Prediction of Optimal Design Parameters for Reinforced Soil Embankments with Wrapped Faces Using a GA–BP Neural Network

Abstract

Under the same geological conditions, the thickness and length of the reinforced strip, the slope ratio of the reinforced embankment, the modulus of elasticity of the fill and the reinforced strip, and the friction angle at the interface between the reinforcement and the soil, are the main design parameters that have an important influence on the stress, deformation, and stability of the encompassing reinforced soil embankment. To quickly and accurately determine the optimal design parameters for reinforced soil embankments with wrapped faces, ensuring minimal cost, while maintaining structural safety, we propose a design parameter prediction model based on a GA–BP neural network. This model evaluates parameters within their specified ranges, using maximum lateral displacement, maximum vertical displacement, maximum stress in the XZ direction, the maximum shear strain increment, and the safety factor, as assessment criteria. The primary objective is to minimize the overall cost of the embankment. A comparison with five machine learning algorithms shows that the model has high prediction accuracy, and the optimal design parameter combinations obtained from the optimization search can significantly reduce the cost of the embankment, while controlling the displacement and stability of the embankment. Therefore, the GA–BP network is suitable for predicting the optimal design parameters of reinforced soil embankments with wrapped faces.

1. Introduction

The addition of a wrapped face to reinforced soil embankments is a novelty in reinforcement structures. The end of the reinforcement material is folded back and wrapped around the embankment at the surface of the slope. The lateral soil pressure is transmitted to the soil through reinforcement strips, anchoring the structure. It has advantages, namely it is light weight, has strong adaptability, and has good seismic resistance; it occupies a small amount of land; and saves on building materials; among other things [1]. In recent years, China has attached great importance to reducing carbon emissions, and reinforced soil embankments with wrapped faces can be utilized in the gaps in geogrid grass planting, giving the effect of greening and landscaping, and can also reduce carbon emissions [2]; this is a considerable development. This soil reinforcement structure has been used more frequently, as the number of deeply excavated and highly filled embankments has increased [3,4]. Scholars have investigated the mechanical properties and stability of these structures using theoretical analysis [5,6], laboratory tests [7,8,9], field monitoring [10,11], and numerical simulations [12,13]. However, due to the complex interaction mechanisms between the geogrid and the soil, determining the impact of the reinforcement parameters on slope stability remains challenging [14]. Consequently, further research is necessary to select design parameters that meet the requirements of engineering practice.

Ji et al. [14] used an orthogonal experimental design to analyze the sensitivity of factors affecting the reinforced slope’s stability and to assess the reinforcement performance. They proposed that slope engineering should make use of materials with high cohesiveness and large internal friction angles. The nonlinear finite element program ABAQUS was utilized by Yang et al. [15] to evaluate the impact of the quantity of the reinforcement layers, the reinforcement material modulus, the soft foundation modulus, the fill height, and the thickness of the soft soil layers, demonstrating that reinforcement inhibited lateral displacement and improved slope stability. The length and horizontal spacing of the reinforcing strips, as well as the internal friction angle and cohesiveness of the fill, were examined by Luo et al. [16] with regard to their impact on slope stability. The stability index of reinforced soil embankments was analyzed by Que et al. [17] using the Monte Carlo method. They found that higher cohesion, the internal friction angle, the tensile strength of the reinforcement material, and the soil-reinforcement friction coefficient, improved the stability. Zheng Lifeng et al. [18] employed a sequential quadratic programming algorithm based on indoor model tests to develop a mathematical model aimed at optimizing the reinforcement deployment scheme for reinforced soil-retaining walls. The objective was to minimize the amount of reinforcement strips, while ensuring no damage occurred. Although numerous studies have been conducted on reinforced soil structures, previous optimal designs have often overlooked the interaction between various design parameters. Furthermore, these studies primarily focused on stability as the main optimization objective, with less emphasis on economic factors.

To achieve efficient and accurate optimal design parameters for reinforced soil embankments that ensure both safety and cost effectiveness, this paper leverages the reconstruction project involving Xianhu Road in Yiling District, Yichang City. The design parameters considered include the wrapping thickness, slope rate, elastic modulus of the filler, reinforcement strip length, modulus of elasticity of the reinforcement strip, and the interfacial friction angle. Constraints are set for the maximum lateral displacement, maximum vertical displacement, maximum XZ direction stress, maximum shear strain increment, and the factor of safety. The objective is to minimize the embankment cost per meter. A BP neural network prediction and optimization model, based on a genetic algorithm, is established to determine the optimal design parameters. These results are then compared with other schemes to verify the effectiveness of the optimization.

2. Overview of the GA–BP Algorithm

BP is a widely used nonlinear artificial neural network that does not require mathematical modeling of the system. It relies solely on sample data to perform nonlinear mapping from the input space (quantity of input neurons) to the output space (quantity of output neurons) [19]. For parameter design optimization, BP can be utilized in place of finite element calculations to map the input–output relationship of the optimization objective. Figure 1 shows the BP neural network model’s organizational layout.

Where {X₁, … X_n} represent the input values and {Y₁, … Y_m} represent the output values. In addition, w_ij denotes the weights within the hidden layer and the input layer, and w_jk denotes the weights separating the output layer from the hidden layer. Empirical equations can be used to calculate the number of nodes in the hidden layer; in general, the number of layers and nodes in the hidden layer is chosen depending on the outcomes of the model training process. Equation (1) is the empirical formula used to calculate the number of hidden layer nodes [19]:

$$n_1=\sqrt{n+m}+a$$

(1)

Here, n₁ represents the hidden layer number. The variables n and m denote the number of input and output units, respectively, and $a$ is a constant within the interval of 1 to 10.

BP uses the mean square error (MSE) as the criterion for evaluating network training. Network training ceases when the MSE of the network falls below a predetermined threshold. The learning steps are as follows:

(1) Variable Definitions

Input: x = (x₁, x₂, …, x_n)

Input vector of the hidden layer: hi = (hi₁, hi₂, …, hi_p)

Output vector of the hidden layer: ho = (ho₁, ho₂, …, ho_p)

Input vector of the output layer: yi = (yi₁, yi₂, …, yi_q)

Output vector of the output layer: yo = (yo₁, yo₂, …, yo_q)

Desired output: d_o = (d₁, d₂, …, d_q)

Connection weights between the input and the hidden layer: w_ij

Connection weights between the hidden and the output layer: w_jk

Thresholds of the hidden layer neurons: b_h

Thresholds of the output layer neurons: b₀

Number of sample data: k = 1, 2, …, m

Activation formulation: f(•)

Error formulation: $e={\displaystyle \frac{1}{2}}{\sum }_{o=1}^q(d_o(k)-y{o}_o(k){)}^2$

(2) Network Initialization: Set the error function e, specify the accuracy ε and the maximum number of learning iterations M, and assign random numbers between −1 and 1 to each connection weight.

(3) Choose the k-th input sample at random, along with the intended output:

$$x(k)=(x_1(k),x_2(k),\dots ,x_n(k))$$

$$d_O(k)=(d_1(k),d_2(k),\dots ,d_q(k))$$

(4) Determine each neuron’s input and output for the buried layer:

$$h{i}_h(k)={\sum }_{i=1}^n{w}_{ij}{x}_i(k)-b_h\hspace{1em}h=1,2\dots ,p$$

$$h{o}_h(k)=f(h{i}_h(k))\hspace{1em}h=1,2,\dots ,p$$

$$y{i}_o(k)=\sum {\displaystyle \frac{p}{h=1}}{w}_{jk}h{o}_h(k)-b_o\hspace{1em}o=1,2,\dots ,q$$

$$y{o}_o(k)=f(y{i}_o(k))o=1,2,\dots ,q$$

(5) For every neuron in the output layer ${\delta }_o(k)$, find the partial derivatives of the error function using the network’s actual and desired outputs:

$${\displaystyle \frac{\partial e}{\partial w_{jk}}}={\displaystyle \frac{\partial e}{\partial y{i}_o}}{\displaystyle \frac{\partial y{i}_o}{\partial w_{jk}}}$$

$${\displaystyle \frac{\partial y{i}_o(k)}{\partial w_{jk}}}={\displaystyle \frac{\partial ({\sum }_h^p{w}_{ho}h{o}_h(k)-b_o)}{\partial w_{jk}}}=h{o}_h(k)$$

$$\begin{gathered} {\displaystyle \frac{\partial e}{\partial y{i}_o}}={\displaystyle \frac{\partial (\frac{1}{2}{\sum }_{o=1}^q(d_o(k)-y{o}_o(k)){)}^2}{\partial y{i}_o}} \\ =-(d_o(k)-y{o}_o(k))f^{\prime }(y{i}_o(k))-{\delta }_o(k) \end{gathered}$$

(6) To determine the partial derivatives of the error function for each neuron in the hidden layer, use the connection weights between the hidden layer and the output layer ${\delta }_o(k)$, as well as the output of the hidden layer ${\delta }_h(k)$ itself:

$${\displaystyle \frac{\partial e}{\partial w_{jk}}}={\displaystyle \frac{\partial e}{\partial y{i}_o}}{\displaystyle \frac{\partial y{i}_o}{\partial w_{jk}}}=-{\delta }_o(k)h{o}_h(k)$$

$${\displaystyle \frac{\partial e}{\partial w_{ij}}}={\displaystyle \frac{\partial e}{\partial h{i}_h(k)}}{\displaystyle \frac{\partial h{i}_h(k)}{\partial w_{ij}}}$$

(7) Use the ${\delta }_o(k)$ values of every neuron in the output layer and the outputs of every neuron in the hidden layer $w_{jk}(k)$ to modify the connection weights:

$$\mathsf{\Delta }{w}_{jk}(k)=-\mu {\displaystyle \frac{\partial e}{\partial w_{jk}}}=\mu {\delta }_o(k)h{o}_h(k)$$

$$w_{jk}^{N+1}=w_{jk}^N+\eta {\delta }_o(k)h{o}_h(k)$$

(8) Utilize the inputs from each neuron in the input layer and the ${\delta }_h(k)$ values of each neuron in the hidden layer $w_{ij}(k)$ to modify the connection weights:

$$\mathsf{\Delta }{w}_{ij}(k)=-\mu {\displaystyle \frac{\partial e}{\partial w_{ij}}}={\delta }_h(k)x_i(k)$$

$$w_{ij}^{N+1}=w_{ij}^N+\eta {\delta }_h(k)x_i(k)$$

(9) Compute the global error:

$$E={\displaystyle \frac{1}{2m}}{\sum }_{k=1}^m{\sum }_{o=1}^q(d_o(k)-y_o(k){)}^2$$

(10) Check whether the error satisfies the specified criteria. When the error exceeds the maximum number of learning iterations or the error reaches the predetermined level of precision, the algorithm stops. Otherwise, select the next sample and continue with a new round of learning.

Sufficient training samples are required to ensure the accuracy of BP neural network outputs. They contain input parameters (design parameters) and the expected output (evaluation indicators). The training samples can be obtained from analytical calculations, empirical data, or computational data generated during the design process [20]. However, the gradient descent method is employed to train BP, resulting in local optima and unstable learning and generalization abilities. As new training samples are added, the network’s pattern may be altered, leading to deviations in the results [21,22]. Hence, traditional algorithms cannot fully exploit the advantages of BP neural networks, necessitating the use of GAs for optimization.

Genetic algorithms are random global search and optimization methods [23] that describe biological evolution using three operators: replication, crossover, and mutation. The GA generates a set of solutions for each iteration. Initially, the solutions are generated randomly, and in subsequent iterations, new solutions are created through evolution and inheritance. An objective function is used to evaluate the solutions. This process repeats until a convergence criterion is met. The new set of solutions selectively retains high-value old solutions, while incorporating new solutions obtained by combination. Since GAs retain competitive genes in each iteration, they consistently seek the optimal value for the evaluation function [24]. The GA–BP prediction model algorithm flow is illustrated in Figure 2.

Genetic algorithms have several advantages over traditional optimization methods: lower computational requirements for problem optimization, the ability to handle various objective functions and constraints, and the capacity to perform a probabilistic global search. The shortcomings of BP neural networks are addressed, and by employing a GA to identify the ideal network weights and thresholds, which are employed as the initial weights and thresholds in the neural network model, the prediction model’s speed and accuracy are increased.

3. Project Overview

The reconstruction project for Xianhu Road in Yiling District, Yichang City, is located in Wangjiaping Village, Letianxi Town, Yiling District, Yichang City, Hubei Province. Provincial Highway 334 borders the project to the north and east, and the Yangtze River is located at the southern and western borders. The length of the reconstructed section is 613 m, and it is basically constructed according to the mountain, adopting the standard for a secondary highway, with a design speed of 40 km/h. The roadbed is 8.5 m wide, and the width of the traffic lane is 7 m. The roadbed in the K0 + 000 to K0 + 130 section is relatively high, with a maximum fill height of 25 m. A photograph of the embankment site is shown in Figure 3.

A 24 m high secondary embankment was selected. The foundation was of moderately weathered limestone and was not replaced during construction to minimize the effect of foundation settlement and enable the design parameters of the embankment to be optimized. The effects of groundwater and surface water were not considered. The material properties of the rock and soil used are shown in

Table 1. Parameters of geotechnical materials.

Parameters	Moderately Weathered Limestone	Landfill
Capacity γ (kN/m3)	2000	1870
Elastic modulus E (MPa)	1520	Awaiting a decision
Poisson’s ratio v	0.23	0.3
Cohesive force c (kPa)	1.53	30
Angle of internal friction φ (°)	37	35

,

Table 2. Parameters of geogrid material.

Thickness t (mm)	5
Length L (m)	Awaiting a decision
Elastic modulus E (MPa)	Awaiting a decision
Poisson’s ratio v	0.33
Interfacial friction angle φ (°)	Awaiting a decision
Interfacial rigidity k (MPa)	2.3

presents the material parameters of the geogrid, and a schematic diagram of the model is shown in Figure 4.

The design parameters of the reinforced soil embankment with a wrapped face due to be optimized include the wrapping thickness, slope ratio, elastic modulus of the fill and reinforcement, reinforcement length, and soil–reinforcement interface friction angle. The evaluation indicators are the maximum lateral and vertical displacement, maximum stress in the XZ direction, maximum shear strain increment, and the safety factor. The design parameters’ range of values are based on engineering experience (

Table 3. Design parameter value range.

Wrapping thickness (m)	0.6–1.2
Slope ratio of reinforced embankment (-)	1:0.5–1:1.5
Modulus of elasticity of the fill (MPa)	15–45
Reinforcement strip length (m)	10–30
Modulus of elasticity of reinforcement strip (GPa)	1–1.8
Interfacial friction angle (°)	1–5

).

4. Establishment of the GA–BP Model

The objective of this paper is to optimize the design parameters to minimize the costs, while ensuring the safety of the dam. This involves developing an optimization design model that treats the design parameters as independent variables. The model uses the cost of the reinforced earth embankment as the objective function and the safety evaluation indices, predicted by the design parameters, as constraints. The aim is to find the minimum value of the objective function. Based on this analysis, the following mathematical model was developed:

$$\left\{\begin{array}{l}\min f=f(x,c)\\ s.t.l_b\le x\le u_b\\ y_i(x)\le a\end{array}\right.$$

(2)

In this equation, f(x,c) represents the cost function of the reinforced soil embankment, x is the vector of the design parameters, and c denotes the vector of the material unit prices, such as the unit price of the reinforcement and filling materials. Moreover, l_b, u_b indicate the upper and lower limits of the design parameters, y_i(x) corresponds to the value of the evaluation indices, and a signifies the upper or lower limits of these evaluation indices.

Based on the above equations, the solution model for the optimal design parameters of the embankment is established.

4.1. Model Training

An input layer, hidden layers, and an output layer make up BP. The study’s neural network topology is 6–10–5, meaning that there are six design parameters in the input layer, ten neurons in the hidden layer, and five evaluation indicators in the output layer. Figure 5 shows a schematic of the neural network structure.

We created a computational model of the secondary embankment using the FLAC3D finite element program. Sixty-eight sets of design parameters were selected for the numerical simulations through orthogonal and control variable tests. The evaluation indicators were used to assess the calculation results. This process yielded 68 sets of sample data for model construction and training. Each data set included six input parameters to be optimized: wrapping thickness, slope ratio of the reinforced embankment, elastic modulus of the fill, reinforcement length, reinforcement elasticity modulus, and soil–reinforcement interface friction angle. The output data consisted of five optimization targets: maximum lateral displacement, maximum vertical displacement, maximum stress in the XZ direction, maximum shear strain increment, and the safety factor. The sample input and output data are listed in

Table 4. Sample input data.

Sample Number	Wrapping Thickness (m)	Slope Ratio of Reinforced Embankment (-)	Modulus of Elasticity of the Fill (MPa)	Reinforcement Strip Length (m)	Modulus of Elasticity of Reinforcement Strip (MPa)	Interfacial Friction Angle (°)
1	0.4	1:2.5	55	30	2600	5
2	0.4	1:2	35	15	1000	4
3	0.4	1:1.5	15	25	1400	3
4	0.4	1:1	45	10	1800	2
5	0.4	1:0.5	25	20	2200	1
6	0.6	1:2.5	15	15	1800	1
7	0.6	1:2	45	25	2200	5
8	0.6	1:1.5	25	10	2600	4
9	0.6	1:1	55	20	1000	3
10	0.6	1:0.5	35	30	1400	2
……
59	1.2	1:0.5	15	10	1000	0
60	1.2	1:0.5	15	10	1000	1
61	1.2	1:0.5	15	10	1000	2
62	1.2	1:0.5	15	10	1000	3
63	1.2	1:0.5	15	10	1000	4
64	1.2	1:0.5	15	10	1000	10
65	1.2	1:0.5	15	10	1000	15
66	1.2	1:0.5	15	10	1000	20
67	1.2	1:0.5	15	10	1000	25
68	1.2	1:0.5	15	10	1000	30

and

Table 5. Sample output data.

Sample Number	Maximum Lateral Displacement (mm)	Maximum Vertical Displacement (mm)	Maximum Stress in the XZ Direction (kPa)	Maximum Shear Strain Increment (-)	Safety Factor (-)
1	−8.413	−39.511	−59.931	12.26	3
2	−14.935	−61.015	−64.914	19.416	2.29
3	−35.776	−140.180	−71.626	46.161	2.27
4	−15.796	−48.925	−77.285	15.229	1.496
5	−35.359	−87.785	−100.78	29.90	1.895
6	−30.564	−141.850	−58.19	43.16	2.71
7	−11.501	−49.167	−64.36	14.94	2.59
8	−24.072	−86.462	−71.78	26.56	1.84
9	−13.051	−40.945	−77.78	12.31	1.918
10	−29.835	−63.329	−107.00	23.53	2.1
……
59	−80.271	−150.520	−115.35	63.888	1.504
60	−80.189	−150.640	−113.7	63.614	1.520
61	−80.217	−150.520	−112.92	63.785	1.520
62	−80.260	−150.500	−114.3	63.616	1.527
63	−80.142	−150.470	−113.51	63.79	1.363
64	−80.202	−150.440	−114.89	64.00	1.496
65	−80.271	−150.520	−115.35	63.89	1.504
66	−80.189	−150.640	−113.70	63.61	1.520
67	−80.217	−150.520	−112.92	63.79	1.520
68	−80.260	−150.500	−114.30	63.62	1.527

, respectively. Due to space constraints, only a portion of the data are presented.

Seventy percent of the sample data were used as the training set to train the neural network prediction model; fifteen percent were used as the test set to confirm the model’s ability to generalize and the accuracy of the predicted data and fifteen percent were used as the validation set to find the number of hidden neurons in the neural network model.

Due to significant differences in the magnitudes of the input and output data, normalization was used to accelerate the solution process and improve accuracy. The normalization principle is expressed in Equation (3):

$$y^{\prime }=lower+(upper-lower)\times {\displaystyle \frac{y-min}{\mathit{max}-min}}$$

(3)

where y means the original data and y′ represents the normalized data in the range [0, 1].

4.2. Establishment Model

As illustrated in Figure 2, the GA–BP model was established as follows:

(1) Use the trained BP prediction model as the GA’s fitness function. Based on the optimization model, an objective function is established to minimize the cost per linear meter of the reinforced soil embankment:

$$\mathit{min}f={\displaystyle \frac{h{c}_1}{x_1}}(x_4+x_1+2)+{\displaystyle \frac{h{c}_2}{2}}(2x_4+{\displaystyle \frac{x_1}{x_2}})$$

(4)

In this equation, h is the slope height, set at 24 m; c₁ and c₂ represent the unit prices of the reinforcement and fill, respectively, valued at 30/m² and 9.49/m³; x₁ denotes the wrapping thickness; x₂ is the slope rate; and x₄ signifies the reinforcement strip length. The remaining parameters have minimal relevance to the embankment cost and are therefore excluded from the objective function.

(2) In setting constraints, the value ranges for the design parameters are presented in Table 3, and the constraint ranges for the evaluation indices are defined by the following equation:

$$\left\{\begin{array}{l}0\ge y(1)=y_{pred}(1)\ge 1\%H\\ 0\ge y(2)=y_{pred}(2)\ge 0.3\%H\\ 0\ge y(3)=y_{pred}(3)\ge -115\\ 0\le y(4)=y_{pred}(4)\le 64\\ y(5)=y_{pred}(5)\ge 1.45\end{array}\right.$$

(5)

In this equation, y(i) represents the evaluation indices, including the maximum lateral displacement, maximum vertical displacement, maximum XZ stress, maximum shear strain increment, and the safety factor. The slope height h is set at 24 m. According to the construction drawings, the maximum lateral displacement and maximum vertical displacement should not exceed 1% and 0.3% of the slope height, respectively. For the maximum XZ stress and maximum shear strain increment, the upper limits are based on the highest values observed during training, as the design drawings and specifications do not provide explicit values. The safety factor, according to current Chinese highway design specifications, must be at least 1.45 in the most unfavorable conditions.

These parameter upper and lower bounds are transformed into normalized data ranges, which serve as constraints for the genetic algorithm model.

(3) To ensure the algorithm identifies a solution that meets the constraints during the search process, a penalty function is constructed using the interior point penalty function method. This approach transforms the constrained minimization problem into an unconstrained minimization problem. The penalty function is as follows:

$$\varphi (x)=f(x)-10\sum _{j=1}^n(abs(y_i(x)))$$

(6)

(4) Set the GA parameters (

Table 6. Main parameters of the genetic algorithm.

Crossover rate	0.8
Variation rate	0.2
Group size	100
Algebraic function	100
Number of variables	6

).

The population’s best individual level of fitness is updated when it exceeds that of the best individual in the previous iteration. This iterative process continues until the individual with the highest fitness is obtained.

(5) Output the optimization results, as shown in

Table 7. Optimal solutions from the GA–BP neural network search.

Wrapping Thickness (m)	Slope Ratio of Reinforced Embankment (-)	Modulus of Elasticity of the Fill (MPa)	Reinforcement Strip Length (m)	Modulus of Elasticity of Reinforcement Strip (GPa)	Interfacial Friction Angle (°)
1.1983	1.9962	31.3183	10.01	1.784	4.986

and

Table 8. Evaluation indicator values derived from the GA–BP neural network.

Maximum Lateral Displacement (mm)	Maximum Vertical Displacement (mm)	Maximum Stress in the XZ Direction (kPa)	Maximum Shear Strain Increment (-)	Safety Factor (-)
−47.0218	−71.8192	−113.5368	32.4313	1.6283

.

Traditional embankment design involves selecting multiple solutions based on specifications and design experience, followed by finite element modeling to choose the optimal solution. This approach is influenced by the time required for finite element modeling and the individual subjectivity of the designer, limiting the number of samples for solution selection. The GA–BP model established in this paper offers two key advantages: it can quickly output evaluation indices based on design parameters, facilitating the comparison of different schemes, and it can provide a more accurate solution tailored to the decision maker’s requirements in terms of cost and embankment stability. This significantly enhances design efficiency.

5. Validation Model

To verify the reliability and capability of the predictive model, the statistical metrics presented in

Table 9. Statistical metrics used for evaluating the ML models.

Abbreviation	Formula	Description
$R^2$	$R^2=1-{\displaystyle \frac{{\sum }_{i=1}^n(A_i-P_i{)}^2}{{\sum }_{i=1}^n(A_i-A_{avg}{)}^2}}$	R squared
$MSE$	$MSE={\displaystyle \frac{{\sum }_{i=1}^n(A_i-P_i{)}^2}{n}}$	Mean squared error
$RMSE$	$RMSE=\sqrt{{\displaystyle \frac{{\sum }_{i=1}^n(A_i-P_i{)}^2}{n}}}$	Root mean squared error
$MAE$	$MAE={\displaystyle \frac{{\sum }_{i=1}^n\left|A_i-P_i\right|}{n}}$	Mean absolute error
$MARE$	$MARE={\displaystyle \frac{{\sum }_{i=1}^n\left|\frac{A_i-P_i}{A_i}\right|}{n}}$	Mean absolute relative error
$MSRE$	$MSRE={\displaystyle \frac{{{\sum }_{i=1}^n\left|\frac{A_i-P_i}{A_i}\right|}^2}{n}}$	Mean square relative error
$RMSRE$	$RMSRE=\sqrt{{\displaystyle \frac{{\sum }_{i=1}^n(\frac{A_i-P_i}{A_i}{)}^2}{n}}}$	Root mean square relative error
$RRMSE$	$RRMSE={\displaystyle \frac{\sqrt{\frac{1}{n}{\sum }_{i=1}^n(A_i-P_i{)}^2}}{{\sum }_{i=1}^n{A}_i}}\times 100$	Relative root mean square error
$MBE$	$MBE={\displaystyle \frac{{\sum }_{i=1}^n(A_i-P_i)}{n}}$	Mean bias error
$erMAX$	$erMAX=\mathit{max}(\left|{\displaystyle \frac{A_i-P_i}{A_i}}\right|)$	Maximum absolute relative error
$SD$	$SD=\sqrt{{\displaystyle \frac{\sum (X_i-Arithmeticmean{)}^2}{total number}}}$	Standard deviation of the difference between actual and predicted values
$t-stat$	$t-stat=\sqrt{{\displaystyle \frac{(n-1)MB{E}^2}{RMS{E}^2-MB{E}^2}}}$	t-statistic
$U_{95}$	$U_{95}=\sqrt{1.96(S{D}^2+RMS{E}^2)}$	Uncertainty at 95%

Note: A represents actual, P stands for predicted.

were used [25]. The GA–BP neural network was compared with five machine learning algorithms, random forest (RF), decision tree (DT), gamma regressor (GR), gradient boosting machine (GBM), and linear regression (LR) [26], each of which outputs statistical metrics for comparison after training is complete. In order to make a more intuitive comparison, each metric is assigned a score of 1–5, according to the ranking, and the final ranking is made according to the total score.

Table 10. Statistical metrics calculation results.

Model	R2	MSE	RMSE	MAE	MARE	MSRE	RMSRE	RRMSE	MBE	erMAX	SD	t-Stat	${\mathit{U}}_{95}$
Evaluation index for prediction of maximum lateral displacement
GA–BP	0.996	3.380	1.839	1.598	0.073	0.012	0.108	0.411	−0.057	0.267	1.838	0.093	3.639
RF	0.575	402	20.064	18.866	1.091	2.351	1.533	−47.685	8.590	3.394	19.113	1.421	38.794
DT	0.835	104.153	10.206	7.289	0.258	0.185	0.430	1.847	−6.159	0.916	8.138	2.270	18.274
GBM	0.977	16.468	4.058	1.922	0.038	0.006	0.079	0.700	1.016	0.242	3.929	0.776	7.908
LR	0.866	59.649	7.723	4.670	0.098	0.018	0.132	1.490	−1.662	0.313	7.542	0.661	15.113
GR	0.882	74.734	8.645	7.379	0.160	0.036	0.189	1.515	−5.608	0.378	6.579	2.557	15.209
Evaluation index for prediction of maximum vertical displacement
GA–BP	0.999	2.323	1.524	1.212	0.018	0.001	0.028	0.134	−0.200	0.076	1.511	0.397	3.005
RF	0.160	1967	44.355	37.244	0.704	0.891	0.944	−47.445	28.154	1.623	36.128	2.464	80.089
DT	0.995	11.220	3.350	2.148	0.033	0.003	0.055	0.287	−0.425	0.137	3.323	0.384	6.605
GBM	0.993	16.519	4.064	2.230	0.050	0.011	0.105	0.362	−1.627	0.310	3.724	1.311	7.718
LR	0.703	359.202	18.953	10.077	0.094	0.035	0.187	1.487	1.139	0.529	18.918	0.181	37.490
GR	0.942	97.877	9.893	4.707	0.040	0.005	0.071	0.807	3.607	0.201	9.212	1.175	18.926
Evaluation index for prediction of maximum stress in the XZ direction
GA–BP	0.990	5.270	2.296	1.649	0.024	0.001	0.037	0.271	0.770	0.105	2.163	1.068	4.415
RF	0.686	153	12.405	10.776	0.131	0.026	0.163	−13.511	0.460	0.357	13.067	0.111	25.224
DT	0.695	119.498	10.932	6.347	0.057	0.008	0.089	0.996	−4.395	0.220	10.009	1.317	20.750
GBM	0.937	13.424	3.664	2.627	0.025	0.001	0.034	0.339	2.247	0.066	2.894	2.329	6.537
LR	0.429	301.296	17.358	9.794	0.095	0.025	0.157	1.788	−1.940	0.458	17.249	0.337	34.259
GR	−0.502	489.915	22.134	13.144	0.124	0.039	0.198	2.195	11.673	0.477	18.806	1.862	40.662
Evaluation index for prediction of maximum shear strain increment
GA–BP	0.993	3.752	1.937	1.694	0.053	0.005	0.071	0.430	−0.016	0.146	1.937	0.024	3.835
RF	0.311	353	18.808	16.721	0.883	1.331	1.153	51.700	−11.306	2.224	15.844	2.257	34.429
DT	0.897	46.503	6.819	5.426	0.220	0.117	0.341	1.412	1.038	0.768	6.740	0.462	13.423
GBM	0.978	9.528	3.087	2.398	0.081	0.013	0.112	0.671	−1.313	0.240	2.793	1.410	5.828
LR	0.872	33.550	5.792	3.592	0.090	0.017	0.129	1.141	0.285	0.238	5.785	0.148	11.461
GR	−0.124	433.634	20.824	12.749	0.271	0.139	0.373	4.197	12.012	0.743	17.010	2.118	37.644
Evaluation index for prediction of safety factor
GA–BP	0.930	0.026	0.161	0.142	0.086	0.012	0.110	0.796	−0.020	0.235	0.160	0.373	0.317
RF	0.430	0.117	0.342	0.245	0.125	0.026	0.161	18.390	−0.027	0.308	0.359	0.235	0.694
DT	0.305	0.099	0.315	0.218	0.132	0.039	0.197	1.931	−0.012	0.529	0.315	0.110	0.623
GBM	0.825	0.025	0.160	0.111	0.079	0.014	0.118	1.049	0.064	0.267	0.146	1.322	0.303
LR	0.444	0.112	0.335	0.232	0.134	0.047	0.216	1.860	−0.076	0.624	0.326	0.702	0.654
GR	−0.475	0.231	0.480	0.280	0.196	0.119	0.344	3.143	−0.172	1.019	0.449	1.148	0.920

shows the comparison of the statistical metrics of the algorithms.

From the results in Table 10, the R² of the GA–BP neural network is higher, which indicates that the algorithm is more accurate. According to the results in

Table 11. Comparison of optimization programs.

Design Parameters	Optimization Solutions	Comparison Program	Evaluation Indicators	Optimization Solutions	Comparison Program
Wrapping thickness (m)	1.1983	1.200	Maximum lateral displacement (mm)	−47.0218	−50.623
Slope ratio of reinforced embankment (-)	1.9962	2.000	Maximum vertical displacement (mm)	−71.8192	−91.658
Modulus of elasticity of the fill (MPa)	31.3183	25.000	Maximum stress in the XZ direction (kPa)	−113.5368	−115.76
Reinforcement strip length (m)	10.010	10.000	Maximum shear strain increment (-)	32.4313	38.68
Modulus of elasticity of reinforcement strip (GPa)	1784.354	1000.000	Safety factor (-)	1.6283	1.410
Interfacial friction angle (°)	4.986	5.000	Embankment cost	7639.05	7625.93

, the GA–BP neural network is the best method, with scores of 17, 15, 18, 13, and 22, respectively. In addition, GBM also has higher accuracy with scores of 23, 46, 30, 31, and 26, making it second only to the GA–BP neural network. The rest of the algorithms have higher accuracy in the prediction of certain metrics, such as the decision tree, which has an R² of 0.995 in predicting the maximum vertical displacement, but only 0.305 in predicting the safety factor, which makes it a weak prediction model.

The best way to better demonstrate the estimation accuracy of each algorithm is to plot the error curves. Figure 6 demonstrates the predicted versus actual values for each algorithm using five sets of samples.

From Figure 6, which shows the prediction of the five evaluation indexes, it can be seen that the predicted values by the GA–BP neural network and GBM are closer to the actual values, which is also the same as for the results analyzed in Table 10. The statistical error data indicate that the conclusion can be drawn that the maximum error value of the GA–BP neural network is 8.62%, which occurs at the maximum shear strain increment of sample no. 3, and that the rest of the error is less than 5%; the maximum error of the GBM is 24.08%, which occurs at the maximum shear strain increment of sample no. 5, there are five errors greater than 10%, and the overall accuracy is lower than that of the GA–BP neural network; the maximum errors of the GR, DT, RF, and LR are 288%, 72%, 258%, and 152%, respectively, and the prediction accuracy is substantially lower than that of the GA–BP neural network.

The analyses in Table 10 and Figure 6 show that the GA–BP neural network has better prediction accuracy, and the advantages and disadvantages of its optimization-seeking scheme are analyzed next. The data on displacement and stress, close to the optimized scheme, were selected from the training set for analysis. The design parameter values, corresponding evaluation indices, and total cost of both the optimized scheme and the comparison scheme are presented in Table 11.

Analyzing the data presented in Table 11, it is evident that the optimized scheme and the comparison scheme have a similar cost, wrapping thickness, slope rate, and reinforcement strip length. However, the comparison scheme exhibits excessive maximum vertical displacement, maximum stress in the XZ direction, and safety factor. This discrepancy arises from significant differences in the elasticity modulus of the fill and the elasticity modulus of the reinforcement strip between the two schemes. Although these parameters are not directly associated with the cost of the embankment, they significantly enhance structural performance. Traditionally, parameter optimization aimed at minimizing costs often considers only cost-related parameters, neglecting those that, while unrelated to cost, influence the structural performance of the embankment. This analysis demonstrates that the GA–BP model established in this paper is suitable for optimizing the parameters of reinforced soil embankments to achieve minimal costs.

6. Conclusions

In this study, a GA–BP model was presented for the optimization of the design parameters of a reinforced soil embankment with a wrapped face. We conducted analysis using a case study to validate the model’s performance. The research conclusions are twofold and are listed below:

(1) Comparing the GA–BP model with the GBM, RF, DT, GR, and LR algorithms, the ranking scores of the model evaluation indexes of the GA–BP model are 17, 15, 18, 13, and 22, in the relevant order, and in the comparison between the predicted and actual values, the maximum error of the GA–BP model was 8.62%, higher than that of the other algorithms, indicating that GA–BP prediction is more accurate;

(2) The optimal scheme obtained through GA–BP optimization is compared with the scheme using the training set. The optimized solution provides better control of the evaluation metrics, such as displacement and stress, while ensuring the lowest cost. This comparison validates the feasibility of the GA–BP neural network model in optimizing the design parameters of packaged reinforced soil embankments.

This study verifies the feasibility of the GA–BP neural network for the optimization of the parameters of a reinforced soil embankment with a wrapped face, but the following shortcomings and outlooks should be considered:

(1) During the examination of the reinforced soil embankment with a wrapped face established and performed in this paper, only the design parameters of the main part of the embankment are considered, but the foundation and groundwater treatments in actual projects also have a certain influence on the deformation and stability of the embankment. This paper simplifies the foundation part when considering the influencing factors in order to consider the influencing factors of the foundation and groundwater;

(2) The data set used in the training model in this paper consists of finite element calculation data, and construction monitoring sample data can be collected to train the prediction model in order to accurately predict the deformation and stability of the embankment during the construction process;

(3) The gradient boosting machine (GBM) algorithm also shows high prediction accuracy when comparing the algorithms, and subsequent research can optimize the GBM algorithm to further compare it with the GA–BP model.

In practical engineering projects, a more compliant solution is achieved by narrowing the range of values of the design parameters and constraints as needed. Additionally, using more sample data to train the prediction model can provide more accurate and suitable optimization schemes.