Application of a Hybrid Machine Learning Model for the Prediction of Compressive Strength and Elastic Modulus of Rocks

Abstract

This paper presents a machine learning-based approach to estimating the compressive strength and elastic modulus of rocks. A hybrid model, GWO-ELM, was built based on an extreme learning machine network optimized by the grey wolf algorithm. The proposed model was carried out on 101 experimental datasets, and four commonly used models were used as benchmarks to evaluate the accuracy of the proposed model. The results showed that the proposed hybrid model can accurately achieve the prediction of elastic modulus and compressive strength with high correlation coefficients and small prediction errors. The prediction performance of the hybrid model is significantly better than the other four original models, and it is an alternative model for predicting the compressive strength and elastic modulus of rocks, which is recommended as an auxiliary tool for real-time prediction of rock mechanical properties.

1. Introduction

When designing and building geotechnical engineering structures, the mechanical properties of rocks are crucial [1,2,3,4]. Uniaxial compression strength (UCS), a crucial rock mechanical property metric, determines the material’s mechanical strength, which is crucial for assessing the adaptability and resilience of rocks to weathering. The modulus of elasticity (E) is essential to the analysis of deformation and can be used to estimate the bearing capacity of rocks. The slope of the stress–strain curves of unconfined compression tests can be used to assess their uniaxial compressive strength and modulus of elasticity of rocks. The steps and tests for determining these parameters have been standardized by the International Society for Rock Mechanics, American Society for Testing and Materials (ASTM), and the International Society for Rock Mechanics (ISRM) [5]. However, laboratory tests are destructive and are time-consuming and uneconomical to implement for a lot of experimental samples. In addition, the results of the tests are influenced by the quality of the rock samples. Therefore, to some extent, there are some limitations in obtaining these two parameters directly from laboratory experiments. For this reason, it is of positive scientific and engineering interest to develop a nondestructive, cost-effective, fast, efficient, and reliable method for estimating mechanical properties [6,7,8].

According to the findings of relevant studies, rock qualities and their indices, such as porosity (n), point loading index (Is₍₅₀₎), p-wave velocity (Vp), Schmidt hammer bounce number (Rn), water absorption, mineral composition, etc., have a significant effect on both UCS and E [9,10,11]. These two mechanical parameters can be determined in a nondirect way from the point of view of the influencing parameters. In this regard, simple models based on statistical regression seem to provide new methods for the prediction of UCS and E. For the prediction of UCS and E, a number of single regression formulations have been created employing variables including the Schmidt hammer rebound number, p-wave velocity (V_p), and point load strength (PLS) [6,12,13,14,15]. In rock engineering, however, the empirical link between a single parameter and UCS or E is not sufficient. Multiple regression analysis was devised to predict UCS, taking into account several criteria related to rock qualities, to overcome this issue [16,17,18,19]. It was discovered that the prediction accuracy of the majority of multiple regression equations was not adequate and the predictions given by these models were still subject to large errors. This is mainly attributed to the fact that the regression equations, while being easy to use, also limit the relationship between the input parameters and the output results. In fact, this nonlinear correlation is complex and the established explicit formulas do not adequately reflect this intrinsic relationship. Moreover, statistical prediction methods are deficient for newly available data, i.e., the form of the obtained equation needs to be refitted and updated if the new data differ from the original data [20].

The advent of various soft technologies, such as intelligent models and algorithms [21,22,23,24], along with the advancement of computer technology has greatly contributed to new prospects for the regression problem. Predictive models based on intelligent algorithms such as the artificial neural network (ANN) [16,25,26], support vector regression (SVR) [27,28,29,30,31,32], gene programming (GEP) [6,33,34,35], random forest (RF) [36,37], long short-term memory (LSTM) [38], and adaptive neuro-fuzzy inference system (ANFIS) [39,40,41] have received considerable attention and application in the regression analysis of rock properties. For example, Danial et al. [42] employed an adaptive neuro-fuzzy inference system on 45 rock samples to predict compressive strength and elastic modulus, and the model reached better prediction results than multiple regression analysis and an ANN. The R² for the prediction of UCS and E were 0.985 and 0.990, respectively. Naseer et al. [43] employed five parameters as input feature vectors to predict the compressive strength and elastic modulus of rocks under different thermal conditions, and the prediction results using multiple linear regression (MLR), ANN, RF, and k-nearest neighbor (KNN) models were compared, where RF achieved the best prediction with R² of 0.97 and MAPE of 0.25% for both UCS and E. Yang et al. [44] developed a particle swarm optimization-based SVR model to predict the rate-dependent compressive strength of rocks, and the results showed that the proposed hybrid model predicted better results than the elm and RF models. Matin et al. [36] developed a nonparametric prediction model for UCS and E using the RF model with variable rock properties including porosity (n), point load index (Is₍₅₀₎), p-wave velocity (V_p), and Schmidt hammer rebound number (R_n) The resulting correlation coefficients were 0.93 and 0.91. respectively, which demonstrates that the RF model is a trustworthy and precise method for assessing and estimating complex interactions in rock mechanics. To compare the prediction results of different models for UCS, Arsalan et al. [31] conducted a comparative analysis of model prediction effectiveness on 170 datasets using several models. The results showed that Gaussian process regression (GPR) obtained the best performance with the highest correlation coefficient of 0.9955 and the smallest RMSE of 0.52169. These studies have shown that the predictions using soft computing methods are more accurate than those of traditional statistical models. This is because these models can capture the nonlinear correlation between the input and output in an implicit form instead of an explicit equation.

Although there have been many studies using different intelligent models to predict UCS and elastic modulus, little research has been reported on the use of extreme learning machines (ELMs) in predicting the mechanical property prediction of rock samples [45,46]. In contrast, ELM modes have been widely used in regression analysis such as solving concrete strength prediction [47,48]. Therefore, this paper aims to investigate the predictability of the ELM model for UCS and E. For the ELM network structure, the combined selection of weights and bias has a great effect on the predictive performance of the model. Considering the optimal performance of the GWO algorithm in the selection of hyperparameters of C and γ for the kernel-based ELM model with Gauss kernel function [46], which is an improved algorithm of the ELM model, the GWO algorithm was also employed to optimize and select the best combination of weights and bias to achieve the accurate prediction of UCS and E in this paper.

2. Methodology

2.1. Extreme Learning Machine Network

The extreme learning machine (ELM) is applicable to the output prediction problem of complex nonlinear systems and has the advantages of fast learning speed, good generalization performance, good scalability, and low computational complexity. The extreme learning machine adopts a three-layer structure, which is shown in Figure 1, with N arbitrary samples, where $x_i\in R^m$ and $y_i\in R^n$. Then, the extreme learning machine model with L hidden layer nodes can be represented as

$${\displaystyle \sum _{j=1}^L{\beta }_{j}g({\omega }_j{x}_i+b_j)=y_i}$$

(1)

where β_j is the output weight, g(x) is the activation function, w_j is the input weight vector, b_j is the bias, and y_i is the network output of the sample.

If the single hidden layer model is able to approximate N samples with zero error, i.e., it satisfies ${\displaystyle \sum _{j=1}^N{\Vert y_i-t_i\Vert }^2}=0$, then there exist parameters w_j, b_j, and β_j that satisfy the following conditions.

$${\displaystyle \sum _{j=1}^L{\beta }_{j}g({\omega }_j{x}_i+b_j)=t_i}$$

(2)

Transformed into matrix form, the equation can be derived as follows.

$$\begin{gathered} H\beta =T \\ H=\left[\begin{array}{c}h(x_1)\\ ⋮\\ h(x_N)\end{array}\right]=\left[\begin{array}{ccc}g({\omega }_1{x}_1+b_1)& \cdots & g({\omega }_L{x}_1+b_L)\\ ⋮& & ⋮\\ g({\omega }_1{x}_N+b_1)& \cdots & g({\omega }_L{x}_N+b_L)\end{array}\right] \end{gathered}$$

(3)

Then, the output weight matrix β can be expressed as:

$$\beta =H^{†}T$$

(4)

$H^{†}$ is the Moore–Penrose generalized inverse matrix of H.

2.2. Grey Wolf Algorithm Optimizer

The social structure and hunting habits of grey wolf populations served as the inspiration for the development of the wolf optimization algorithm, a new pack intelligence optimization method [49,50]. To find the best solution to the issue at hand, the algorithm mimics the behavior of grey wolves in nature with regard to task division, hunting, and pouncing on prey [51,52]. The wolf hunting process is described as follows.

(1) Social hierarchical stratification. Grey wolves are good at running, like to live in groups, and follow a strict social hierarchy among wolves. As shown in Figure 2, wolves can be divided into 4 grades according to the hierarchy, from high to low, α, β, δ, and ω. Calculate the physical fitness of each grey wolf individual, and mark the 3 grey wolves with the best physical fitness as α, β, and δ in turn, and the remaining grey wolves as ω. The chase is initiated by α, β, and δ, ω tracks and rounds up prey, and the position of the prey corresponds to the global optimal solution of the optimization problem [53].

(2) Surrounding prey. When hunting, grey wolves will first evaluate the distance between themselves and their prey before approaching slowly and then circling.

$$D=\left|C\cdot X_P(t)-X(t)\right|$$

(5)

$$C=2r_1$$

(6)

where t denotes the number of current iterations; $X_P(t)$ denotes the position vector of the prey; $X(t)$ denotes the position vector of the grey wolf; C is the coefficient vector; r₁ is a random vector taking values between [0,1]; D denotes the distance between the wolf and the prey. Subsequently, the wolves have to update their positions according to the prey orientation.

$$X(t+1)=X_P(t)-A\cdot D$$

(7)

$$A=2a\cdot r_2-a$$

(8)

where A is a vector of coefficients; r₂ is a random vector taking values between [0,1].

(3) Hunting. When the grey wolf determines the location of the prey, the alpha wolf will lead the beta wolf and the δ wolf in pursuit. α, β, and δ are the most capable among the wolves and are closest to their prey. Use the position relationship of the three to judge the position of the prey, guide the wolf to attack the position of the prey, and finally capture the prey after multiple positioning and movement. The process of position updating is as follows.

$$\begin{array}{l}\{\begin{array}{l}{D}_{\alpha }=\left|C_1\cdot X_{\alpha }(t)-X(t)\right|\\ D_{\beta }=\left|C_2\cdot X_{\beta }(t)-X(t)\right|\\ D_{\delta }=\left|C_3\cdot X_{\delta }(t)-X(t)\right|\\ X_1=X_{\alpha }-A_1\cdot D_{\alpha }\\ X_2=X_{\beta }-A_2\cdot D_{\beta }\\ X_3=X_{\delta }-A_3\cdot D_{\delta }\\ X(t+1)=\frac{X_1+X_2+X_3}{3}\end{array}\end{array}$$

(9)

where X(t + 1) is the updated individual position of the grey wolf. The ω wolves are guided by the 3 head wolves towards the prey position.

To improve the model performance, the grey wolf algorithm is introduced in this paper to optimize the selection of weights w and bias b. The model framework established is presented in Figure 3.

3. Database

Referring to the model input features studied above, in this paper, four effective rock features, namely porosity (n), p-wave velocity (V_p), Schmidt hammer rebound number (R_n), and point loading index (Is₍₅₀₎), are used as input variables of the model to predict UCS and E. For intelligent models, a rich and representative database is necessary. For this purpose, 101 datasets with four input variables and two output variables each were gathered from the literature [16,54], in which 71 granite block samples were collected from the face of the tunnel and 30 Travertine samples were extracted from the mine. More detailed information about the rock samples can be found in References [16,54]. The distribution of these variables and their correlation coefficients with each other are shown in Figure 4 and Figure 5, respectively. The statistical characteristics of these variables are summarized in

Table 1. Mathematical statistical characteristics of the database.

Variable	n	Rn	VP	Is(50)	UCS	E
Unit	%	-	m/s	MPa	MPa	GPa
Number	101	101	101	101	101	101
Max	10.27	61	7943	7.1	211.9	183.3
Min	0.1	25.63	2823	0.89	22.7	3.05
Mean	2.21	43.13	5517.80	3.28	93.03	63.45
Median	0.46	46	5450	3.16	95.6	66.7
SD	3.24	11.13	930.94	1.28	50.15	47.80
Type	Input	Input	Input	Input	Output	Output

. The training process used 10-fold cross-validation [55,56], and the dataset was randomly divided into a training set (80 samples) and a test set (21 samples). For comparison purposes, the dataset was fixed after the first random selection. The training set and the test set are thus the same for different models.

4. Analysis of Results

Correlation coefficients (R²), root mean square error (RMSE), and mean absolute percentage error (MAPE) were adopted to measure the prediction outputs of each model in order to quantitatively compare the predictive capabilities of the models [57,58]. Since the results of UCS and E are independent, the two output targets are discussed separately.

4.1. Analysis of Compressive Strength Prediction Results

Figure 6 displays the predictions provided by individual models. Further, the training and testing of the models are presented separately as shown in Figure 7. It can be seen that the suggested GWO-ELM model achieves the maximum value both for the training and testing phases, with the correlation coefficient R² of the prediction outcomes of the machine learning models being greater than those of the MLR model. Additionally, the correlation coefficient shows that the optimized GWO-ELM model performs more predictably than the original ELM model.

Further, the distribution of prediction errors for each model is calculated as shown in Figure 8. Then, the relative error for the test set is calculated and plotted in the box plot shown in Figure 9. In addition, the evaluation metrics are listed in

Table 2. Evaluation index of different models for UCS.

Model	Training			Test
	R2	RMSE	MAPE	R2	RMSE	MAPE
MLR	0.871	17.914	19.444%	0.891	17.711	19.960%
ELM	0.917	14.334	15.890%	0.913	16.914	19.855%
DT	0.945	11.673	10.039%	0.939	14.817	17.191%
SVR	0.934	12.750	14.379%	0.939	12.754	18.844%
GWO-ELM	0.946	11.519	12.424%	0.951	11.446	16.131%

, where for the test set, the hybrid GWO-ELM model achieves an RMSE of 11.446 and a MAPE of 16.131%. This further demonstrates the excellent prediction accuracy of the optimization model suggested in this paper and its ability to capture the nonlinear relationship between the UCS and the four input variables.

4.2. Analysis of Elastic Modulus Results

Similarly, the comparison between the predicted results of the different models for the elastic modulus and the actual values is shown in Figure 10. Based on the test set’s prediction findings, which are shown in Figure 11, the correlation coefficient of the GWO-ELM model achieved 0.87, which is still a small value compared to the prediction of compressive strength, although it is the maximum among the five models. From the error analysis in Figure 12 and Figure 13, it is clear that the optimized model improved the prediction accuracy compared to that before optimization. Among the evaluation indexes of the model prediction results listed in

Table 3. Evaluation index of different models for E.

Model	Training			Test
	R2	RMSE	MAPE	R2	RMSE	MAPE
MLR	0.839	19.269	42.293%	0.843	18.275	26.298%
ELM	0.899	15.204	31.114%	0.839	18.927	48.527%
DT	0.889	16.014	36.778%	0.854	17.801	37.133%
SVR	0.889	16.005	35.212%	0.869	17.203	30.282%
GWO-ELM	0.903	14.949	26.781%	0.870	17.571	23.649%

, the optimized model achieved the smallest RMSE of 17.571 and MAPE of 23.649%, which were better than the prediction results of the other four models.

5. Discussion

Comparing the prediction results of different models, it can be found that the other four models all have better prediction results than the MLR model, which is mainly due to the fact that the MLR model restricts the mapping relationship between multiple input parameters and outputs, and this inherent mapping relationship is often nonlinear. Compared with the other three original machine learning models, the hybrid model proposed in this paper improves the predictive performance of the model due to the optimal combination of hyperparameters (weights and bias) obtained by using the GWO algorithm.

As far as the prediction results of compressive strength and elastic modulus are compared, the intrinsic correlation between the four input features used in this paper and compressive strength is stronger, while that with elastic modulus is weaker. Therefore, future research can consider input feature diversity and introduce more input parameters, which have a stronger intrinsic correlation with compressive strength and elastic modulus, to build the model. Moreover, the number of data samples in this study is limited and increasing the number and diversity of samples as much as possible can help to improve the prediction accuracy and generalization ability of the hybrid model.

6. Conclusions

In this paper, a hybrid GWO-ELM model was developed to predict the UCS and E of rocks. The main findings are as follows.

(1) In terms of evaluation metrics of prediction results, the proposed hybrid model reached the R² of 0.951, RMSE of 11.446, MAPE of 16.131% for UCS, and R² of 0.870, RMSE of 17.571, MAPE of 23.649% for E. The model achieved a better predictive performance than the other four models with the highest correlation coefficient R² and the lowest error indicator.

(2) Compared with the prediction results of the original ELM model, the GWO algorithm was validated as an efficient optimization algorithm that can improve the prediction performance of the ELM model by selecting the best combination of weights and bias.

(3) The proposed GWO-ELM model could be recommended as a prediction auxiliary tool for mechanical properties to provide support and assistance for real-time analysis of rock mechanical properties.

(4) The input feature diversity and the number of dataset samples considered in this paper limit the prediction performance of the model to some extent and increasing the effective input parameters and the number of dataset samples as much as possible can help improve the prediction performance of the hybrid model.