A Coalmine Water Inrush Prediction Model Based on Artificial Intelligence

China boasts a wealth of coal resources. All varieties of coal have been detected in mines across the country. The supply of coal, the leading primary energy source, directly affects industrial development and even social stability [1-6]. Many coalmines in China face complex hydrogeological conditions. If a water inrush occurs, the roadway will be quickly flooded if the accident is not handled properly, posing a serious threat to the lives of workers and the safety of coalmine properties [7-9]. Therefore, the prevention of water inrush is recognized as the key to the work safety in coalmines. An effective way to prevent water inrush is to build a water inrush safety model for the target coalmine [10-12].

The causes of water inrush vary from mine to mine, because each coalmine has unique hydrogeological and storage conditions. Considering the variation, many scholars have explored the mechanism of coalmine water inrush, constructed evaluation index system of coalmine water inrush safety, and introduced artificial intelligence (AI) into the analysis and prediction of coalmine water inrush [13-16].

From the perspective of physics, Winkler [17] considered the defect and local instability of the rock floor as the main causes of coalmine water inrush, and proposed to judge the rock floor instability by detecting whether the stress of the water barrier is imbalanced. Kumari and Om [18] combined mathematics with structural mechanics, and systematically analyzed the causes of water inrush fault in coalmines, based on catastrophe theory and limit bending moment theory. Based on some coalmine water inrush samples, Henriques and Malekian [19] discretized the continuous water inrush prediction information with support vector machine (SVM), and solved the local minimum problem of traditional neural networks (NNs). Zhang et al. [20] combined the genetic algorithm (GA) with backpropagation (BP) algorithm into a coalmine water inrush prediction model based on artificial neural network (ANN); the model has a high training accuracy, but consumes too much time in adjusting the network parameters.

The prediction of coalmine water inrush must be rapid, accurate, and effective. Hence, it is important to improve the operating speed of the prediction model, while ensuring the prediction accuracy [21-23]. Bhattacharjee et al. [24] developed a hybrid prediction model based on extreme learning machine (ELM) and the principal component analysis (PCA), and proved that the hybrid model is more accurate and faster than the least squares (LS) SVM and traditional BP model. Liu et al. [25] constructed a real-time monitoring system for water inrush from coalmine floor, and optimized the input weights and hidden layer bias, which are assigned randomly by the ELM, with the adaptive difference algorithm.

Traditionally, the evaluation index systems for coalmine water inrush overlook the fact that the causes of water inrush might change in real time with external forces in coalmining. Moreover, the existing ELM-based prediction algorithms cannot achieve desirable classification accuracy or generalization, despite their rapid prediction. To solve these problems, this paper develops an AI-based coalmine water inrush safety prediction model, making coalmine water inrush prediction more accurate, real-time, and robust. Firstly, the causes of coalmine water inrush were combed, and a reasonable evaluation index system was established. Next, the feasibility of optimizing the ELM with particle swarm optimization (PSO) algorithm and ant colony optimization (ACO) algorithm was demonstrated, and the coalmine water inrush safety prediction model was created based on the ELM optimized by the PSO and ACO. The dimensionality reduction in subset classification was introduced in great details. Finally, the effectiveness and real-time performance of our model were proved through experiments.

3.1 The ELM algorithm

The ELM algorithm is a learning algorithm based on a feedforward NN with a single hidden layer (Figure 2). The most notable advantage of the algorithm is its fast speed, which arises from the fact that the algorithm can find the unique optimal solution from the generalized reverse by adjusting the number of nodes, without needing to change the input weights or biases of the hidden layer. The structure of the feedforward NN with a single hidden layer is shown in Figure 2 below.

For a dataset of N random samples, suppose the input matrix is X_i=[x_i1, x_i2, …, x_in]^T$\in$R^n×N, where ω_i=[ω_i1, ω_i2, …, ω_in] is the connection weight between input layer and hidden layer, β_i is the connection weight between hidden layer and output layer, b_i is the bias of the i-th hidden layer node, and g(x) be the activation function of the hidden layer. Then, the j-th element in the output matrix T of the ELM network with M hidden layer nodes can be expressed as:

${{t}_{j}}=\sum\limits_{i=1}^{M}{{{\beta }_{i}}g\left( {{\omega }_{i}}{{X}_{i}}+{{b}_{i}} \right)}$          (1)

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Figure 2. The structure of the feedforward NN with a single hidden layer

If the activation function g(x) is infinitely differentiable, ω_i and b_i can remain unchanged during the training after being randomly initialized. The value of β_i can be obtained by solving the following system of linear equations:

$\left\| H\left( {{{\hat{\omega }}}_{i}},{{{\hat{b}}}_{i}} \right){{{\hat{\beta }}}_{i}}-{{T}^{T}} \right\|=\min \left\| H\left( {{\omega }_{i}},{{b}_{i}} \right){{{\hat{\beta }}}_{i}}-{{T}^{T}} \right\|$          (2)

where, H is the output matrix of the hidden layer; T^T is the transposed matrix of T. The above formula is equivalent to a minimal loss function:

$E=\frac{1}{2}{{\sum\limits_{j=1}^{P}{\left( \sum\limits_{i=1}^{M}{{{\beta }_{i}}g\left( {{\omega }_{i}}{{X}_{i}}+{{b}_{i}} \right)-{{o}_{j}}} \right)}}^{2}}$          (3)

In the traditional gradient descent method, a set of parameter values are randomly initialized, and updated constantly in the solving process, such that the loss function is reduced iteratively to the minimum. In the ELM algorithm, once the input and output values of the objective function are given, the parameters can be viewed as independent variables, while reducing the original value of the objective function. In this way, the loss function can be minimized. Since H remains the same through the training, the generalized inverse matrix of H can be denoted as H^-1. Then, β_i can be uniquely determined as the least squares solution of formula (2):

${{\hat{\beta }}_{i}}={{H}^{\text{-}1}}{{T}^{T}}$          (4)

3.2 Combinatory optimization of the ELM

In coalmines, water inrush may be induced by various complex factors. That is why a dozen of secondary indices were selected in our evaluation index system. The more the variables used in prediction, the higher the dimensionality of the ELM parameters. However, the traditional way to explore the correlations between influencing factors, that is, updating ELM parameters through gradient descent, could easily fall into the local minimum trap, and even leads to an incorrect prediction model. Here, the classification performance and running speed are optimized by the PSO algorithm, drawing on the metaheuristic search method. The workflow of the PSO-optimization of the ELM algorithm is explained in Figure 3.

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Let P=(P₁, P₂, …, P_n) be a particle swarm in a D-dimensional space, and P_i=(p_i1, p_i2, …, p_iD) and V_i=(v_i1, v_i2, …, v_iD) be the position and velocity of the i-th particle, respectively. The particles in the swarm iteratively share information with each other, in search of the optimal solution. During the iteration, the i-th particle updates its position and velocity constantly based on the best-known individual position pBest, which is expressed as a matrix P_i=(p_i1, p_i2, …, p_iD), and the best-known global position gBest, which is expressed as another matrix P_g=(p_g1, p_g2, …, p_gD):

$V_{id}^{t+1}=\delta V_{id}^{t}+{{a}_{1}}{{\varepsilon }_{1}}\left( P_{id}^{t}-X_{id}^{t} \right)+{{a}_{2}}{{\varepsilon }_{2}}\left( P_{gd}^{t}-X_{id}^{t} \right)$          (5)

$p_{id}^{t+1}=p_{id}^{t}+p_{id}^{t+1}$          (6)

where, d=1, 2, …, D is the dimension of the optimal solution obtained through iterations; t is the number of iterations needed to find the optimal solution; δ is the inertia weight that balances the individual extreme with the global extreme by adjusting the search range; ε₁ and ε₂ are two random numbers in [0, 1] designed to increase the randomness of the search; a₁ and a₂ are two acceleration factors that balances the individual extreme with the global extreme by adjusting the number of iterations adaptively in real time. To make the iterative search more pertinent, the particle position and velocity were limited within [-P_max, P_max] and [-V_max, V_max], respectively. The acceleration factors a₁ and a₂ can be respectively updated by:

${{a}_{1}}=\left( {{a}_{1h}}-{{a}_{1k}} \right)\frac{t}{{{t}_{\max }}}+{{a}_{1k}}$           (7)

${{a}_{2}}=\left( {{a}_{2h}}-{{a}_{2k}} \right)\frac{t}{{{t}_{\max }}}+{{c}_{2k}}$          (8)

where, a_1h, a_2h, a_1h, and a_2h are constants; t is the current number of iterations; t_max is the maximum number of iterations.

During the feature extraction of coalmine water inrush, most evaluation indies can be characterized by Yes or No. The eigenvalues of such indices can be defined as 0 or 1. In this case, the particle velocity can be updated by formula (5), while the particle position can be updated based on the probability that the velocity is mapped to 1 in the interval of [0, 1]. In this way, each evaluation index can be represented by continuous numerical values. Then, corresponding particle velocity can be updated by the sigmoid function:

$\text{sig}V_{id}^{t+1}=\frac{1}{1+{{e}^{-V_{id}^{t+1}}}}$           (9)

The particle position can be updated by:

${{P}_{id}}=\left\{ \begin{align}  & 1,if\text{ }\lambda <sig\left( {{V}_{id}} \right) \\  & 0,if\text{ }\lambda \ge sig\left( {{V}_{id}} \right) \\ \end{align} \right.$           (10)

For better computing power and operating efficiency, the ELM algorithm was further improved by the ACO algorithm, which is based on distributed computing. Since the coalmine water inrush prediction involves multiple evaluation indices, the input weights and hidden layer biases were ranked as the nodes of the path traversed by each ant, according to the importance of each evaluation index.

It is assumed that there are R nodes and A ants, and every two adjacent nodes are connected by two paths. Then, an ant faces two path options after arriving at a node: Yes (1) or No (0). Hence, the probability P_i,j for an ant at the i-th node to choose the j-th node can be calculated by:

${{P}_{ij}}=\frac{{{\tau }_{ij}}}{\sum\limits_{i=0}^{R}{{{\tau }_{ij}}}}$          (11)

where, τ_ij pheromone concentration between the i-th node and the j-th node; i=1, 2, …, R; j=0 or 1.

Taking the path of each ant as a feasible solution to coalmine water inrush prediction, the paths of the entire ant colony constitute the solution space of the problem to be optimized. Then, the fitness of the path traversed by an ant, i.e. the absolute error between the ideal output and the real output, can be computed by:

$fitnes{{s}_{ij}}=\frac{1}{R}\sum\limits_{i=1}^{R}{\left| {{I}_{ij}}-{{T}_{ij}} \right|}$           (12)

Then, the obtained solutions are ranked by fitness. The paths corresponding to the optimal solution (optimal paths) are allocated into set O. After that, the path of each ant in the next cycle can be adjusted by formulas (13) and (14), and the pheromone concentration corresponding to the optimal paths, as well as the increment of pheromone concentration, can be updated by the enhancement coefficient ρ:

$\begin{align}  & {{\tau }_{ij}}\left( t+1 \right)={{\tau }_{ij}}\left( t \right)+\rho \times  \\  & \left[ {\left( fitnes{{s}_{\max }}-fi{{t}_{ij}} \right)}/{\left( fitnes{{s}_{\max }}-fitnes{{s}_{\min }} \right)}\; \right] \\ \end{align}$          (13)

$\Delta {{\tau }_{ij}}=\left\{ \begin{align}  & fitnes{{s}_{ij}}/\left( \mu \times A \right),If\text{ }the\text{ }path\in O \\  & 0,\text{                        }If\text{ }the\text{ }path\notin O \\ \end{align} \right.$          (14)

where, fitness_max and fitness_min are the maximum fitness and minimum fitness, respectively; μ is the proportion of the ants traversing the optimal paths in the colony. The iteration is terminated at the maximum number of iterations t_max, and the maximum fitness in the iteration process can be obtained:

$fitness=\max \left( fitnes{{s}_{t-\max }} \right)$          (15)

where, t=1, 2, …, t_max. The paths corresponding to the maximum paths are recorded to obtain the optimal input weights and hidden layer biases of the ELM algorithm.

So far, this paper relies on combinatory optimization to improve the computing power and operating efficiency of the ELM algorithm. If there are many evaluation indices for coalmine water inrush prediction, there will be a huge number of candidate solutions. In this case, the fitness function should be properly optimized to enhance the generalization ability and classification accuracy of the ELM through training.

To reveal the generalization ability and classification accuracy of each sub-classifier, the original dataset was split into multiple subsets, without changing the properties of the classifier. The last subset was taken as the test subset, and the other subsets as the training subsets. For each subset, the L2-norm of the connection weight β between the hidden layer and the output layer in the ELM network was calculated. Then, the fitness function for the w-th subset can be adjusted into:

$fitnes{{s}_{w}}=\frac{Accurac{{y}_{w}}}{{\overline{\left\| {{\beta }_{w}} \right\|}}/{\overline{\left\| {{\beta }_{all}} \right\|}}\;}$          (16)

where, Accuracy_w is the mean classification accuracy of the w-th subset; the denominator is the mean L2-norm ratio between the weight of each subset and the output weights of all subsets of the ELM network. It can be seen that, with the growing mean classification accuracy of each subset, the L2-norm will decrease, and the fitness will increase. The workflow of the ACO-optimization of the ELM algorithm is explained in Figure 4.

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Figure 4. The workflow of the ACO-optimization of the ELM algorithm

To verify its effectiveness, the proposed combinatory optimized ELM prediction model was tested on 150 sets of data collected from the working face in a typical coalmine in Shanxi province, northern China.

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Figure 6. The error change curve of our model

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Figure 7. The comparison of prediction errors

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Figure 8. The comparison of MSEs

Figure 6 provides the error change curve of our model. It can be seen that, with the growing number of iterations, the mean squared error (MSE) of our model gradually decreased from 0.01246 to 0.08432.

Figures 7 and 8 compare the errors and MSEs of our algorithm and those of the traditional ELM algorithm in predicting water inrush of the said coalmine. The comparison shows that the mean prediction error and MSE of the traditional ELM algorithm were 15.56 and 2.85, respectively; those of our algorithm were 11.47 and 0.67, respectively. Hence, the proposed algorithm has better prediction accuracy than the traditional ELM algorithm.

Table 1. The comparison of prediction errors and training durations of different algorithms

Table 1 compares the prediction errors and training durations of our algorithm with the BP algorithm, the GA-BP algorithm, the long short-term memory (LSTM) algorithm, the traditional ELM algorithm, the PSO-ELM algorithm, and the ACO-ELM algorithm.

It is obvious that our algorithm, as a combinatory optimized algorithm, consumed slightly more time than the traditional ELM algorithm, the PSO-ELM algorithm, and the ACO-ELM algorithm in training, but achieved faster speed than BP, GA-BP, and LSTM algorithms.

Figure 9 compares the prediction accuracies before and after the dimensionality reduction in subset division. It is clear that the dimensionality reduction improves the prediction effect of our model by removing the redundant evaluation indices for coalmine water inrush.

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Figure 9. The comparison of the prediction accuracies before and after the dimensionality reduction