gushing water seepage
d3	Ground overload around the pit		None	There is a locally overload greater than 10%P and less than 20%P	Local overloads greater than 20%P and less than 30%P	There is a locally overload greater than 30%P
a2	Groundwater level rise and fall/mm	Quantitative factors	Measured
b1’	Active earth pressure of excavated part/kN		Measured
b2’	Passive earth pressure in passive zone/kN		Measured
b3	In the air time/d		Measured
b6	Axial force of each support/kN		Measured
c1	Rigidity of enclosure structure/m4		Measured
c2	Support stiffness/m4		Measured
d1	Daily rainfall/mm		Measured
d4	Daily average temperature/°C		Measured

, because the factor a1 (physical parameters of rock and soil) is complicated and difficult to be expressed directly, it is suggested to be combined with factors b1 (the depth of current excavation) and b2 (the depth of enclosure under current excavation surface) to form an active excavation area, and finally form the active soil pressure $\mathrm{b}{1}^{\prime }$ and passive soil pressure $\mathrm{b}{2}^{\prime }$. Table 2 also provides the quantified expression of each factor with B and P, which denote the width of the pit and designing allowable load, respectively.

3. Research on Filtering Methods of Influencing Factors

Before applying the neural network to predict the displacement of the foundation pit, it is necessary to solve the selection of the model input variables since, when some irrelevant independent variables are input into the neural network, the complexity of the model will be undoubtedly increased, and the accuracy is decreased. Therefore, it is necessary to choose meaningful and relevant independent variables as input variables when using the neural networks to predict displacement. However, which variables should be selected as input variables, and what principles should be adopted for the selection are issues that need to be studied.

Two steps are adopted in this section to filter the parameters affecting the deformation of the foundation pit. One step is eliminating the redundant and repeated factors in the influencing factor set to obtain the “one-time reduction set”. Another step is to quantify the correlation between the influencing factors in the “one-time reduction set” and the predicted displacement, and then strike out the influencing factors with minimal correlation in the ‘one-time reduction set’ according to the correlation analysis results, and the “simplest set” is finally obtained. Therefore, the “simplest set” can be used for the input variable set for the prediction model.

3.1. Filtering of the Redundant Factors According to the RSARM

One of the advantages of RSARM is that it can effectively remove redundancy and simplify the expression of knowledge. Hence, the attribute reduction function of the rough set is adopted to reduce the factors influencing the deformation of subway foundation pits supported by bracing piles (walls) [33]. Hence, each condition attribute $C_{\mathrm{i}}$ is an equivalence relationship (indistinguishable relationship) in the original decision information table, while the division of the universe $U$ is expressed as $U|C_i$, and the set $C$ of all the condition attributes in the decision information table is expressed as $U|C$. Similarly, the division of the domain $U$ formed by the decision attribute $D$ is expressed as $U|D$. However, it is noticeable that not all the conditional attributes of the decision information table are equally important or necessary since some of the conditional attributes may be redundant and repeated. Therefore, the purpose of attribute reduction is to find some necessary condition attributes in the condition attribute set, so that the division of these condition attributes of the decision attributes is the same as the division of all condition attributes of the decision attributes.

At present, many researchers have been concentrating on the research about the RSARM, and the reduction algorithm based on GA is adopted in this research.

3.2. Factor Filtering Based on the Average Influence Value Algorithm

When the redundant repetitive factors are deleted through the rough-set attribute reduction function, it cannot guarantee the greater correlation between the influencing factors in the “one-time reduction set” and the predicted displacement. Therefore, the mean impact value (MIV) algorithm is adopted in this paper to analyze the correlation between each influencing factor and the displacement of the foundation pit.

As [34] set out, the mean impact value (MIV) algorithm can be used to express the degree of influence of each input on each output. As shown in Figure 1, the solution process of the MIV method is: ① Model calculation. Using the original training data to train to obtain the initial prediction model. ② Increase or decrease of independent variables. Each input variable in the original training data is respectively added and subtracted by a certain percentage (usually 10%; one variable is changed each time, and other variables remain unchanged) based on its original value to form two new training samples. ③ Finding the prediction difference. This is because the difference between the two prediction results means the influence of the variable on the output value and the average influence value of the input variable on the output variable can be obtained by averaging multiple observations. Finally, by repeating the above process, the average influence value of all input variables can be effectively obtained. It should be noticed that the absolute value of MIV indicates the importance of the influence of input variables on the network output, and the symbol indicates the direction of the correlation. Therefore, the input variables can be filtered based on MIV.

4. Multi-Intelligence Prediction Model for Foundation Pit Displacement

According to the aforementioned research, a new model is proposed and is combined with the ELM model in this section.

4.1. ELM Model

Compared with the traditional feedforward neural network that uses the gradient descent method to adjust the weight, the ELM model has its own advantage [35]. According to Figure 2, the ELM model includes three layers: the input layer, hidden layer, and output layer.

As shown in Figure 2, there are n independent variables corresponding to n input-layer neurons, and m dependent variables corresponding to m output-layer neurons, with the hidden layer containing one neuron. The expressions of the connection weight $\omega$, $\beta$, and the hidden layer threshold b are as follows:

$$\mathbf{\omega }={\left(\begin{array}{cccc}{\omega }_{11}& {\omega }_{12}& ...& {\omega }_{1n}\\ {\omega }_{21}& {\omega }_{22}& ...& {\omega }_{2n}\\ ...& ...& ...& ...\\ {\omega }_{l1}& {\omega }_{l2}& ...& {\omega }_{ln}\end{array}\right)}_{l\times n}={\left(\begin{array}{c}{\mathbf{\omega }}_1\\ {\mathbf{\omega }}_2\\ ...\\ {\mathbf{\omega }}_l\end{array}\right)}_{l\times 1}$$

(1)

$$\mathbf{\beta }={\left(\begin{array}{cccc}{\beta }_{11}& {\beta }_{12}& ...& {\beta }_{1m}\\ {\beta }_{21}& {\beta }_{22}& ...& {\beta }_{2m}\\ ...& ...& ...& ...\\ {\beta }_{l1}& {\beta }_{l2}& ...& {\beta }_{lm}\end{array}\right)}_{l\times \mathrm{m}}$$

(2)

$$\mathit{b}={\left(\begin{array}{c}{b}_1\\ b_2\\ ...\\ b_l\end{array}\right)}_{l\times 1}$$

(3)

where ${\omega }_{ji}$ is the weight of the connection between the i neuron of the input layer and the j neuron of the hidden layer, ${\beta }_{jk}$ means the weight of the connection between the j neuron of the hidden layer and the k neuron of the output layer, and $b_m$ denotes the threshold of the neuron of the hidden layer.

The input x and output y corresponding to the training set containing N samples are:

$$\mathit{x}={\left(\begin{array}{cccc}{x}_{11}& x_{12}& ...& x_{1N}\\ x_{21}& x_{22}& ...& x_{2N}\\ ...& ...& ...& ...\\ x_{\mathrm{n}1}& x_{\mathrm{n}2}& ...& x_{\mathrm{n}N}\end{array}\right)}_{n\times N}={\left(\begin{array}{cccc}{\mathit{x}}_1& {\mathit{x}}_2& ...& {\mathit{x}}_N\end{array}\right)}_{1\times N}$$

(4)

$$\mathit{y}={\left(\begin{array}{cccc}{y}_{11}& y_{12}& ...& y_{1N}\\ y_{21}& y_{22}& ...& y_{2N}\\ ...& ...& ...& ...\\ y_{\mathrm{m}1}& y_{\mathrm{m}2}& ...& y_{\mathrm{m}N}\end{array}\right)}_{\mathrm{m}\times N}={\left(\begin{array}{cccc}{\mathit{y}}_1& {\mathit{y}}_2& ...& {\mathit{y}}_N\end{array}\right)}_{1\times N}$$

(5)

Supposing $g\left(x\right)$ represents the activation function of the hidden layer neuron, the output of the neural network t in Figure 2 is:

$$\mathit{t}={\left(\begin{array}{cccc}{\mathit{t}}_1& {\mathit{t}}_2& ...& {\mathit{t}}_N\end{array}\right)}_{1\times N}={\left(\begin{array}{cccc}{t}_{11}& t_{12}& ...& t_{1N}\\ t_{21}& t_{22}& ...& t_{2N}\\ ...& ...& ...& ...\\ t_{\mathrm{m}1}& t_{\mathrm{m}2}& ...& t_{\mathrm{m}N}\end{array}\right)}_{\mathrm{m}\times N}$$

(6)

$$t_{pq}={\displaystyle \sum _{j=1}^l{\beta }_{jp}}g({\mathit{\omega }}_j\cdot x_q+b_j)$$

(7)

where ${\omega }_j\cdot x_q$ means the inner product of ${\mathit{\omega }}_j$ and ${\mathit{x}}_q$. Equation (6) can be expressed as

$$\mathit{H}\mathit{\beta }={\mathit{t}}^{\prime }$$

(8)

where $t^{\prime }$ denotes the transposition of t, and H is the hidden layer output of the neural network, which is:

$$\mathit{H}={\left(\begin{array}{cccc}g({\mathit{\omega }}_1\mathit{\cdot }{\mathit{x}}_1+b_1)& g({\mathit{\omega }}_2\mathit{\cdot }{\mathit{x}}_1+b_2)& ...& g({\mathit{\omega }}_l\cdot {\mathit{x}}_1+b_l)\\ g({\mathit{\omega }}_1\cdot {\mathit{x}}_2+b_1)& g({\mathit{\omega }}_2\mathit{\cdot }{\mathit{x}}_2+b_2)& ...& g({\mathit{\omega }}_l\mathit{\cdot }{\mathit{x}}_2+b_l)\\ ...& ...& ...& ...\\ g({\mathit{\omega }}_1\cdot {\mathit{x}}_N+b_1)& g({\mathit{\omega }}_2\cdot {\mathit{x}}_N+b_2)& ...& g({\mathit{\omega }}_l\mathit{\cdot }{\mathit{x}}_N+b_l)\end{array}\right)}_{N\times l}$$

(9)

Since $\omega$ is fixed, and the threshold b is also fixed, the network training is equivalent to seeking the least square solution problem corresponding to the following formula:

$$\underset{\mathit{\beta }}{\min }\Vert \mathit{H}\mathit{\beta }-t^{\prime }\Vert$$

(10)

whose solution is:

$$\widehat{\mathit{\beta }}={\mathit{H}}^{+}{\mathit{t}}^{\prime }$$

(11)

where ${\mathit{H}}^{+}$ is the Moore–Penrose generalized inverse of H.

4.2. Optimization of Model Parameters of ELM

There is one parameter of neurons 1 in the hidden layer in the ELM that needs to be determined artificially, which means there is only one hyperparameter that has an important impact on the training accuracy and generalization capability of the ELM model. Hence, this section adopts the cross-validation (CV) method to optimize the number of neurons in the hidden layer of the ELM model. K-fold cross-validation is used to obtain the optimal number of neurons l in the hidden layer of ELM. In reality, the high precision and good generalization ability can be obtained by the ELM model after the number of neurons in the hidden layer is determined by cross-validation optimization.

4.3. IPM Based on ELM

Based on the maximum horizontal displacement and depth of inclination of subway foundation pits, the IPM based on the ELM is proposed according to the two-step filtering method and the ELM model, whose process is shown in Figure 3.

5. Application of Intelligent Forecasting Model

5.1. Project Overview

One foundation pit in Nanchang Rail Transit Line 2 with an 800 mm thick and 21.4 m deep underground continuous wall is chosen as the example, whose supporting system is a reinforced concrete support plus two steel supports. The vertical distance between the supports is 6 m. The reinforced concrete supporting section is 800 mm × 1000 mm, which was set with the horizontal distance 9 m. The outer diameter of the rigid support reaches 609 mm. The wall thickness and support distance are 16 mm and 3 m, respectively.

5.2. Model Application

The application of the model is shown in Figure 3, and the analysis and quantification of the displacement are provided in Table 2. The training set data involves 50 complete data (including all influencing factors) of the measurement points, ZQT15 (mileage YCK27+314.700) and ZQC16 (mileage YCK27+330.500), of the completed part of the excavation, which include the data from the start to the completion of the excavation. The testing set data involves measuring point ZQT18 (mileage YCK27+349.000) and 25 data of all influencing factors.

5.2.1. Filtering of the Influencing Factors

The filtering of the factors influencing the displacement of the foundation pit is carried out according to a two-step filtering method. Firstly, the RSARM is used to eliminate redundant and repeated factors, and, hence, a reduction set is obtained. Secondly, the relevance of the influencing factors in the reduction set is evaluated, and the influencing factors with small correlation are eliminated. As a result, the simplest set of influencing factors is obtained. Finally, the simplest set is used for the input variable of the prediction model.

It is suggested to discretize the maximum horizontal displacement u_max and then carry out reduction work on the attributes shown in Table 2 through the attribute reduction algorithm, from which the one-time reduction set, including the rise and fall of groundwater level a2, the active earth pressure of the excavated part $\mathrm{b}{1}^{\prime }$, the flying time b3, the axial force b6-1 of support 1, the axial force b6-2 of support 2, the support stiffness c2, the daily rainfall d1, and the daily average temperature d4, can be effectively obtained.

After removing the redundant influencing factors by the rough set attribute reduction algorithm, the MIV of each influencing factor in the primary reduction set and the maximum horizontal displacement of the inclinometer can be obtained by the average influence value algorithm, which can be used for the evaluation of the correlation between the influencing factors of the primary reduction set. In order to eliminate the influence of randomness on the results, 30 repeated experiments were carried out and presented in

Table 3. Results of filtering by MIV method.

Factor	Sequence	Correlation (MIV)
b6-2	1	0.290
$\mathrm{b}{1}^{\prime }$	2	0.217
b6-1	3	0.175
c2	4	0.125
b3	5	0.095
d1	6	0.006
d4	7	−0.028
a2	8	−0.065

. It can be concluded that a2, b3, b6-1, b6-2, and c2 have large correlation with the maximum horizontal displacement of the inclination, with the absolute value of the cumulative weighted value exceeding 95%. Therefore, it can be used as the main factor that affects the maximum horizontal displacement of the inclinometer. However, the correlation between d1 and d4 is very small, which can be filtered.

5.2.2. Model Construction

According to the optimization of the four-fold cross-validation on the number of hidden layer neurons, there are 30 neurons in the hidden layer. The comparison of the training time and accuracy of the ELM model with different numbers of hidden layer neurons are shown in Figure 4. It is seen that the training time increases slightly as the increasing of the number of hidden layer neurons and, both, the root mean square difference of the training set (RMSE) and the average relative error (MRE) decrease with the increase of the number of hidden layer neurons. When the number of hidden layer neurons reaches 80, the predicted value of the training set is almost the same as the true value, which showed an over-fitting phenomenon. It is also notable that the mean square error and average relative error of the test set decrease firstly, and then increase with the increase of the number of hidden layer neurons, and finally reach a minimum value when the number of hidden layer neurons is 30, which states the correction of the number of neurons in the hidden layer according to the four-fold cross-validation.

Based on the above analysis, the number of neurons in the hidden layer is determined to be 30, and the ELM network structure is also completely determined.

5.3. Model Evaluation

Figure 5a and b show the prediction results after the model is trained. It can be seen that most of the prediction values of the training set are well matched with the true value, with the average relative error of the maximum horizontal displacement of the inclinometer at 4.38% and the mean square error at 0.49 mm. In addition, the errors of average relative and mean square of the position in the same depth are 3.18% and 0.36 m, respectively. In addition, the prediction result of the testing set can also reflect the changing trend of the data of the testing set, even though the average relative error and mean square error are slightly larger than those of the training set. Furthermore, it is notable that the average relative error of the maximum horizontal displacement of the inclination is 8.68%, and the mean square error reaches 0.87 mm. Moreover, the errors of the average relative and the mean square of the position at the same depth are 7.59% and 0.71 m, respectively. Therefore, the precision of the IPM is accurate enough to meet the requirement of engineering, which also has high accuracy and good generalization ability.

6. Conclusions

The main conclusions are as follows:

(1) It is feasible to use the RSARM and AIVA to simplify the influence factors of the displacement of the foundation pit. It is also advised to apply the ELM to the foundation pit displacement prediction. It is suggested to optimize and determine the hyperparameters of the ELM model by the K-fold cross-validation method.

(2) The average relative error of the maximum horizontal displacement of the inclination predicted by the foundation pit displacement prediction model is 8.68%, with the mean square error of 0.87 mm. In addition, the average relative error of the corresponding depth is 7.59%, with the mean square error of 0.71 m, which is accurate enough to meet the requirements in actual engineering. Moreover, the IPM of the foundation pit displacement has high precision and good generalization ability, which is recommended for the pre-controlling measures of foundation pits in advance.

(3) The IPM of foundation pit displacement is developed by combining RSARM, AIVA, and limit learning machine. Theoretically, the analysis of the impact of the excavation of the foundation pit on the adjacent tunnel and the analysis of the impact of tunnel construction on the surrounding environment can also be carried out using the theories. In addition, the geotechnical parameters of the foundation pit have great uncertainty, which is very important to form a geotechnical parameter uncertainty analysis method based on the whole construction process to guide engineering practice. It is suggested that the subsequent integration of multi-output support vector machine algorithm and Bayesian theory can be considered so as to establish a probabilistic inverse analysis method of rock parameters combined with monitoring data to study the uncertainty of rock parameter and dynamics with construction excavation, to better guide actual engineering.