Research on a BP Neural Network Slope Safety Coefficient Prediction Model Based on Improved Sparrow Algorithm Optimization

Abstract

Through the stability evaluation of a slope, a landslide geological disaster can be identified, and the safety and risk control of a project can be ensured. This work proposes an improved sparrow search algorithm to optimize the slope safety factor prediction model (ISSA–BP) of a BP neural network, through an improvement in two aspects: introducing dynamic weight factors and reverse learning strategies to realize adaptive searches. The optimal value improves a defect in the traditional model, preventing it from easily falling into the local minimum. First, combined with 352 sets of actual slope data, three machine learning models were used to predict the safety factor of the slope. Then, the accuracy index was used for evaluation. Compared with other models, the MAPE, RMSE, and R² of the ISSA-BP model were 1.64%, 0.0296, and 0.99, respectively, and the error was reduced by 78% compared with the BP neural network, showing better accuracy. Finally, the three models were applied to the slope stability analysis of Tianbao Port in Wenshan Prefecture. The research shows that the predicted value of the ISSA–BP model was the closest to the actual safety factor, which verified the experimental results. The improved ISSA–BP model can effectively predict the safety factor of slopes under different conditions, and it provides a new technology for slope disaster warning and control.

1. Introduction

In recent years, with the increasing economic development and urbanization around the world, landslides have become one of the main types of geological hazards that cause huge economic losses and human casualties [1]. According to statistics on landslide work in Iran, approximately 187 people die from landslides every year, and the total economic loss is approximately USD 12.7 billion [2].

In landslide geological hazard prevention, traditional research methods are complex and require consideration of soil–interface interaction parameters, soil stiffness, and consolidation coefficients [3]. Existing methods only address the effects of specific slope conditions (e.g., height, slope angle, and soil properties) and do not propose a general solution. In general, factors such as soil conditions, geological conditions, and slope type influence the initial slope and horizontal displacement of each node [4,5]. Some scholars believe that the safety factor of slopes is influenced by key parameters [6,7], such as the initial ground properties (e.g., soil weight ($\gamma$), angle of internal friction ($\phi$), height ($H$), cohesion ($c$), slope angle ($\alpha$), and pore water pressure ($ru$)) [8]. It has also been suggested that specific formulas and methods can be used to predict reliable values for the occurrence of landslides [9,10,11,12]. However, since none of the previous studies considered the input layer with the greatest influence, the abovementioned formulas and methods have certain limitations, resulting in deviations in the results.

Although many mathematical functions have been proposed by previous authors, the factor of safety of side slopes (FOS) is a highly nonlinear problem that is more suitable to be solved by artificial neural network methods [13,14,15]. In the 1980s, Boltzmann machines and back propagation (BP) algorithms [16] made neural networks a research hotspot again. Back propagation (BP) is a common method for training artificial neural networks [17], which can solve some complex engineering problems, although its initial weights and thresholds have a great impact on the prediction effect [18], while the defects of a slow learning speed and easily falling into the local minima [19,20,21], mean the initial weights and thresholds must be optimized. Recently, genetic algorithms (GAs) [22,23] have been used to train neural networks, although it was found that when the network is complex and the training samples are large, the convergence speed of the GAs becomes slower and, therefore, the convergence accuracy is affected and reduced. Scholars have developed a large number of population intelligent optimization algorithms to address the flaws of artificial neural networks, such as the particle swarm (PSO) algorithm [24,25], ant colony algorithm (ACO) [26,27,28], and cuckoo search algorithm (CS) [29,30,31]. The population intelligence optimization algorithm can continuously perform an integrated search, while it has a good search capability because of its strong parallelism and stability in solving optimization problems.

To optimize artificial neural networks, as much as possible, scholars have also proposed combinations of algorithms to improve their performance [32,33]. Zhang et al. [34] trained PSO and BP together and concluded that the PSO–BP model improved the speed and accuracy of the suboptimization compared with a single model. In 2020, a new population intelligence optimization algorithm, the sparrow search algorithm (SSA), was proposed [35]. It was quickly optimized by researchers. Liu and Shu [36] used chaotic optimization methods to improve the convergence speed of SSA models. Zhu et al. [37] proposed a novel sparrow algorithm to improve the weights and thresholds of the reactor model. Zhang et al. [38] improved the SSA with sinusoidal chaotic mapping and applied it to the study of wind speed prediction models; the optimization results showed that, compared with the single BPNN algorithm, the prediction accuracy of the improved model was significantly improved. The machine learning method provided a new solution for the stability of geotechnical engineering. Reviewing the literature, few scholars have paid attention to the overall performance comparison of different models, and almost no work has evaluated the BP neural network optimized based on the improved sparrow algorithm (ISSA–BP) method for slope stability analysis.

This study introduces an improved sparrow algorithm to optimize the BP neural network model. The sparrow algorithm was improved by introducing dynamic weight factors and reverse learning strategies to realize an adaptive search for the optimal value and to improve the traditional model’s defect of easily falling into the local minimum. Furthermore, 352 sets of actual slope cases were evaluated using three machine learning models, and the model accuracy index was used to determine the best neural network model. Finally, the slope safety coefficient was predicted using field survey data from the Tianbao Port, Wenshan Prefecture, and the accuracy of the prediction results was verified by the ISSA–BP model.

2. Improved Sparrow Search Algorithm and PSO Optimization Algorithm

2.1. PSO Algorithm Optimization Model

Particle swarm optimization (PSO) is a stochastic global search technique that simulates the migratory and aggregation behavior of bird flocks during foraging using simulation models of group activity patterns to solve various optimization issues [39,40].

Suppose there is a population of N particles in a D-dimensional space, then, the dimension $D$ can be determined according to Equation (1).

$$D=l\times m+m\times n+m+n$$

(1)

where (1) denotes the neural network’s input layer; $m$ denotes the neural network’s hidden layer; $n$ denotes the number of nodes in the neural network’s output layer.

In particle swarm space, to avoid blind search, the particle’s position and velocity are usually limited to within a reasonable range; if the dimension is $D$, the particle’s location in population $X_i$ is denoted as:

$$X_i=\left(x_{i1},x_{i2},\cdot \cdot \cdot ,x_{iD}\right),i\in \left[1,2,\cdot \cdot \cdot ,N\right];X_i\in \left[-X_{\max },X_{\max }\right]$$

(2)

the particle velocity $V_i$ in the population can be represented as:

$$V_i=\left(v_{i1},v_{i2},\cdot \cdot \cdot ,v_{iD}\right),i\in \left[1,2,\cdot \cdot \cdot ,N\right];V_i\in \left[-V_{\max },V_{\max }\right]$$

(3)

each particle searches for its optimal position and the searched optimal position, i.e., the individual extreme value is noted as $P_{best}$:

$$P_{best}=\left(p_{i1},p_{i2},\cdot \cdot \cdot ,p_{iD}\right),i\in \left[1,2,\cdot \cdot \cdot ,N\right]$$

(4)

in the whole particle swarm, each particle shares information with other particles, and the global optimal position reached by all particles during the successive generations of search (i.e., the global extremum) is noted as $G_{best}$:

$$G_{best}=\left(p_{g1},p_{g2},\cdot \cdot \cdot ,p_{gD}\right),i\in \left[1,2,\cdot \cdot \cdot ,N\right]$$

(5)

where all particles in the particle swarm update parameters according to the individual and global extremums. The particle optimization algorithm is formulated as follows [41]:

$$v_{i+1}=\omega v_i+c_1{r}_1\left(P_{best}-x_i\right)+c_2{r}_2\left(G_{best}-x_i\right)$$

(6)

$$x_{i+1}=x_i+v_{i+1}$$

(7)

where (6) $c_1$ is the empirical coefficient of a single particle; $c_2$ is the empirical coefficient of a particle swarm; $r_1$ and $r_2$ are arbitrary values in the [0, 1] range; $\omega$ is the non-negative inertia weight, which can control the speed of the particles and reduces linearly as the number of repetitions increases. Its value is proportional to the performance of global optimization and inversely proportional to the performance of local optimization. Its update is shown in Equation (8):

$$\omega \left(t\right)={\omega }_{\max }-\frac{\left({\omega }_{\max }-{\omega }_{\min }\right)}{t_{\max }}$$

(8)

2.2. Optimization of the Sparrow Search Algorithm

2.2.1. Sparrow Search Algorithm

Inspired by sparrow populations’ foraging and antipredatory behaviors, a novel population intelligence optimization algorithm with a bionic principle of the detector–follower–alarm model, called the sparrow algorithm, was proposed in 2020. Each sparrow in the population has a matching solution in the SSA. The detector is responsible for finding areas with high food reserves and adaptability and for directing the population to food-producing regions and directions. To get more fitness, the followers choose to forage using the detector with the highest fitness value. Some followers watch the detectors, search for food around them, and even compete for food. In addition, they alert and move quickly to a safe area when danger is recognized, while the sparrows in the middle area move freely and approach other sparrows. At the same time, when the alarm value is greater than the safety threshold, the detector will take other sparrows to other areas to find food [42,43].

The sparrow collection matrix is as follows:

$$X={\left[x_1,x_2,\cdots ,x_n\right]}^T,x_i=\left[x_{i,1},x_{i,2},\cdots ,x_{i,d}\right]$$

(9)

a sparrow’s fitness can be represented by the following mixture:

$$F_x={\left[f\left(x_1\right),f\left(x_2\right),\cdots ,f\left(x_N\right)\right]}^T$$

(10)

$$f\left(x_i\right)=\left[f\left(x_{i,1}\right),f\left(x_{i,2}\right),\cdots ,f\left(x_{i,d}\right)\right]$$

(11)

where (10) $F_x$ represents the fitness matrix of individuals, while the individuals with high fitness values become detectors and search for food.

The updated formula for the detector is as follows:

$$X_{i,j}^{t+1}=\left\{\begin{array}{c}{X}_{i,j}^t\cdot \exp \left(\frac{-i}{\alpha \cdot ite{r}_{\max }}\right),R_2<ST\\ X_{i,j}^t+Q\cdot L,R_2⩾ST\end{array}\right.$$

(12)

where (12) $t$ is the number of iterations; $j=\left(1,2,\cdot \cdot \cdot ,d\right)$, and $X_{i,j}^t$ is the i-th individual’s location in the j-th dimension; $\alpha$ is a random number distributed between (0 and 1); $ite{r}_{\max }$ is the number of iterations that can be performed; $R_2$ is a number chosen at random between (0 and 1); ST denotes a warning threshold taking the value interval (0.5 and 1).

When $R_2<ST$, the alarm value is less than the safe value, indicating that the population is in a safe position, and the detector conducts an extensive search and approaches the best position. When $R_2>ST$, the alert value exceeds the safe value, the detector issues an alarm, stops foraging, and then, moves to the current best position.

The formula for updating the location of the followers is as follows:

$$X_{i,j}^{t+1}=\left\{\begin{array}{c}Q\cdot \exp \left(\frac{X_{\mathrm{worst}}-X_{i,j}^t}{i^2}\right),i>\frac{n}{2}\\ X_p^{t+1}+\left|X_{i,j}^t-X_p^{t+1}\right|\cdot A^{+}\cdot \mathrm{L},\mathrm{otherwise}\end{array}\right.$$

(13)

where (13) $X_p$ represents the sparrow’s best position in the current population, and $X_{worse}$ represents its worst position.

Finally, during foraging, some sparrows in the population will be responsible for vigilance, comprising 10–20% of the entire population. When danger is sensed, all sparrows will stop foraging and relocate to a different place. The formula for updating these sparrows’ positions is as follows:

$$X_{i,j}^{t+1}=\left\{\begin{array}{c}{X}_{\mathrm{best}}^t+\beta \cdot \left|X_{i,j}^t-X_{\mathrm{best}}^t\right|,f_i>f_g\\ X_{i,j}^t+k\cdot \left(\frac{\left|X_{i,j}^t-X_{\mathrm{worst}}^t\right|}{\left(f_i-f_w\right)+\epsilon }\right),f_i=f_g\end{array}\right.$$

(14)

where (14) $\beta$ is the step adjustment factor, and it follows the standard normal distribution; $f_g$ and $f_w$ denote the global ideal fitness value and global worst fitness value of the sparrow, respectively, and $k$ is the sparrow’s movement direction.

2.2.2. Improvement of the Sparrow Search Algorithm (ISSA)

Owing to the idea of following the optimal solution throughout the procedure, all particles may converge to the ideal solution more quickly. This non-two-way feedback information flow leads to a single change in the particle’s position, so the PSO algorithm tends to cause early convergence late in the repetition. The updated method of the SSA at different positions diversifies the individual displacement, which has the advantages of a good convergence effect, fast speed, and strong robustness, yet when it approaches the global optimum, it will have the disadvantage of decreasing the population diversity. Therefore, in this paper, we considered the following two aspects for the improvement of the sparrow algorithm, to result in a higher performance global search capability and find more meaningful regions in the solution space.

(1)

Dynamic adaptive weights

All sparrows in the population rely on detectors with initial fitness values to forage, and the amount of food available at each sparrow’s present position determines its fitness value. If the location of the detectors is not updated in a timely manner, it is easy for the entire population to fall into a local optimum.

In this paper, we introduce a dynamic weight factor in the detector update position so that it can supply a higher weight increase in the convergence speed at the start of the iteration and adaptively lower its value at the conclusion of the iteration to accomplish a better local search.

The following is the weight calculation formula:

$$\omega =\frac{e^{2\left(1-t/ite{r}_{\max }\right)}-e^{-2\left(1-t/ite{r}_{\max }\right)}}{e^{2\left(1-t/ite{r}_{\max }\right)}+e^{-2\left(1-t/ite{r}_{\max }\right)}}$$

(15)

the sparrows’ position update formula is as follows:

$$X_{i,j}^{t+1}=\left\{\begin{array}{c}{X}_{i,j}^t+\omega \left(f_{j,l}^t-X_{i,j}^t\right)\cdot \mathrm{rand},R_2<ST\\ X_{i,j}^t+Q\cdot L,R_2⩾ST\end{array}\right.$$

(16)

where $f_{j,l}^t$ is the global optimal solution.

(2)

Reverse learning strategy

This study utilized the reverse learning approach to improve the sparrow algorithm and achieve the inverse solution using the inverse learning mechanism, which increases the search range of the algorithm and improves the fault, whereby the algorithm falls easily into the local region.

$$X_{best}^{\prime }(t)=ub+r\oplus \left(lb-X_{best}(t)\right)$$

(17)

$$X_{i,j}^{i+1}=X_{best}^{\prime }(t)+b_1\oplus \left(X_{best}(t)-X_{best}^{\prime }(t)\right)$$

(18)

$$b_1={\left(ite{r}_{\max }-t/ite{r}_{\max }\right)}^t$$

(19)

where $b_1$ is the control parameter; $X_{best}^{\prime }$ is the inverse solution of the best solution in the $t$-th generation.

The reverse learning strategy improves the sparrow algorithm’s global search capabilities. For insurance purposes, it is necessary to determine whether the new position is superior to the prior one, and a greedy rule is introduced after performing the update to decide whether to update the position by comparing the magnitude of the new fitness value with the old fitness value. The greedy rule is as follows:

$$X_{best}=\left\{\begin{array}{c}{X}_{i,j}^{t+1},f\left(X_{i,j}^{t+1}\right)<f\left(X_{best}\right)\\ X_{best}(t),f\left(X_{i,j}^{t+1}\right)\ge f\left(X_{best}\right)\end{array}\right.$$

(20)

3. Algorithm Optimization BPNN

3.1. BP Neural Network

The BP neural network is a multilayer feed-forward supervised artificial neural network that identifies the nonlinear connection between input and output information by simulating a real brain’s processing mechanism. The weights and thresholds between each neuron are constantly adjusted during the training process, and the reverse transmission of the input signal and the error signal is carried out [44], as shown in Figure 1 [45].

Suppose the input of the neuron from one to n in the BP neural network is $x_1,x_2,\cdot \cdot \cdot ,x_n$, while the j-th neuron after being stimulated is expressed as as $w_{j1},w_{j2},\cdot \cdot \cdot ,w_{jn}$ [46]. Then, the input value of the j-th neuron is shown in Formula (21):

$$S_j={\displaystyle \sum _{i=1}^n{w}_{ji}}\cdot x_i+b_j=w_{j}X+b_j$$

(21)

where $X={\left[x_1,x_2,\cdot \cdot \cdot ,x_i,\cdot \cdot \cdot ,x_n\right]}^T$ and $W_j=\left[w_{j1},w_{j2},\cdot \cdot \cdot ,w_{ji},\cdot \cdot \cdot ,w_{jn}\right]$. If $x_0=1$ and $w_{j0}=b_j$, whereby the input signal and the input offset $b_j$ are contained as weight elements; then, $X={\left[x_0,x_1,x_2,\cdot \cdot \cdot ,x_i,\cdot \cdot \cdot ,x_n\right]}^T$ and $W_j=\left[w_{j0},w_{j1},w_{j2},\cdot \cdot \cdot ,w_{ji},\cdot \cdot \cdot ,w_{jn}\right]$. The following expression is the result of the activation function’s action:

$$y_i=f\left(s_j\right)=f\left({\displaystyle \sum _{i=1}^n{w}_{ji}}\cdot x_i\right)=F\left(W_{j}X\right)$$

(22)

$$Ne{t}_i={\displaystyle \sum _{j=1}^M{w}_{ij}}{x}_j+{\theta }_i$$

(23)

where (23) ${\theta }_{{}_i}$ is the buried layer’s bias vector; thus, $\theta ={\left({\theta }_1,{\theta }_2,\cdot \cdot \cdot ,{\theta }_q\right)}^T$. The output layer’s input and output equations are shown in Equations (24) and (25):

$$Ne{t}_k={\displaystyle \sum _{j=1}^M{w}_{ki}}{y}_i+{\alpha }_k$$

(24)

$$O_k=\psi \left(Ne{t}_k\right)$$

(25)

where (24) ${\alpha }_k$ is the input layer’s offset vector, which is $\alpha ={\left({\alpha }_1,{\alpha }_2,\cdot \cdot \cdot ,{\alpha }_L\right)}^T$; and (25) $O=\left(o_1,o_2,\cdot \cdot \cdot ,o_L\right)$ is the output vector:

$$f=\frac{1}{1+e^{-x}}$$

(26)

the error ($e$) is calculated using the ideal output value, and the output value is achieved by the BP neural network, as shown in Equation (27):

$$e_k=Y_k-O_k$$

(27)

where $Y_k$ is the ideal input value. The BP neural network is tuned with the weights $w_{jk},w_{ij}$ and thresholds a and b, as shown in Equations (28)–(31):

$$w_{ij}=w_{ij}+\eta y_j\left(1-y_j\right)x(i){\displaystyle \sum w_{jk}}{e}_k$$

(28)

$$w_{jk}=w_{jk}+\eta y_i{e}_k$$

(29)

$$a_j=a_j+\eta y_i\left(1-y_j\right){\displaystyle \sum w_{jk}}{e}_k$$

(30)

$$b_k=b_k+e_k$$

(31)

the training process of the BPNN algorithm is shown in Figure 2.

The weight size has a significant impact on the BPNN training outcome and is optimized using the additional momentum method, as follows:

$$\Delta W^{\prime }\left(t+1\right)=\Delta W\left(t+1\right)+{\partial }^{*}\Delta W\left(t\right)$$

(32)

where (32) $t$ denotes the number of generations, and $\partial$ is the influence factor, which is the additional momentum.

In this paper, we used a constant adaptive learning rate. Taking too large a value, the BP neural network will be turbulent; taking too small a value, the expression effect is poor. Thus, the adaptive learning rate can be expressed as:

$$\eta (t+1)=\left\{\begin{array}{c}1.05\eta (t)E(t+1)>E(t)\\ 0.7\eta (t)E(t+1)<E(t)\\ \eta (t)\mathrm{other}\end{array}\right.$$

(33)

for the trained BP neural network, it can be thought of as a nonlinear function with input information $x$ and output information $y$.

$$y=f\left\{x,w,b\right\}$$

(34)

where (34) the set of weights is represented by $w$, and the set of biased items is represented by $b$.

According to the topology of the neural network, $x_1,x_2,\cdot \cdot \cdot ,x_6$ is the network’s input value and $y_1$ is the network’s output value. In this paper, to avoid introducing additional constraints and to guarantee the numerical accuracy of forward propagation in the neural network, the input information, x, of the network is each geological parameter (e.g., soil weight ($\gamma$), angle of internal friction ($\phi$), height ($H$), cohesion ($c$), slope angle ($\alpha$), and pore water pressure ($ru$)), and the output information, y, is the slope safety factor, the structure of which is shown in Figure 3.

To guarantee numerical accuracy, the hidden layer was configured via two layers. After numerous training sessions, the training impact was the greatest when the number of nodes in the hidden layer was 10. Other training settings included 10,000 iterations of the BP neural network for the slope safety coefficient prediction, a learning rate of 0.005, and a goal error of 0.00000008.

3.2. PSO–BP Neural Network Model

Although the BP neural network has a strong learning capacity, its application’s impact on the real slope safety coefficient prediction scheme is non-ideal and expected. The PSO algorithm can make the convergence faster by calculating the fitness value to search for in a larger space [47].

The PSO–BP neural network operates as follows:

(1)

Determine the BP neural network’s topology and related parameters.

(2)

Perform the data normalization processing, whereby the fitness value is the error between the BPNN output value and the actual value.

(3)

Initialize the particle swarms and build the individual particles and particle swarm networks.

(4)

Carry out iterative optimization and calculate the particle fitness value.

(5)

Select the individual extremes and population global extremes based on particle fitness.

(6)

Use the following equation to update the locations and velocities of all particles.

(7)

The ideal result is output if the method optimization termination condition is met. Otherwise, proceed to step 4.

(8)

Use the optimization result as the BP neural network’s starting weight and threshold.

(9)

Train the BP neural network to recognize the termination situation.

The flowchart of the PSO–BP neural network model construction is shown in Figure 4.

In this paper, the number of populations was taken as 50 and the velocity range was taken as [−0.8, 0.8].

3.3. ISSA–BP Neural Network Model

BP neural network starting weights and thresholds have a profound influence on model prediction accuracy, and the PSO algorithm tends to lead to premature convergence in the late iteration. The goal of utilizing ISSA to improve BP neural networks is to acquire the ideal weights, thresholds, and fitness values for better prediction outcomes. Therefore, the ISSA algorithm is used in this study to improve the BP neural network.

The following is the specific implementation procedure.

(1)

Determine the BPNN’s topology and the important parameters.

(2)

Initialize the population, number of iterations, and the relevant parameters.

(3)

The sparrow’s beginning location is utilized as the BPNN’s initial weight and threshold.

(4)

The neural network is trained by data processing, and the adaptive value of the sparrow individual population is the training error.

(5)

Update the position separately, according to the ISSA algorithm flow.

(6)

If the algorithm’s termination condition is met, the optimal result is produced. Otherwise, proceed to step (4).

(7)

The algorithm training optimization results are used as the BP neural network’s initial weights and thresholds.

(8)

The BP neural network is trained to the termination condition.

Figure 5 depicts the ISSA–BP neural network model creation procedure.

After the training settings, the population of the sparrows was limited to 50, with a maximum of 500 training iterations.

4. Experimental Preparation

The experiments in this study were performed using a Windows 64-bit operating system with MATLAB version 2022.

4.1. Data Description

In this model for forecasting slope stability, the parameters influenced by geological and topographical conditions were soil weight ($\gamma$), angle of internal friction ($\phi$), height ($H$), cohesion ($c$), slope angle ($\alpha$), and pore water pressure ($ru$). The sample consisted of 352 sets of slope data based on the literature and a field survey [48,49,50,51,52,53,54,55,56,57,58].

The goal of this study was to assess the accuracy of the safety factor prediction using three intelligent methods: BP neural network, PSO–BP, and ISSA–BP. The database was randomly divided into two portions, similar to prior studies [59,60]: the training set and test set had ratios of 80% and 20%, correspondingly. The training set randomly selected 80% of the data, and the remaining 20% was for the test set. Some of the experimental data are listed in

Table 1. Slope safety factor dataset.

No.	$\mathit{\gamma }$	$\mathit{c}$	$\mathit{\phi }$	$\mathit{\alpha }$	$\mathit{H}$	$\mathit{r}\mathit{u}$	Safety Factor
1	22.4	10	35	45	10	0.4	0.9
2	20	20	36	45	50	0.5	0.83
3	20.6	16.3	26.5	30	40	0	1.25
4	20	0.1	36	45	50	0.5	0.67
5	18	24	30.15	45	20	0.12	1.12
6	23.4	15	38.5	30.3	45.2	0	1.587
7	20	0	24.5	20	8	0.35	1.37
9	27.3	14	31	41	110	0.25	1.249
10	27.3	31.5	29.7	41	135	0.25	1.245
11	18.84	14.36	25	20	30.5	0	1.875
12	31.3	68.6	37	47	270	0.25	1.2
13	19.6	22	20	42.2	21	0	0.989
14	28.4	39.23	38	35	100	0.38	1.99
15	19.06	11.7	28	35	21	0	1.09
16	19.72	30	30	20	20	0.5	1.54

.

Due to the possibility of equipment failure and other reasons, some of the original data have abnormal values, which will affect the model’s prediction effect if directly entered into the model, and these data were processed considering the small impact of a small part of the data on the overall data. To avoid the inconsistency of each data parameter producing bias in the results, the sample data were normalized, and the calculation formula is as follows:

$$X_{norm}=2\times \frac{X-X_{\min }}{X_{\max }-X_{\min }}-1$$

(35)

where (35) $X$ is the initial vector without sample normalization, $X_{norm}$ is the normalized vector of the $X$ vector, and $X_{\min }$ and $X_{\min }$ represent the minimum and maximum values of the sample, respectively.

Violin plots were used to show the data structure, combined with density estimation and box line plots, and to visualize the distribution of the input parameters, the parameters were normalized and plotted in Figure 6.

It can be seen from Figure 6 that the cohesion values are more concentrated, the internal friction angle and slope angle are more widely distributed, and there is no large correlation among the six parameters, which means that each sampled parameter had an independent role in assessing the slope safety factor. To evaluate the most suitable model, we compared the different models with each other.

4.2. Model Parameter Settings

In this study, 352 groups of actual slope examples were divided into a training set and test set, and the ISSA–BP model was established by inputting six indicators, including slope angle, slope height, cohesion, bulk density, pore pressure ratio, and bulk density. The data processing and optimization operation of the algorithm can obtain the required slope safety factor, as shown in Figure 7.

The preset functions of the MATLAB neural network box greatly facilitated the learning training of the BP neural network, so all program parts could be effectively integrated into MATLAB. For learning training, the activation function was set to transig, the output function was set to pureline, the mean square error (MSE) was chosen as the error function, the goal error for training the network was set at 0.00000008, and the neural network’s learning rate was set to 0.005.

4.3. Evaluation Criteria

Before analyzing the results of each model, four metrics for evaluating the accuracy of the model, namely, $MAE$(mean absolute error), $MAPE$(mean absolute percentage error), $RMSE$(root mean square error), and $R^2$(coefficient of determination) [61], were used to calculate the differences between the projected and actual values. The lower the value of the first three indicators, the better the forecast model’s performance, and the calculation method is presented below:

$$MAE=\frac{1}{N}{\displaystyle \sum _{i=1}^N\left|y_i-y_i^{*}\right|}$$

(36)

$$MAPE=\frac{1}{N}{\displaystyle \sum _{i=1}^N\left|\frac{y_i-y_i^{*}}{y_i}\right|}\times 100\%$$

(37)

$$RMSE=\sqrt{\frac{1}{N}{\displaystyle \sum _{i=1}^N{\left(y_i-y_i^{*}\right)}^2}}$$

(38)

The coefficient of determination ($R^2$) reflects how much the percentage of the fluctuation in the dependent variable can be described by the fluctuation in the independent variable, which is also the correlation coefficient squared. The larger the coefficient of determination, the greater the degree to which the independent variable interprets the dependent variable, and the closer the observation points are to the regression line. The closer $R^2$ is to 1, the higher the correlation is, which was calculated as follows:

$$R^2=\frac{{\displaystyle \sum _{\mathrm{i}=1}^N{\left(y_i-\overline{y}\right)}^2}}{{\displaystyle \sum _{\mathrm{i}=1}^N{\left(y_i^{*}-\overline{y}\right)}^2}}$$

(39)

5. Experimental Results

In this study, a BP neural network and PSO–BP neural network were used as comparisons against the proposed ISSA–BP model, which was used to assess the model’s prediction accuracy and performance.

Fitness is an index that measures the pros and cons of individuals in the population. The fitness convergence curve can reflect the algorithm’s prediction performance and convergence efficiency, and the convergence curve of the ISSA–BP after the model’s run is shown in Figure 8.

In Figure 8, the adaptation curve gradually flattens out and steadies into a development condition, which reduced the instability of the model in this paper. Thanks to its two improvements (i.e., use of adaptive weights and backward learning strategy), the ISSA–BP model showed good performance and great improvement in the performance of the BPNN.

In this section, other neural networks are compared with the ISSA–BP proposed in this paper, and we chose the PSO–BP and BP models as the benchmark models and plotted the predicted results against the true values, as shown in Figure 9.

Figure 9 shows that compared with the other prediction models, the FOS predicted by the ISSA–BP model was very consistent with the actual FOS. The absolute error distribution was approximately 0.02. The operating results of the PSO–BP model were quite different from the actual value, and there was no tendency to fit the actual value, while for the BP neural network, the data offset was too large. When the error generated by the training samples was smaller than the real error or almost the same, it proves that the ISSA–BP model in this study had a strong generalization ability and the best prediction effect.

The scatter plots for the ISSA–BP, PSO–BP, and BPNN model prediction results are shown in Figure 10.

Figure 10 shows the regression coefficients R² for each model. The ISSA–BP model with an R² value of 0.99 had the best-fit line between the projected and actual safety coefficient values, and the correlation between the output data and the actual data was large. The R² value of the PSO–BP was 0.8197, the R² value of the BPNN was 0.6283, and the fitting effect needs to be improved. Meanwhile, it is not difficult to observe that the prediction results of the unimproved BPNN show that some data also performed well, although it can be seen from the figure that there were deviations in the data that were too large, which also reflects the network characteristic of a grid structure that is unstable, and it easily slips into the local minima, requiring further exploration and development.

The FOS comparison results for the training set of the slope predicted by the ISSA–BP model is shown in Figure 11, and the anticipated values were almost identical to the actual values. According to the above figure, the FOS errors produced by most of the datasets were close to 0. Therefore, the average absolute error value was approximately 0.021541, which indicates that the ISSA–BP model works very well and confirms the high prediction accuracy of the ISSA–BP model.

According to the prediction performance assessment indices of the neural network models in

Table 2. Comparison of the evaluation indexes of the three neural network models.

Model	MAE	MAPE (%)	RMSE	R2
BP	0.1009	25.95	0.2879	0.6283
PSO-BP	0.1586	4.28	0.1259	0.8197
ISSA-BP	0.0215	1.64	0.0296	0.9900

, the ISSA–BP model had the best prediction effect, with an average absolute percentage error of 1.64%, root mean square error of 0.0215, and coefficient of determination of 0.99. The average absolute error was 0.0215 and the average absolute error of the ISSA–BP was reduced by 78% compared with the BPNN. From the table, it can be concluded that the model’s error was reduced and achieved the expected goal, and it correlated more strongly with the actual value, indicating that the BP network established by the improved sparrow algorithm to optimize the nonlinear relationship was more accurate.

In summary, the comparison of the predicted and actual values and the study of each of the model’s error outcomes show that the ISSA–BP’s projected values were closer to the real values and had a better prediction performance. Therefore, among the intelligent prediction models of the slope safety coefficient, the ISSA–BP had the best prediction effect, followed by the PSO–BP’s prediction effect, while the BPNN had the worst prediction effect due to its uncertainty.

6. Engineering Applications

6.1. Regional Geological Situation

Land resources are the basis for human survival and development, and it is very important to formulate effective land planning. The site of the Tianbao border Police Station in the Wenshan Border Management Detachment is proposed to be built at Tianbao Port, Malipo County, Wenshan Prefecture, Yunnan Province, which belongs to the mid-mountain landform. The monitoring area has a subtropical plateau monsoon climate, with an annual average temperature of 17.6 °C and precipitation of 1068 mm. The elevation in the proposed project area is between 70 and 128.65 m, and the slope of the terrain is approximately 25–50°. The position of the 3–3′ engineering geological section is shown in Figure 12a. The top stratum of the site is mainly an artificial accumulation layer of the Quaternary system, and the underlying layer is a limestone layer of the Xiechang Formation of the upper Cambrian system, with well-developed joints and fissures. The current situation of the site is shown in Figure 12b, where the vegetation is lushly distributed. The stability of the slope in the monitoring area directly affects the safety of the proposed Tianbao border Police Station, so it is necessary to analyze the slope’s stability.

6.2. Application Result

According to the geological data extracted from the field investigation and drilling, the stability of the slope was evaluated, and a two-dimensional model of the slope in the monitoring area was established using the Lizheng slope stability calculation software. The established model is shown in Figure 13.

This work adopted the technical code for building slope engineering and did not consider earthquakes. The method for the calculations was to automatically search for the most dangerous sliding surface, and using a center step of 1 m, the actual safety factor of the monitoring area was calculated. The results are shown in

Table 3. Calculation results of composite soil slope stability.

Dangerous Sliding Center (m)	Dangerous Sliding Radius (m)	Sliding Force (kN)	Anti-Skid Force (kN)	Safety Factor
(131.69, 51.34)	83.85	19,938.40	24,235.51	1.216

.

Table 4. Prediction results of the three neural network models.

Model	$\mathit{\gamma }$	$\mathit{c}$	$\mathit{\phi }$	$\mathit{\alpha }$	$\mathit{H}$	$\mathit{r}\mathit{u}$	Predicted FOS
BP	18.5	9.50	26.50	30	70	0	1.349
PSO-BP	18.5	9.50	26.50	30	70	0	1.406
ISSA-BP	18.5	9.50	26.50	30	70	0	1.231

shows the input parameters of the slope in the monitoring area and the slope safety factors predicted by the three machine learning methods. Compared with the actual safety factors in Table 3, the ISSA–BP’s results were very close to the actual results. The error was only 0.02, and the result of the ISSA–BP model remained the best, which was the same as the above experimental results, as verified by multiple slope examples.

In addition, due to the structural characteristics of the artificial neural network, the results of the model operations were closely related to the dataset, and a high-quality dataset can improve the prediction accuracy of the model. Therefore, in the following research, more strategies can be considered to achieve an accurate prediction of the slope safety factor.

7. Conclusions

This study proposed a slope safety factor prediction model (ISSA–BP) based on an improved sparrow search algorithm to optimize the BP neural network, and it was optimized in two aspects: the introduction of dynamic weight factors and the ability to coordinate local mining and global searches. Combined with the reverse learning strategy, to obtain the reverse solution, the search range of the algorithm was expanded and its shortcoming of being easily trapped in a local area was improved. Based on 352 sets of slope examples, three machine learning methods were used to predict the safety factor of the slopes.

Through the calculation of evaluation indicators, the root mean square error of the ISSA–BP prediction model was 1.64%, the average absolute error was 0.0215, and the coefficient of determination was 0.99; compared with the BP neural network, the error was reduced by 78%. The BP neural network model optimized by the improved sparrow algorithm showed higher accuracy.

Finally, using three models to predict the slope safety factor at Tianbao Port in Wenshan Prefecture and comparing it with the actual value, the result of the ISSA–BP model’s operation was the closest to the actual safety factor, which verifies the experimental results and provides a basis for establishing the Tianbao border Police Station.