The implementation process of the BGD-MSR-DNN rockburst intensity grade prediction model includes data pre-processing, model establishment, and model prediction, as shown in Fig. 1.

The data pre-processing part takes typical rockburst sample data worldwide as data sets, selects the optimal based on multiple rockburst evaluation indexes, and normalizes the training samples before inputting them into the model. This step saves the time cost of the rockburst intensity grade and helps improve the accuracy of predicting the rockburst intensity grade. The model establishment part builds and trains the BGD-MSR-DNN model, calculates the Accuracy and Loss indicators of the model, and determines whether the two indicators meet the requirements. If the requirements are met, the model enters prediction step. If not, the model is retrained. The model predictions part solidifies the model, ensuring it meets the requirements. It inputs the rockburst sample data to be tested into the model and outputs the rockburst intensity grade.

Fig. 2 presents the overall framework of the BGD-MSR-DNN rockburst intensity grade prediction model established in this paper.

The BGD-MSR-DNN model adds an MSR module to the hidden layer of the DNN. Different connection methods in the module can make al jump to the next layer or several layers to achieve the purpose of conveying information to deeper layers. The BGD module is included to optimize the model algorithm, and Softmax is adopted as the activation function. Following this, 10-fold cross-validation is employed to output the prediction results of the model. The specific steps of model prediction are as follows:

Step 1: Take the rockburst evaluation indexes as the model input, normalize the data, improve gradient descent efficiency to determine the optimal solution, and accelerate the convergence speed. The specific equation is as follows:

where xn is the original data; xn′ is the normalized data; xmax and xmin are the maximum and minimum values in the original data sets.

Step 2: In the DNN forward propagation algorithm, parameter xi is used as the input and σ(z) is the activation function. Assuming that there are m neurons in the l−1 layer and n neurons in the l layer, the linear coefficient wi of l layer consists of an n×m weight coefficient matrix Wl. Bias parameter b of the l layer forms an n×1 bias vector Bl, output a of the l−1 layer forms an m×1 vector al−1, and the l layer is Before activation, the linear output z forms an n×1 vector zl, and the output a of the l layer forms an n×1 vector al. The linear relationship between each layer z , the output of the l layer al is described as:

(2) z=∑i=1mwixi+b

(3) al=σ(zl)=σ(Wlal−1+Bl)

To improve the convergence speed of the model, this paper uses the BGD module to optimize the backpropagation algorithm and selects the cross-entropy loss function. Then the loss function L and the gradient δl of the output layer are expressed as:

(4) L=−1n∑x[yln⁡a+(1−y)ln⁡(1−a)]

where n is the total number of samples, x is the degree of the prediction vector, y is the label on the corresponding x dimension, and a is the prediction label. When cross-entropy loss function is used, the relationship between gradient δl and δl+1 is deduced by mathematical induction as:

(6) δl=∂J(W,b,al,y)∂zl=(∂zl+1∂zl)T∂J(W,b,al,y)∂zl+1=(∂zl+1∂zl)Tδl+1

Obtaining ∂zl+1∂zl=Wl+1diag(σ′(zl)) through Eq. (2), The relationship between δl and δl+1 can be obtained as:

(7) δl=(Wl+1)Tδl+1⊙σ′(zl)where ⊙ is the Hadamard product, The gradient δl+1 of layer l+1 can be found by formula (7), the learning rate is then set as α, and the gradients of Wl and Bl of the l layer are updated as follows:

(8) Wl=Wl−α∑i=1mδl(al−1)T

(9) Bl=Bl−α∑i=1mδl

Step 3: In a deep neural network, the more network layers, the larger the function space. In theory, the model’s performance will improve as the neural network’s depth deepens. Therefore, when constructing a DNN structure, as the number of network layers increases, the extracted features will be more prosperous and more abstract, and the classification accuracy will be higher [27]. However, the network’s performance does not improve with the increase of network layers. On the contrary, there will be a phenomenon of network degradation in which the accuracies of the training and test sets decrease. This results in problems such as decreased model accuracy and low generalization ability [28]. In this paper, MSR module is added to the model to solve the problem of network degradation when there are too many hidden layers in the DNN. In the multiscale residual module, al can skip one or several hidden layers to pass information to deeper layers of the neural network. Fig. 3 depicts the MSR module structure in the model.

Step 4: The rockburst intensity grade prediction is a multi-classification problem, and thus the output component of the proposed model employs the Softmax activation function [29], which is described as follows:

(10) sjl=exp⁡(zjl)∑i=1nexp⁡(zil)

In Eq. (10), n is the number of neurons in the l layer of the output layer; and sjl is the output of the j neuron in the output layer.

Step 5: To improve the model’s generalization ability, this paper first adopts a 10-fold cross-validation method [30] to randomly divide the rockburst sample data into 10 parts, nine of which are for training, and one is for verification. The training and verification process is performed 10 times and the optimal of the 10 prediction results Y is taken as the output.

To verify the performance of the BGD-MSR-DNN model, this paper collected 350 sample data of rockburst from global research related to rockburst, including 149 data from [31], 129 data from [32], 68 data from [33], and 4 data from [34] (Table 1), including projects related to mine engineering, deep tunnels, and large underground caverns worldwide. The data includes the maximum tangential stress of the surrounding rock of the cave wall (σθ), the uniaxial compressive strength of the rock (σc), the uniaxial tensile strength of the rock (σt), the elastic energy index of the rock (Wet), and the corresponding rockburst intensity grade. The rockburst intensity grade is divided into four categories: none rockburst; light rockburst; moderate rockburst; and strong rockburst, represented by letters N, L, M, and S, respectively.

Among the 350 rockburst samples, there were 51 cases (14.6%) of none rockburst, 104 cases (29.7%) of light rockburst, 138 cases (39.4%) of moderate rockburst, and 57 cases of strong rockburst (16.3%). The proportion of samples of each category is shown in Fig. 4. The data quantity of the moderate rockburst sample is significantly higher than that of the other three sample data. This results in an unbalanced sample category, which leads to inaccurate experimental results. To reduce the impact of this phenomenon on the experiment, this paper uses random undersampling to randomly select samples from the moderate rockburst category to form data sets [35].

Rockburst occurrence is a function of numerous factors and its formation mechanism is complex. Table 2 reports the rockburst intensity grade criteria proposed by scholars based on the factors affecting rockburst [36], where σθmax is the tangential stress of the cavity, σL is the axial, tangential force, and σ1 is the maximum principal stress.

It can be seen from Table 2, all the proposed approaches use a single evaluation index, and thus their prediction effect is not ideal. Due to the complex mechanism of rockburst and many influencing factors, the correlation between evaluation index will affect the prediction results of rockburst intensity grade. In order to improve the prediction performance of the model, this paper uses the Spearman coefficient method to obtain and analyze the correlation coefficient between the evaluation index, as shown in Table 3.

The correlation coefficient indicates the degree of linear correlation between each index in the rockburst sample data. Table 3 reveals that the majority of the correlation coefficients between each index are below 0.5 after the analysis using the Spearman coefficient method, with a moderate, weak, very weak, or no correlation. To further improve the accuracy of model prediction, this paper selects σθ/σc, σc/σt, and Wet with the lowest correlation coefficients as the optimal rockburst evaluation indexes.

According to the optimal rockburst evaluation indexes selected in Section 3.2, with σθ/σc as the X-axis, σc/σt as the Y-axis, and Wetas the Z-axis, 350 groups of rockburst sample data are used to create a three-dimensional distribution graph for analysis (Fig. 5). The rockburst intensity grade is observed to increase with the value of each index. However, there is no apparent linear relationship, and the boundary distinction between none rockburst, light rockburst, moderate rockburst, and strong rockburst is not obvious. This is particularly true for the difference between minor and moderate rockburst. As the data between these two grades account for a large proportion of the total data, the difficulty of predicting the rockburst intensity grade is consequently increased.

To further evaluate the rockburst sample data, outliers were detected using boxplots of the three evaluation indexes σθ/σc, σc/σt, and Wet are shown in Fig. 6. Each evaluation index exhibits outliers, which were replaced or deleted from the sample data in the previous rockburst intensity grade prediction work [37]. However, due to the complex mechanism of rockburst, outliers in actual engineering applications often require special attention, and thus the outliers were not processed in the experiment. Instead, outliers and the original sample data were input into the model to focus on the experiment and analysis.

This paper uses several evaluation metrics to evaluate the comprehensive performance of the proposed model, including Accuracy, Precision, Recall, and F₁-score [38]. The higher the evaluation metrics, the better the test model’s performance. The formula and descriptions of each indicator are described in Table 4.

To verify the effectiveness of the BGD-MSR-DNN model in the prediction of rockburst intensity grade, this paper first divides the rockburst sample data into training and test sets at a ratio of 9:1. These sets are input into the k-nearest neighbors (KNN) model, probabilistic neural network (PNN) model, SVM model, particle swarm optimization-support vector machine (PSO-SVM) model, DNN model, and BGD-MSR-DNN model for training. The test sets are then used to evaluate the prediction performance of each model. Fig. 7 presents the confusion matrix derived from the prediction results of each model.

Fig. 7 shows the distribution of the real category and the predicted category when each model predicts. The darker the color of the diagonal line of the confusion matrix, the more the number of predicted categories is the same as the real category. It can be seen that the BGD-MSR-DNN model predicts the best results. The model evaluation metrics detailed in Section 4.1 are employed to analyze the prediction results. Accuracy, Precision, Recall, and F₁-score of rockburst intensity grade prediction models were calculated by confusion matrix, and the performance of different models was analyzed based on four evaluation metrics. Table 5 reports the evaluation metrics of the six rockburst intensity grade prediction models. The best results are indicated in bold.

It can be seen from Table 5 that four evaluation metrics of the BGD-MSR-DNN model are the best compared with other models under the same test sets. Four evaluation metrics of the BGD-MSR-DNN model surpass those of the other models. The Accuracy rate reached 97.1%, exceeding the KNN model (51.4%), PNN model (60%), SVM model (62.9%), PSO-SVM model (71.4%), and the DNN (82.9%) model by 45.7%, 37.1%, 34.2%, 25.7%, and 14.2%, respectively. It shows that the prediction performance of the deep learning model has been dramatically improved compared with the machine learning model. Moreover, the BGD-MSR-DNN model outperformed the original DNN model by 9.4%, 19.8%, and 15.4% in three evaluation metrics of Precision, Recall, and F₁-score, and corresponding values were much higher than the other comparable models. It proves that adding the BGD module and MSR module to the DNN network structure has achieved ideal results and improved the model’s prediction accuracy. In addition, comparing the evaluation metrics of the SVM model and the PSO-SVM model, the DNN model, and the BGD-MSR-DNN model, it can be found that the prediction performance of the combined classifier is much more excellent than that of a single classifier. Combination classifiers can also make up for the limitations of single classifiers in predicting rockburst intensity grade.

To prove the necessity of the BGD module and MSR module, In the ablation experiments in this section, the model performance after adding the BGD module and MSR module is verified by comparing the prediction performance of the DNN model, BGD-DNN model, MSR-DNN model, and BGD-MSR-DNN model. The results of the ablation experiments are shown in Table 6, and the best results are indicated in bold.

The experimental results in Table 6 show that when the BGD-DNN model is added with the BGD module in the DNN network structure, the two evaluation metrics of Recall and F₁-score are respectively increased by 12.6% and 9.6% compared with the DNN model, because the BGD module can update parameters faster when processing data sets with fewer samples, thus significantly improving the efficiency of the network; When the MSR-DNN model of the MSR module is added to the DNN network structure, the DNN model has increased by 8.5%, 7.7% and 6.6% in the three evaluation metrics of Accuracy, Recall and F₁-score, respectively, because the MSR module effectively solves the problem of network degradation when there are too many hidden layers of DNN, it dramatically improves the accuracy of model prediction; When the BGD-MSR-DNN model with both the BGD module and the MSR module is added to the DNN network structure, the accuracy of prediction is significantly improved compared with the other three models. It shows that the model’s prediction accuracy is related to the BGD and MSR modules. It also proves the effectiveness of the BGD-MSR-DNN model in predicting the rockburst intensity grade.

In addition, ablation experiments found that the BGD-MSR-DNN model with BGD module and MSR module has significantly improved in various evaluation metrics compared with the original DNN model. Therefore, the DNN model and the BGD-MSR-DNN model are selected for further analysis and comparison, and the comparison results are shown in Fig. 8 and Fig. 9. It can be seen from the figure that with the increase of iteration rounds, the Accuracy curve and Loss curve of the two models tend to be stable, but the curve of the BGD-MSR-DNN model fluctuates less. At the same time, the training time of the BGD-MSR-DNN model is significantly lower than that of the DNN model, indicating that the BGD-MSR-DNN model has better convergence speed and better prediction performance.

To verify the sensitivity of the hyperparameters existing in the BGD module during model prediction, we continuously search and obtain values manually and then conduct prediction experiments. The values during the experiment are shown in Table 7.

Finally, through experiments, the evaluation metrics curves of the hyperparameter α under different values are obtained, as shown in Fig. 10. The curves of four evaluation metrics of Accuracy, Precision, Recall, and F₁-score obtained under different values of hyperparameter α exhibit a normal distribution. At α = 0.06, four evaluation metrics all reach the maximum value. Furthermore, the floating trend of the curve in Fig. 10 indicates that the value of the hyperparameter a affects the speed at which the model determines the optimal value. If hyperparameter α is too small, the optimization efficiency may be too low, causing the model to fail to converge. In contrast, if hyperparameter α is too large, the optimal value may be missed, resulting in a decline in model performance.