Model for Predicting the Rockburst Intensity Grade in Gently Dipping Rock Strata via MIPSO-RF

Abstract

This study aims to improve the prediction accuracy of rockburst intensity grades in gently dipping rock strata, and provide reliable technical support for risk prevention, long-term stable production and sustainable development in underground engineering construction. Therefore, a rockburst intensity grade prediction model combining multi-strategy improved particle swarm optimization (MIPSO) with random forest (RF) is proposed, and the stress coefficient (SCF), brittleness coefficient (B) and elastic energy index (W_et) are selected as input indicators. After the algorithm and model are validated using benchmark test functions and the five-fold cross-validation method, their performance is compared with that of the other four models based on evaluation metrics, and the Shapley interpretability analysis (SHAP) is conducted. The results show that the performance of the model is superior to that of other models, and the importance ranking of the prediction indicators is SCF, W_et, and B. Finally, the application software developed based on the model is used for rockburst intensity grade prediction; rockburst prediction indicators are obtained through experiments and numerical simulations, and the prediction results obtained after importing them into the software are consistent with the actual situation, which proves that the rockburst prediction framework constructed in this paper has practicality.

1. Introduction

Since modern times, the demand for mineral resources has increased, mining operations have gradually advanced to deep strata with more complex geological conditions, the occurrence environment of rock masses has become increasingly complex, and rockburst disasters have occurred frequently [1,2]. Rockbursts are geodynamic disasters triggered by the sudden release of accumulated elastic potential energy within the surrounding rock of underground structures, and are characterized by rock fracturing and ejection [3]. The sudden onset and extreme destructive force of this hazard pose severe threats to the safety of personnel and equipment, potentially rendering support measures ineffective and rendering the work area unusable [4]. The long-term stable operation of mines constitutes a core prerequisite for the sustainable utilization of mineral resources. Its significance extends beyond the production guarantee of individual mines, but rather represents a systematic project spanning the three dimensions of economy, ecology, and society.

With increasing mining depth, the surrounding rock stress gradually increases, and the occurrence frequency of rockburst disasters also increases accordingly [5]. Given that gently dipping rock strata account for a significant proportion of mining operations, scholars have conducted relevant studies to investigate the relationship between such strata and rockbursts and ensure the sustainable development of mines. Lin et al. carried out a study where the energy accumulated in the structural planes of gently dipping rock strata under gradient stress was observed to increase with the increase in dip angle, and verified that the higher the maximum infrared radiation temperature of the unloading surface, the stronger the rockburst intensity [6]. Feng et al. conducted studies on the rockburst occurrence mechanism and found that rock masses containing structural planes are associated with greater hazards and higher frequencies when rockbursts occur, and that the boundary morphology of rockburst craters is controlled by the surrounding rock structural planes [7,8]. Zhou et al. used similar materials in direct shear tests and concluded that gently inclined structural planes would undergo shear failure or shear-tension composite failure, which in turn induces rockburst hazards [9]. Studies on single structural planes under triaxial conditions conducted by Su et al. demonstrated that the larger the dip angle of the exposed structural plane is, the higher the rockburst intensity and the local energy release rate [10]. Chen et al. showed in a study that with the increase in structural plane dip angle, the failure mode of gently dipping bedded shale specimens gradually transformed from tensile failure to tension-shear composite failure, and the failure degree was enhanced [11]. Feng et al. adopted a numerical simulation method to study the influence of structural planes on the failure characteristics of surrounding rock in deep hard rock roadways, and the results showed that the slab splitting failure of surrounding rock under the action of structural planes would lead to stress concentration and energy accumulation near the structural planes, which could easily induce rockbursts under the action of tangential stress [12]. Studies conducted by scholars have demonstrated that rockburst occurrence in gently dipping rock strata is primarily controlled by structural planes. The presence of structural planes induces changes in the original stress distribution of rock masses, leading to the generation of shear stress in the vicinity of structural planes. Consequently, the failure mode of rock masses is transformed from tensile failure to combined shear-tensile failure, which increases the probability of rockburst disasters and exacerbates the degree of rockburst-induced damage.

At present, rockburst prediction methods are classified into empirical indicator prediction methods and multi-indicator comprehensive evaluation models [13]. The prediction of rockburst intensity grade via empirical indicator prediction methods is achieved through empirical rockburst criteria, such as the Hoek criterion, the Russenes criterion [14], and the brittleness criterion [15]. However, such methods are characterized by a single discriminant indicator, are applicable only to rockburst prediction under specific conditions, and have significant limitations [16]. The accurate prediction of rockburst intensity grade via multi-indicator comprehensive evaluation models is achieved through mathematical methods, such as the fuzzy mathematics comprehensive evaluation method [17]. Examples include the rough sets-normal cloud model [18], the subjective and objective weighting-matter-element extension model [19], and the extended multi-attributive border approximation area comparison method [20]. These methods rely on the users’ subjective understanding of rockbursts and impose high requirements on users.

With the development of big data technology, machine learning models have gradually become alternative solutions for rockburst prediction. Therefore, on the basis of existing studies, the exploration of better algorithms and the establishment of more accurate and practical rockburst intensity grade prediction models are valuable for engineering applications. For example, Jin et al. optimized the support vector machine (SVM) via the whale optimization algorithm (WOA) to develop the WOA-SVM [21]. Wang et al. optimized back propagation (BP) via the sparrow search algorithm (SSA) to obtain the SSA-BP, which is applied to rockburst prediction [22]. Li et al. optimized the convolutional neural network (CNN) via multiple imputation chained equations (MICE) to obtain the MICE-CNN [23]. Wang et al. optimized the extreme gradient boosting (XGBoost v1.7.6) via the genetic algorithm (GA) to obtain the GA-XGBoost [24]. Wu et al. optimized the SVM via the gray wolf optimizer (GWO) to obtain the GWO-SVM [25]. Mu et al. optimized the combination of CNN and SVM via the black-winged kite algorithm (BKA) to obtain the BKA-CNN-SVM [26]. Li et al. optimized the back propagation neural network (BPNN) via the Newton–Raphson-based optimizer (NRBO) to obtain the NRBO-BPNN [27].

The random forest (RF) is an advanced machine learning algorithm proposed by Leo in 2001. It has the advantages of fewer parameter adjustments, high training efficiency, strong robustness, and low susceptibility to overfitting. To improve the prediction performance and generalization ability of a model, optimization algorithms are usually introduced to optimize it [28]. Guo et al. [29] proposed an algorithm that combines mean particle swarm optimization (MPSO) and the GA to optimize the random forest, which was applied to evaluate landslide susceptibility. Studies have shown that this optimization algorithm effectively improves the prediction accuracy of the model [29]. Man et al. [30] optimized the RF via the SSA and applied it to rockburst intensity grade prediction. This optimization scheme effectively addresses the problem of insufficient randomness of the RF and improves the prediction accuracy and generalization ability of the model [30]. Gao et al. [31] optimized the particle swarm optimization (PSO) with simulated annealing (SA) and then integrated it with the radial basis function (RBF) to construct an SA-PSO-RBF model, which was applied to predict rockburst intensity grades. The results show that the SA can effectively avoid the premature convergence problem of the PSO [31]. Su et al. [32] established a rockburst grade prediction model using the Multiscale Graph Convolutional Neural Network (MGCNN). By integrating similarity perception and inter-indicator correlation, the accuracy of the prediction results was effectively improved [32]. Dong et al. optimized the Generalized Relevance Matrix Machine (GRMM) model via PSO for rockburst early warning, which effectively solved the fitting problem encountered during parameter adjustment and significantly enhanced the prediction accuracy and generalization ability [33]. Mei et al. adopted the combination of Shapley interpretability analysis (SHAP) and XGBoost for rockburst prediction, which effectively improved the accuracy and interpretability of rockburst prediction [34]. Gao et al. combined the genetic projection pursuit algorithm (GPPA) with the geothermal correction coefficient for rockburst prediction, thereby improving the model’s accuracy and optimization efficiency under geothermal conditions [35].

In recent years, various models have been applied to engineering projects by many scholars, and good results have been achieved. However, relatively few studies have been conducted on the rockburst intensity grade prediction for gently dipping rock strata. In this study, a rockburst intensity dataset is first established, and then the multi-strategy improved particle swarm optimization (MIPSO) is combined with the RF to construct the MIPSO-RF rockburst intensity grade prediction model. After the model is validated via fivefold cross-validation, the performance of this model and the comparison models is evaluated through four evaluation indicators and SHAP interpretability analysis in this study. Finally, an intelligent prediction framework for the rockburst intensity grades of gently dipping rock strata is constructed through software development. The prediction indicators obtained from laboratory rock mechanics tests and numerical simulations are imported into the software for engineering verification to verify the applicability of the model in practical engineering.

This research goes beyond model training by developing software that directly outputs rockburst intensity grades from input parameters, enabling rapid on-site assessment for significant practical utility, which facilitates the long-term stable operation and sustainable development of mines.

2. Dataset Establishment and Analysis

2.1. Determination of the Rockburst Intensity Grade Prediction Indicator

Rockbursts are prone to occur in hard and intact rock masses with high stress concentrations and large amounts of stored energy, and their occurrence is affected mainly by stress, brittleness and energy indicators. Therefore, in the selection process of rockburst intensity grade prediction indicators, these three indicators also need to be selected [36]. Specifically, the maximum tangential stress of the surrounding rock can reflect the magnitude of stress acting on the surface of the surrounding rock, and the uniaxial compressive strength can reflect the hardness of the rock.

Table 1. Overview of machine learning models for rockburst prediction.

Source	Model	Prediction Indicator	Number of Dataset Samples
Jin et al., 2023 [21]	WOA-SVM	σc, σt, σθ, Wet	120
Wang et al., 2024 [22]	SSA-BP	σc, σt, σθ, Wet, σc/σt, σθ/σc	100
Li et al., 2023, [23]	MICE-CNN	σc, σt, σθ, Wet, σc/σt, σθ/σc	120
Wang et al., 2024 [24]	GA-XGBoost	σc, σt, σθ, Wet	471
Wu et al., 2023 [25]	GWO-SVM	σc, σt, σθ	153
Zhou et al., 2025 [26]	BKA-CNN-SVM	σc, σt, σθ, Wet, σc/σt, σθ/σc	284
Li et al., 2025 [27]	NRBO-BPNN	σc/σt, σθ/σc	100

shows that in the empirical indicator prediction method, the stress concentration factor (SCF) is a widely used prediction indicator, and it is exactly the ratio of the maximum tangential stress to the uniaxial compressive strength. Its calculation formula is as follows:

$$SCF={\displaystyle \frac{{\sigma }_{\theta }}{{\sigma }_{\mathrm{c}}}}$$

(1)

where σ_θ represents the maximum tangential stress, MPa; σ_c represents the uniaxial compressive strength, MPa.

During the loading process, energy can accumulate in brittle rock masses in the form of elastic strain energy instead of being consumed through plastic deformation. When the applied stress exceeds the strength limit of the rock itself, the rock will suddenly fail, and the stored elastic strain energy will be instantly released along with fragments and ejected rock blocks. The calculation formula of the brittleness coefficient (B) is as follows:

$$B={\displaystyle \frac{{\sigma }_{\mathrm{c}}}{{\sigma }_{\mathrm{t}}}}$$

(2)

where σ_t represents the Brazilian tensile strength, MPa.

Rockburst lies in the instantaneous imbalance release of the internal energy system of rock triggered by external loading [37]. The occurrence of a rockburst is accompanied by the bursting and ejection of rock blocks, so its genesis is also related to the characteristics of the energy stored in rock masses [38]. The elastic energy index (W_et) is obtained from the stress–strain curve via the integral method, and its calculation formula is as follows:

$$W_{\mathrm{et}}={\displaystyle \frac{E_{\mathrm{e}}}{E_{\mathrm{p}}}}$$

(3)

where E_e represents the retained elastic strain energy, and E_p represents the dissipated strain energy, as shown in Figure 1.

Therefore, SCF, B and W_et are selected as the prediction indicators of rockburst intensity in this study. Specifically, the SCF can reflect the degree of stress concentration of the surrounding rock in underground engineering, the B can characterize the brittle characteristics of rock mass during failure, and the W_et can reflect the performance of rock strata in storing and releasing energy. The classification criteria for the three prediction indicators are presented in

Table 2. Grading criteria of rockburst intensity.

	SCF	B	Wet
None	<0.3	>40	<2.0
Light	0.3–0.5	26.7–40	2.0–3.5
Moderate	0.5–0.7	14.5–26.7	3.5–5.0
Intense	>0.7	<14.5	>5.0

[39].

2.2. Establishment of a Rockburst Sample Dataset

Rockbursts are classified into four grades according to their intensity at the time of occurrence, namely none, light, moderate and intense rockbursts [40]. On the basis of the selected prediction indicators and classification standards, after literature retrieval and elimination of abnormal and duplicate values, 157 sets of rockburst sample data from references [13,41,42,43,44,45,46] are selected to establish a dataset. The dataset includes 36 non-rockburst samples, 31 light rockburst samples, 46 moderate rockburst samples, and 44 intense rockburst samples. Among these, 94 sets of rockburst sample data were selected from rockburst records of gently dipping coal seams, with a burial depth ranging from 50 to 1500 m. The samples cover multiple lithologies dominated by sedimentary rocks, and the rock mass structure spans a range from intact to fragmented and the sample data are shown in

Table 3. Rockburst sample dataset.

Number	SCF	B	Wet	Burial Depth	Lithology (Rock Structure)	Rockburst Intensity Grade
1	0.14	16.89	5.2	204	Rock shale (fragmented)	I
2	0.22	25.00	3.00	412	Sandstone (intact)	II
3	0.55	14.72	6.43	600	Limestone (relatively intact)	III
4	0.78	11.94	7.30	1316	Limestone (relatively intact)	IV
…	…	…	…	…	…	…
154	0.18	34.18	2.45	115	Limestone (fragmented)	I
155	0.29	29. 51	4.10	326	Limestone (intact)	II
156	0.56	33.09	5.62	617	Rock shale (intact)	III
157	0.83	14.44	7.32	987	Granite (intact)	IV

.

2.3. Data Visualization Analysis

The rockburst sample data are visually analyzed via three-dimensional spatial distribution diagrams and boxplots. The data distribution of each sample in the sample dataset is shown in Figure 2. The sample data of non-rockburst grade and light rockburst grade have a few discrete values. Specifically, the sample data of non-rockburst grade are relatively evenly distributed within the interval, whereas the samples of light rockburst grade have a clustered distribution. The samples of moderate rockburst grade and intense rockburst grade contain a small number of discrete values. Specifically, the samples of moderate rockburst grade are dispersed aggregated, whereas the samples of intense rockburst grade are relatively scattered. From the overall perspective of the rockburst sample dataset, the data distribution of each rockburst grade has relatively obvious dividing boundaries, showing significant clustering distribution characteristics.

The boxplots of each prediction indicator are shown in Figure 3. The SCF has few outliers and relatively obvious classification boundaries; with respect to the W_et, the data interval span of non-rockburst and intense rockburst is large, and the data distribution shows an obvious stratification phenomenon; for B, there are individual large discrete values in the moderate rockburst data. All three prediction indicators show obvious data classification phenomena in the interval distribution: the data of SCF and W_et increase with increasing rockburst grade, whereas the data of B decrease with increasing rockburst grade. This finding indicates that the data distribution of the sample dataset established in this study is reasonable and can be used for rockburst intensity grade prediction.

2.4. Analysis of Relationships

The correlations among the variables in the rockburst sample dataset are visually analyzed via the Pearson correlation coefficient to obtain the linear correlation strength and direction among the three prediction indicators. Figure 4 shows that SCF has a weak negative correlation with B, whereas W_et has almost no correlation with the other variables. Moreover, the absolute values of the correlation coefficients among the three prediction indicators are all less than 0.2, which proves that the three prediction indicators selected in this study are scientific and reasonable for the rockburst intensity grade prediction model.

3. Construction of the MIPSO-RF Prediction Model

3.1. Random Forest

RF is a decision tree algorithm based on ensemble learning theory. Its core advantages include: weakening the impact of randomness and one-sidedness of a single decision tree on the overall result by constructing multiple decision trees, thus reducing the model’s sensitivity to parameters; the training process of individual decision trees is independent of each other, supporting parallel computing, which can fully utilize multicore processors or distributed cluster resources to significantly shorten the training time, which is especially suitable for large-scale dataset; effectively evaluating the importance of each feature value by integrating the results of multiple decision trees; and the model can effectively handle high-dimensional data with good adaptability and relatively strong robustness to noise and overfitting [47]. Its structure diagram is shown in Figure 5, and the classification formula is as follows:

$$f\left(x\right)={\displaystyle \sum _{\mathrm{m}=1}^U{c}_{\mathrm{m}}I}\left(x\in R_{\mathrm{m}}\right)$$

(4)

where f(x) is the model classification value; U is the number of decision tree subsets; c_m is the response mean value in the subset samples; R_m is each subset; and I is the indicator function (takes a value of 1 when, otherwise 0).

3.2. Particle Swarm Optimization

PSO is a classic metaheuristic optimization algorithm that is based on a swarm structure and is widely used in various hyperparameter optimization scenarios. By simulating the cooperative behaviors of biological swarms such as bird flocking for predation and fish schooling for migration, the algorithm abstracts the search space of the problem into the activity area of a “particle swarm”; each candidate solution corresponds to a particle in the swarm, and the algorithm finds the global optimal solution by simulating information sharing and cooperative search among particles. Each individual particle records the optimal solution found during its own search process, and the entire particle swarm synchronously shares the optimal position found by all the particles in the swarm. During the iteration process, each particle dynamically adjusts its movement trajectory by combining its own velocity, position, and fitness, continuously approaching a better solution, and finally achieving the approximation of the global optimal solution [48]. Assuming that there are N particles searching for the optimal solution in a D-dimensional space, the velocity update formula is as follows:

$$v_{id}^{k+1}=w{v}_{id}^k+c_1{r}_1\left(p_{id}^k-x_{id}^k\right)+c_2{r}_2\left(p_d^k-x_{id}^k\right)$$

(5)

The position update formula is as follows:

$$x_{id}^{k+1}=x_{id}^k+v_{id}^{k+1}$$

(6)

where i is the particle serial number (i = 1, 2, 3⋯, N); d is the particle dimension (d = 1, 2, 3⋯, D); k is the number of iterations; ω is the inertia weight; c₁ is the individual learning factor; c₂ is the group learning factor; r₁ and r₂ are random numbers within [0, 1], used to increase the randomness of the search; $v_{id}^k$ is the velocity component of particle i in the d-th dimension during the k-th iteration; $x_{id}^k$ is the displacement component of particle i in the d-th dimension during the k-th iteration; $p_{id}^k$ is the historical optimal position of particle i in the d-th dimension during the k-th iteration (the optimal solution of individual search); and $p_d^k$ is the group historical optimal position of the particle in the d-th dimension during the k-th iteration (the optimal solution of the entire particle swarm). Its structure diagram is shown in Figure 6.

3.3. Optimization Strategy

To address the shortcomings of the PSO, the following optimization strategies are adopted after comprehensive consideration: optimizing the initial state of the population through a one-dimensional compound chaotic map; improving the search ability via dynamic self-adaptive feature weighting, Levy flight, and the Cauchy–Gaussian hybrid mutation mechanism; and using the step length factor dynamic adjustment strategy to perform compensation optimization for the early and late stages of the algorithm iteration.

3.3.1. One-Dimensional Compound Chaotic Map

Chaos theory is a research theory that targets deterministic nonlinear dynamic systems, which generate random-like and unpredictable complex motions due to their sensitive dependence on initial conditions, and its applications have been widely extended to fields such as machine learning and artificial intelligence. Both the Logistic map and Tent map are discrete-time chaotic systems. By taking the output of the Tent map as the input of the Logistic map for series combination, the disadvantages of the Logistic map in uneven distribution in the search space, as well as the defects of the Tent map, such as sensitivity to parameters and easy fall into periodic sequences, are avoided. This combination method can prevent the particle swarm from aggregating during generation and enable particles to be more uniformly distributed in the search space [49,50]. The Logistic chaos mapping formula is as follows:

$$x_{\mathrm{n}+1}=p{x}_{\mathrm{n}}\left(1-x_{\mathrm{n}}\right),p\in \left[0,4\right]$$

(7)

where p is the control parameter; when p ∈ [3.5699, 4], the system transitions from a stable state to a chaotic state; and x_n is a nonperiodic and nonconvergent sequence generated under the chaotic map.

The Tent chaos mapping formula is as follows:

$$x_{\mathrm{t}+1}=\left\{\begin{array}{c}2x_{\mathrm{t}},0\le x_{\mathrm{t}}\le 0.5\\ 2\left(1-x_{\mathrm{t}}\right),0.5\le x_{\mathrm{t}}\le 1\end{array}\right.$$

(8)

where x_t is the Tent map value of the current iteration; x_t+1 is the tent map value of the next iteration.

3.3.2. Dynamic Self-Adaptive Feature Weighting

Dynamic self-adaptive feature weighting is introduced into the particle position update process of the particle swarm. The difference between the current position of the particle and the current global optimal position can reflect the gap between the particle and the global optimal solution. The larger the difference is, the greater the gap between the particle and the global optimal solution; in this case, the inertia weight should take a larger value to enhance the global search ability of the current particle; otherwise, the inertia weight should take a smaller value. Assigning different inertia weights to each particle according to its gap from the global optimal solution can balance global and local searches and improve the search performance of the PSO [51]. Its calculation formula is as follows:

$$S_i^k={\displaystyle \frac{1}{X_{\max }-X_{\min }}}\cdot {\displaystyle \frac{1}{D}}{\displaystyle \sum _{d=1}^D\left|{\mathrm{g}}_d^k-x_{id}^k\right|}$$

(9)

$${\omega }_i^k={\omega }_{max}-\left({\omega }_{max}-{\omega }_{min}\right){\left(S_i^k-1\right)}^2$$

(10)

where $S_i^k$ is the difference between the i-th particle and the global optimal solution at the k-th iteration; ${\omega }_i^k$ is the inertia weight of the i-th particle at the k-th iteration; D is the spatial dimension; ω_max and ω_min are the maximum and minimum values of the inertia weight, respectively; $g_d^k$ is the position value of the i-th particle in the d-th dimension within the spatial D at the k-th iteration; and $x_{id}^k$ is the global optimal position value in the d-th dimension within the spatial at the k-th iteration.

3.3.3. Levy Flight

Levy flight is a special random walk strategy proposed by Paul Levy, which introduces the concept of Brownian motion into the non-Gaussian random distribution step size. As a special random walk model, Levy flight can assist particles in searching for targets in different environments through its step size mechanism with a long-tailed distribution. This mechanism ensures the comprehensive search of the particle’s surrounding environment (local search) through short-distance jumps, and reaches another area for large-scale search through longer-distance walks, which can significantly enhance the global search ability of the PSO and help the algorithm jump out of local optimal traps [52,53]. The particle position update formula based on Levy flight is as follows:

$$X_n\left(t+1\right)=X_n\left(t\right)-\alpha \oplus Levy\left(\beta \right)$$

(11)

where X_n(t) is the particle position of the previous iteration; α is the random number for the particle position; $\oplus$ denotes elementwise multiplication; Levy(β) represents the random search path; and β is the characteristic exponent, whose expression is as follows:

$$Levy\left(\beta \right)\sim 0.01{\displaystyle \frac{u}{{\left|v\right|}^{\frac{1}{\beta }}}}\left(X_n\left(t\right)-X_{nbest}\right), \beta \in \left(1,3\right)$$

(12)

$$Levy\left(\beta \right)\sim {\displaystyle \frac{u\sigma }{{\left|v\right|}^{\frac{1}{\beta }}}}, \beta \in \left(1,3\right)$$

(13)

where X_nbest is the optimal solution position in the particle’s motion trajectory; u and v are random variables following a normal distribution, whose expressions are as follows:

$$u\sim N\left(0,{\sigma }_u^2\right);v\sim N\left(0,{\sigma }_v^2\right)$$

(14)

$${\sigma }_u={\left[{\displaystyle \frac{\mathrm{\Gamma }\left(1+\beta \right)\sin \left(\frac{\pi \beta }{2}\right)}{\beta \mathrm{\Gamma }\left(\frac{1+\beta }{2}\right)2^{\frac{\beta -1}{2}}}}\right]}^{{\displaystyle \frac{1}{\beta }}},{\sigma }_v=1$$

(15)

3.3.4. Cauchy–Gaussian Hybrid Mutation Mechanism

To address the problems that the PSO is prone to falling into local optima and has low convergence accuracy in the late iteration stage, a Cauchy–Gaussian hybrid mutation mechanism dynamically adjusted with the number of iterations is adopted. Cauchy mutation can help algorithm jump out of local optimal traps and expand the search range, but the mutation step size of this operator is relatively large, making it easy to miss the optimal position of the particle. Gaussian mutation can narrow the search range and search for the optimal solution in the local area, but it is difficult to search for the global optimal solution. Therefore, Cauchy mutation and Gaussian mutation are combined: in the early stage of the PSO process, the weight of the Cauchy mutation operator is increased to search for possible optimal solutions in the global range with a large step size; in the late iteration stage, the weight of the Gaussian mutation operator is increased to accurately search for the optimal solution in the local area with a small step size. The Cauchy Gaussian hybrid mutation mechanism performs an adaptive mutation operation on the top S particles with the best fitness in the particle swarm, and the selected particles enter the next iteration [54]. Its expression is as follows:

$$u_{k,j}\left(t+1\right)=x_{k,j}^{\ast }\left(t\right)\left[1+{\lambda }_{1}cauchy\left(0,{\sigma }^2\right)+{\lambda }_{2}gauss\left(0,{\sigma }^2\right)\right]$$

(16)

where u_k,j (t + 1) is the position after mutation; x^*_k,j is the position of the k-th optimal individual in the j-th dimensional space at the t-th generation, k = (1, 2, ⋯, S); S is the total number of particles in the particle swarm; cauchy (0,σ²) is a random variable following the Cauchy distribution; gauss (0,σ²) is a random variable following the Gaussian distribution; λ₁ = 1 − (t⁵/T⁵), λ₂ = t⁵/T⁵ are the dynamic adjustment parameters of Cauchy mutation and Gaussian mutation, respectively; and the value of the standard deviation of σ is as follows:

$$\sigma =\left\{\begin{array}{cc}1& f\left(X_k^{\ast }\right)<f\left(X_l^{\ast }\right),k,l\in [1,S],k\ne l\\ exp\left({\displaystyle \frac{f\left(X_k^{\ast }\right)-f\left(X_l^{\ast }\right)}{\left|f\left(X_k^{\ast }\right)\right|+\eta }}\right)& otherwise\end{array}\right.$$

(17)

where f(x) is the fitness function of individual particles in the particle swarm; η is an extremely small positive number, which is used to avoid a zero denominator.

3.3.5. Step Length Factor Dynamic Adjustment Strategy

In the PSO, the selection of the iteration step length affects the convergence speed and solution accuracy of the algorithm. When the step length is too large, although the global search ability is enhanced, it is easy to miss the optimal value; when the step length is too small, although the local search ability is improved, the algorithm will fall into local optimal traps, resulting in reduced convergence speed and decreased solution accuracy [55,56]. To address this, a dynamic step length factor is introduced to dynamically adjust the search step length, and its calculation formula is as follows:

$$w\left(n\right) =w_0\cdot e^{-20\left({\displaystyle \frac{n}{M}}\right) a},a\in [1,2,3\dots 30]$$

(18)

$$Ran{d}_{value}=w\left(n\right)\cdot \left(2\cdot rand-1\right),rand\in [0,1]$$

(19)

where Rand_value is the random step size; w(n) is the dynamic step size adjustment factor; n is the current number of iterations; M is the maximum number of iterations; a is the adjustment parameter; rand is the random function. The expressions for the direction and distance of the particles during the optimization process are as follows:

$$\left\{\begin{array}{c}{X}_i=X+w\left(n\right)\cdot \left(2rand-1\right)\\ Y_i=Y+w\left(n\right)\cdot \left(2rand-1\right)\end{array}\right.$$

(20)

where (X_i,Y_i) is the randomly initialized particle swarm position.

During the iteration process, the step size factor gradually increases in the early stage, and the global optimization ability of the particles is stronger than the local optimization ability at this time. Moreover, the iteration parameter is controlled to be maintained at a small value to prevent the algorithm from falling into local optimal solutions. In the late iteration stage, the global optimal solution has attracted all the particles, but the global space has not been fully explored, which is prone to premature convergence. At this time, the control parameter is kept at a large value to enhance the global search ability and enable particles to explore the global space as fully as possible. In summary, the step length factor dynamic adjustment strategy applied in the iteration process can enhance the global search ability in the early iteration stage and accelerate the convergence speed in the late iteration stage.

3.4. Algorithm Validation

To verify the optimization performance of the MIPSO, six different benchmark test functions were employed for iterative testing, and comparisons were made with the PSO, WOA, and SSA, where the population size and number of iterations for all algorithms were uniformly set to 30 and 1000, respectively. The benchmark test functions utilized are presented in

Table 4. Benchmark test functions.

Number	Function Types	Benchmark Test Functions	Range
F1	unimodal function	$F_1\left(x\right)={\displaystyle {\sum }_{i=1}^n{x}_i^2}$	[−100, 100]
F2	unimodal function	$F_2\left(x\right)={\displaystyle {\sum }_{i=1}^n\left|x_i\right|+{\displaystyle {\prod }_{i=1}^n\left|x_i\right|}}$	[−10, 10]
F3	unimodal function	$F_3\left(x\right)={\displaystyle {\sum }_{i=1}^n{\left({\displaystyle {\sum }_{j=1}^i{x}_j}\right)}^2}$	[−100, 100]
F4	unimodal function	$F_4\left(x\right)={\max }_i\left\{\left|x_i\right|,1\le i\le n\right\}$	[−100, 100]
F5	multimodal function	$F_5\left(x\right)={\displaystyle {\sum }_{i=1}^n\left[x_i^2-10\cos \left(2\pi x_i\right)+10\right]}$	[−5.12, 5.12]
F6	multimodal function	$F_6\left(x\right)={\displaystyle \frac{1}{4000}}{\displaystyle {\sum }_{i=1}^n{x}_i^2-{\displaystyle {\prod }_{i=1}^n\cos \left(\frac{x_i}{\sqrt{i}}\right)+1}}$	[−600, 600]

.

The three-dimensional spatial schematic diagrams of each function are presented in Figure 7.

The performance indicators of each algorithm on different test functions, including the mean value (Mean) and standard deviation (Std), were statistically summarized in

Table 5. Test Results of Benchmark Test Functions.

Function	Index	MIPSO	PSO	SAA	WOA
F1	Mean	4.87 × 10−196	9.10 × 101	1.91 × 101	1.32 × 10−11
	Std	0.00 × 100	3.77 × 101	3.29 × 100	1.04 × 10−11
F2	Mean	5.20 × 10−101	3.17 × 100	1.93 × 101	1.59 × 10−5
	Std	1.56 × 10−100	3.16 × 100	1.22 × 100	6.03 × 10−6
F3	Mean	1.71 × 10−161	3.63 × 103	5.67 × 101	8.14 × 102
	Std	5.01 × 10−161	1.03 × 103	2.19 × 101	1.96 × 103
F4	Mean	1.45 × 10−86	1.26 × 101	1.72 × 100	3.16 × 10−3
	Std	1.97 × 10−86	1.69 × 100	1.82 × 10−1	3.06 × 10−3
F5	Mean	0.00 × 100	8.16 × 101	2.46 × 102	1.79 × 10−9
	Std	0.00 × 100	1.65 × 101	3.36 × 101	1.52 × 10−9
F6	Mean	0.00 × 100	1.63 × 100	5.96 × 10−1	1.26 × 10−12
	Std	0.00 × 100	2.63 × 10−1	6.18 × 10−2	9.45 × 10−13

based on 30 repeated experiments.

In the performance evaluation of the above six benchmark test functions, the MIPSO algorithm is proven to exhibit superior performance and search optimization capability on unimodal and multimodal functions compared with other algorithms. And the curves of the number of iterations are presented in Figure 8.

Overall, the MIPSO algorithm is demonstrated to exhibit excellent performance across all categories of test functions, with particularly outstanding performance in unimodal and multimodal functions. It not only demonstrates fast convergence speed and excellent robustness but also verifies the advantage of possessing both exploration and exploitation capabilities in complex search spaces.

3.5. Prediction Framework Construction

The construction process of the intelligent rockburst intensity grade prediction framework is shown in Figure 9. First, a dataset is established, and the SCF, B, and W_et are selected as the model prediction indicators. The data samples in the rockburst sample dataset are divided into a training set and a test set at a ratio of 7:3, and the consistency of the data distributions between the training set and the test set is ensured to avoid additional deviations. The one-dimensional composite chaotic map is subsequently adopted to optimize the initial state of the population in the PSO, improving the diversity and randomness of the initial population; the dynamic self-adaptive feature weighting, Levy flight, and Cauchy–Gaussian hybrid mutation mechanism are subsequently used to optimize the search ability of the algorithm, avoiding the algorithm falling into local optimal traps, premature convergence, and excessively fast convergence speed; and the step length factor dynamic adjustment strategy is utilized to improve the step size in the early and late iterations of the algorithm, enhancing the overall solution accuracy of the algorithm. The MIPSO is combined with the RF to construct the MIPSO-RF prediction model. Five-fold cross-validation is subsequently adopted to verify the generalization ability of the model. The test set data are input into the rockburst intensity prediction model; the model performance is evaluated in terms of accuracy, F1-score, kappa, precision, recall and AUC; and the average values of each group of model evaluation indicators are calculated for comparison. The SHAP interpretability analysis method is adopted to evaluate the contribution of each prediction indicator through feature importance analysis, feature impact distribution analysis, and feature dependency analysis, further verifying the scientific validity of the prediction indicators selected in this study. Finally, on the basis of the MIPSO-RF constructed in this study, application software is developed to form a complete intelligent rockburst intensity grade prediction framework, and a gently dipping rock strata in Southwest China is taken as an engineering example to carry out engineering application verification of the intelligent prediction framework.

4. Performance Testing

4.1. Test of Predictors

Five-fold cross-validation is used to verify the generalization ability of the models, and the specific process is shown in Figure 10. On the basis of the established rockburst dataset, the test set data are equally divided into five groups, which are sequentially input into the MIPSO-RF, PSO-RF, SSA-RF, WOA-RF, and RF for comparative analysis. To ensure the fairness of the model performance evaluation, the experiment is conducted on a computer with the hardware configuration: an i7-14650HX CPU, an RTX 4060 laptop GPU (8 GB), and 16 GB of memory. The software environment is built on Python 3.12 for rockburst intensity grade prediction. The experimental parameters are set as follows: population size of 50 and 42 iterations. The five groups of data are alternately trained and verified to obtain the corresponding model evaluation indicators for each group, and finally, the average values of the evaluation indicators of all the models are calculated, with the results shown in

Table 6. Model performance evaluation results.

Model	Accuracy	F1-Score	Precision	Recall	Kappa	AUC
RF	70.83%	0.7070	71.98%	72.36%	0.6118	0.9397
WOA-RF	81.25%	0.7903	79.58%	79.66%	0.7472	0.9450
SSA-RF	81.25%	0.7952	79.58%	79.83%	0.7479	0.9418
PSO-RF	83.33%	0.8079	83.05%	82.61%	0.7749	0.9399
MIPSO-RF	95.83%	0.9574	96.15%	95.83%	0.9444	0.9736

.

The test sets of each prediction model are used for rockburst intensity prediction under the same computational environment, and confusion matrix diagrams of the test sets of each model are obtained, as shown in Figure 11. The MIPSO-RF achieved a high accuracy of 95.83%, demonstrating superior performance over other benchmark models.

The performance evaluation results of each model are organized and plotted into a radar chart to compare the performance of the five models, as shown in Figure 12. All six evaluation indicators of the MIPSO-RF are higher than those of the other four prediction models, indicating that the MIPSO has a stronger global search ability, is less likely to fall into local optimality, and has excellent generalization ability.

The receiver operating characteristic (ROC) curves of each prediction model are shown in Figure 13. The ROC curve of the MIPSO-RF model can achieve a high true positive rate (TPR) even at a low false positive rate (FPR). Compared with the other four models, it exhibits an earlier attainment of the maximum TPR, as evidenced by a larger area under the curve with an AUC value of 0.9736. This demonstrates that the MIPSO-RF model possesses high classification efficiency and the ability to accurately identify true positive results, thereby exhibiting excellent performance and outstanding generalization ability. Additionally, it performs well in both global search and local search processes.

4.2. Shapley Interpretability Analysis

4.2.1. Feature Importance Analysis

The SHAP method is used to evaluate the contributions of the three prediction indicators of the MIPSO-RF rockburst intensity prediction model. Feature importance analysis determines the degree of importance by calculating the absolute value of the average Shapley value of each feature variable in the model—higher Shapley values indicate that the feature has a more significant impact on the prediction results [57]. As shown in Figure 14, the SCF has the highest Shapley value and the most prominent feature importance, followed by W_et and B. These three prediction indicators have all demonstrated a certain degree of contribution in the prediction of four rockburst intensity grades through the model, so their application in rockburst intensity prediction is scientific.

4.2.2. Feature Impact Distribution Analysis

To clarify the contribution characteristics of different prediction indicators to rockburst intensity grade prediction, an analysis is conducted via feature scatter density diagrams, with the results shown in Figure 15. The abscissa in the diagram represents the Shapley value of each sample: points on the positive semiaxis indicate a positive contribution to the prediction results, whereas points on the negative semiaxis indicate a negative contribution. The ordinate represents the feature value of each sample: the redder the color is, the larger the feature value; the bluer the color is, the smaller the feature value; and the purple color indicates an intermediate feature value. By combining the distribution position of the data points and the trend of color change in the diagram, the positive and negative impacts of each prediction indicator’s value on the prediction results can be determined.

As shown in Figure 15, overall, SCF has the most significant impact on the prediction results, with its smaller feature values showing a negative contribution; the impact of the W_et is secondary to that of SCF, with smaller feature values presenting a negative contribution and larger feature values presenting a positive contribution; B has the smallest contribution, and its larger feature values show a negative contribution.

4.2.3. Feature Dependency Analysis

To better understand the contributions of different feature values of the three prediction indicators to rockburst intensity grade prediction, feature dependence plots are used for analysis [58]. Each data point in the plot represents a rockburst sample: the abscissa is the feature value of the sample, and the ordinate is the Shapley value. Rockburst samples above the 0-axis of the ordinate positively contribute to the prediction results, and the feature dependence plots of some rockburst intensity grade prediction indicators are shown in Figure 16.

Figure 16 shows that under different rockburst intensity grades, the data points of each prediction indicator with Shapley values greater than 0 have a significantly clustered distribution in the feature interval. Specifically, the clustering phenomenon of the SCF data points is the most prominent, followed by that of the W_et data points, and the clustered distribution feature of the B data points is the least obvious.

4.3. Application Software Development

To avoid the catastrophic consequences caused by rockbursts and ensure the safety of the lives and properties of underground operators, it is highly practical to propose an economical and efficient rockburst prediction method. The dataset training of existing rockburst prediction models mostly covers various working conditions, such as tunnels, hydraulic tunnels, traffic tunnels, deep underground roadways, and hydropower stations. However, there are relatively few special studies carried out on the special working conditions of gently dipping rock strata, resulting in poor prediction performance in such rock strata. In addition, most of these models only remain in the training stage and are difficult to apply effectively for prediction under actual working conditions. Therefore, this study proposes a rockburst prediction model framework that considers the working conditions of gently dipping rock strata to fill the relevant research gap. As shown in Figure 17, the software is based on the MIPSO-RF with a modular design, accepts either direct data input or Excel files containing key rock mechanics parameters (SCF, W_et, B), thereby enabling rapid determination of rockburst intensity grade. The software integrates user-friendly operability, excellent performance, and intuitive result presentation: it features a streamlined interface for accessible operation, delivers efficient and stable computational performance, and presents outputs via clear visualizations to facilitate quick interpretation, and is suitable for engineering projects operating in gently dipping rock strata.

5. Engineering Examples

Structural planes in natural rock masses have a significant effect on the mechanical strength of rock masses. In underground engineering, roadway excavation and mining operations destroy the initial stress balance. To achieve a new equilibrium state, the rock surrounding the roadway will deform or be damaged under the action of stress. After being affected by stress, gently dipping rock strata are prone to deformation along structural planes and stress concentration, which in turn leads to the expansion of rock mass cracks and rock block detachment. At this time, the elastic potential energy accumulated in the rock mass is suddenly released, triggering typical rockburst dynamic disasters such as rock block ejection and spalling. To verify the rockburst intensity prediction framework constructed in this study, a mine in Southwest China is selected for engineering application verification. The data obtained from laboratory rock mechanics tests and numerical simulations are input into the application software developed in this study to test the engineering practicality of the framework.

5.1. Project Background

The case mine is a lead-zinc mine in Southwest China, with terrain sloping from south to north and west to east, located at the southern end of a north–south-trending seismic belt. The orebody is entirely composed of gently dipping limestone formations with a maximum mining elevation of 1630 m (burial depth of 193 m) and a minimum mining elevation of 790 m (burial depth of 1033 m). The mine is divided into levels every 60 m, and FLAC3D is used to construct the orebody structure diagram, as shown in Figure 18.

With increasing roadway burial depth, the number of rockburst disasters caused by stress concentrations in the operation area becomes increasingly severe. Different degrees of rockburst disasters have occurred in multiple existing middle sections of the mine, posing a significant threat to safe production. As shown in Figure 19, affected by the characteristics of gently dipping rock strata, a typical rockburst dynamic phenomenon of surrounding rock detachment and ejection along structural planes occurred at this level, resulting in delays in production plans and losses of personnel and property.

5.2. Rock Burst Prediction Indicator Parameter Acquisition

5.2.1. Laboratory Rock Mechanics Tests

Combined with the actual situation of the mine, levels with sampling conditions were selected for onsite sampling to determine the physical and mechanical parameters of the rocks. It was established during onsite investigations that the 970 m level had not yet been developed and temporarily lacked sampling conditions. The lithology of the collected rock samples is limestone. After coring and grinding, standard rock samples were prepared and grouped. The average density of the rock samples was obtained through weighing with a horizontal electronic balance and measuring with a vernier caliper, which was 2.68 g/cm³, and some samples are shown in Figure 20.

The uniaxial compressive strength test, Brazilian splitting test, and uniaxial one-time loading–unloading test were conducted via a YAW4206T microcomputer-controlled electrohydraulic servo pressure testing machine to obtain the required rock mechanical parameters. Figure 21 shows the test process of some samples and photographs of the samples after failure.

5.2.2. Numerical Simulation of the Maximum Tangential Stress

Owing to the complex geological conditions of the mine, to maximize the restoration of the geological stratum conditions where the mine is located, geological modeling was carried out via FLAC3D 7.0 software, which is based on the Mohr–Coulomb model, and relies on the results of rock mechanics tests and the geological data provided by the mine, to restore the underground orebody as accurately as possible. A three-dimensional model with a length of 3500 m, width of 3000 m, and bottom elevation of 500 m was established. Initial stress calculations were performed to obtain the stress distribution characteristics under the undisturbed state without excavation, as shown in Figure 22.

According to the current mining conditions of the mine, the mining of ore bodies at and above the 970 m level requires four mining cycles. The maximum tangential stress parameters of each level were selected as the maximum tangential stress values of the surrounding rock after the completion of each mining cycle, and the specific mining sequence is shown in

Table 7. Mining cycle table.

First Cycle	Second Cycle	Third Cycle	Fourth Cycle
1330–1390 level	1450–1510 level 1150–1270 levels	1570–1630 level 1030–1090 level	970–1030 level

.

Numerical simulation of the filling mining method was carried out using FLAC3D software. The maximum tangential stress distribution on horizontal sections at key mined levels was obtained, as shown in Figure 23.

The maximum tangential stress after each mining cycle was extracted to obtain the maximum value of the maximum tangential stress of each level during the entire mining process and the average value of the rock mechanics test results. The sorted data are shown in

Table 8. Statistical table of rock parameters.

Cycle Division	Level/m	σc/MPa	σt/MPa	Wet	σθ/MPa
Third cycle	1630	30.45	2.80	1.17	4.2
	1570	36.29	2.65	2.28	5.84
Second cycle	1510	44.35	2.74	1.47	6.21
	1450	50.40	2.49	1.93	11.11
First cycle	1390	52.48	2.12	2.24	12.55
	1330	46.69	2.55	2.64	10.26
Second cycle	1270	48.26	2.34	2.31	8.24
	1210	50.71	2.34	3.39	11.67
	1150	60.36	2.53	3.49	11.4
Third cycle	1090	60.73	2.96	3.21	8.54
	1030	64.84	3.14	3.92	13.70
Fourth cycle	970	68.17	3.26	3.52	19.63

.

5.3. Engineering Applications

The rockburst intensity grade evaluation indicators were calculated from the test results in Table 5. The application software developed based on the model in this study was used to predict the rockburst intensity grades of 12 levels of the mine, and the predictions were compared with the actual results, as shown in

Table 9. Application software prediction results.

Cycle Division	Level/m	SCF	B	Wet	Truth	Prediction
Third cycle	1630	0.14	10.88	1.17	I	I
	1570	0.16	13.69	2.28	I	I
Second cycle	1510	0.14	16.19	1.47	I	I
	1450	0.22	20.24	1.93	II	II
First cycle	1390	0.24	22.40	2.24	II	II
	1330	0.22	18.31	2.64	II	II
	1270	0.17	20.62	2.31	II	II
Second cycle	1210	0.23	21.67	3.39	III	III
	1150	0.19	23.86	3.1	II	II
Third cycle	1090	0.14	20.52	3.21	II	II
	1030	0.21	20.65	2.85	II	II
Fourth cycle	970	0.29	20.91	3.52	III	III

. The predicted results are completely consistent with the actual situation, verifying the reliability and applicability of the MIPSO-RF for rockburst intensity grade prediction in gently dipping rock strata.

6. Conclusions and Future Plans

6.1. Conclusions

To address rockburst disasters in gently dipping rock strata, this study constructs an intelligent prediction framework for rockburst intensity based on MIPSO-RF, which is applied to the engineering practice of a lead-zinc mine in Southwest China, achieving phased results. The specific research contents are as follows:

1. Three prediction indicators—SCF, W_et and B were selected to construct the dataset. Three-dimensional spatial distribution diagrams were drawn, and data analysis was performed via boxplots and correlation heatmaps. The results show that the selected prediction indicators have low correlation, and the dataset data are highly reasonable with few discrete values, indicating significant clustered distribution characteristics and clear grading phenomena.

2. We developed a rockburst intensity grade prediction framework based on an MIPSO-RF. The MIPSO was enhanced through several strategies, including a one-dimensional compound chaotic map, dynamic self-adaptive weighting, Lévy flight, a Cauchy–Gaussian hybrid mutation mechanism, and a dynamic adjustment strategy for the step length factor. This improved MIPSO was then integrated with the RF to form the final model.

3. The performance of the MIPSO-RF constructed in this study was evaluated via four evaluation indicators—accuracy, F1-score, kappa, and AUC—combined with confusion matrix diagrams, ROC curves, and SHAP interpretability analysis. The results show that the MIPSO has a stronger global search ability, a lower probability of falling into local optimal traps, and more easily achieves global optimization, which verifies the scientific validity and rationality of the selected prediction indicators.

4. Application software development was carried out on the basis of the MIPSO-RF constructed in this study, forming a complete intelligent rockburst intensity grade prediction framework. This framework provides a fast and portable rockburst prediction tool for onsite engineers, with the following application process: through the collection of rock samples and the construction of geological models of gently dipping rock strata, combined with laboratory rock mechanics tests and numerical simulation calculations, the rockburst intensity grade prediction indicators of each middle section were obtained through sorting and analysis, which were imported into the software for prediction. The predicted results are completely consistent with the actual situation, fully verifying the engineering practicality of the rockburst prediction framework.

5. Based on the geological conditions of the mine, studies have indicated that the rockburst prediction framework established in this paper exhibits favorable prediction performance for gently dipping limestone strata with a burial depth of 150–970 m, and it also demonstrates good applicability to the currently widely adopted filling mining method.

6.2. Future Plans

1. In the future, the existing dataset is planned to be developed to include as many working conditions as possible. To further advance the generalization capability of the proposed intelligent prediction framework for rockburst intensity grades, this study recommends extending its application to a wider spectrum of special operating conditions. Concurrently, the integration of deep learning methodologies is used to further enhance the intelligence and adaptive capacity of the rockburst intensity grades prediction model.

2. Prediction software will be further developed to enable it to perform rockburst prediction for different geological conditions and conduct prediction for combinations of multiple geological conditions in practical applications.

For future model expansion, the database is intended to be further expanded to enhance the generalization ability of the model. Meanwhile, data in the database will be labeled to clarify the specific geological conditions corresponding to each dataset. In addition, software improvement will be implemented to incorporate the option of selecting specific geological conditions, such as steeply dipping rock strata and strata with high geostress and deep burial depth, enabling accurate prediction for these scenarios. This improvement is expected to ensure that the model maintains excellent generalization ability and applicability under various complex geological conditions.

3. Further refinement of the validation procedures and error reporting mechanisms for the developed application software is recommended to enhance its credibility and reliability.