An Intelligent Approach to Predicting Dilution, Overbreak and Costs in Underground Mining Using Kolmogorov-Arnold Networks

Drill and blast continue to stand out as one of the most versatile and widely employed techniques for excavation in mining, quarrying, and tunneling in hard rock, thanks to its relatively low cost, efficiency, and ease of implementation [1-3].

However, this method has some disadvantages, such as the destruction of the surrounding rock mass, which can result in unwanted cavities due to blasting, referred to as "overbreak"[4]. Furthermore, in mining, it is confronted with dilution, defined as the percentage of material or waste with economic value removed beyond the delimited area of extraction, whether planned or unplanned. Unplanned dilution not only reduces the quality of the extracted ore but also increases processing and transportation costs, which negatively impacts profits, underlining the need for effective management [5, 6].

Mine production capacity and profitability are directly dependent on ore loss and dilution indicators, which are key factors in assessing the development prospects of a mining company. Therefore, assessing the geological and economic factors associated with ore dilution and loss is critical at the feasibility stage of mine design, as well as during mining operations [7]. The rate of ore dilution varies based on the mining method employed for underground extraction of mineral resources [8-10].

Ensuring integrity and quality in mineral resource utilization is the key driver of annual productivity growth in mining enterprises and the expansion of industrial reserves. However, factors such as mine structural complexity, ore and surrounding rock instability, and adverse hydrogeological conditions negatively impact mining productivity. These factors can lead to an increase in ore dilution of up to 35%-40% and ore loss of up to 25%, highlighting the need for mitigation measures [11].

The problem arises in an underground mine, where traditional approaches are severely constrained, as evidenced by the large differences between predicted values and those observed in the field. Some critical factors behind this problem include errors in drill and blast design, such as incorrect drill hole placement, inappropriate application of explosives, and excessive charge length. In general, empirical estimates tend to be insufficient for properly modeling rock mass behavior because they do not include detailed analyses, such as numerical modeling or in-depth in-situ testing, which negatively affects operational planning and overall profitability.

Such discrepancies lead to uncontrolled and excessive fragmentation of the rock mass, causing overbreak, which leads to dilution and unwanted extraction of waste material, increasing the operational costs associated with clearing, support, loading, and hauling.

Therefore, predicting and mitigating overbreak and dilution during the mining process are considerable challenges for any mining project. To improve these predictions, there have been attempts to incorporate more advanced approaches that employ artificial intelligence along with more sophisticated mathematical modeling techniques [12, 13].

Recent studies have indicated substantial improvements in accuracy. Jorquera et al. [14] used various machine learning (ML) models to estimate dilution, achieving a precision score of 0.835, an accuracy of 0.804, and an area under the curve of 0.942 with the Random Forest model.

On the other hand, Chimunhu et al. [15] employed geological, geotechnical, and mine design variables available in the early stages of sub level open stopping to predict dilution. They achieved 93% accuracy in dilution prediction using the principal component analysis-classification and regression trees (PCA-CART) model.

Zhao and Niu [16] used a backpropagation neural network to estimate unplanned mineral dilution, overcoming the limitations of the empirical ELOS graph method and achieving a determination coefficient (R²) greater than 0.95. Jang et al. [17] developed a neuro-fuzzy system to predict unplanned dilution and mineral loss, obtaining a correlation coefficient (R) of 0.72.

Models ranging from simple to hybrid approaches have been used for overbreak prediction, with the latter showing superior performance. Jang and Topal [18] utilized 49 datasets of Rock Mass Rating (RMR) and overbreak as input and output variables, respectively, to predict overbreak, achieving an R² of 0.945 using an Artificial Neural Network (ANN).

To forecast overbreak and manage it prior to drilling and blasting activities, they implemented an ANN alongside a hybrid system combining a genetic algorithm and an artificial bee colony algorithm with an ANN, yielding an R² exceeding 0.90 [19-22]. Furthermore, models incorporating fuzzy logic and adaptive neuro-fuzzy inference systems integrated with particle swarm optimization have been developed, delivering predictions with an R² above 0.94 [23, 24].

Due to the non-linear and multifactorial nature of mining operations such as blasting, multi-objective intelligent optimization algorithms have been implemented to address these challenges. Bakhtavar et al. [25] mainly reduced fragmentation size and costs through a multi-objective stochastic planning model. The multi-objective particle swarm optimization method (MOPSO) has evolved as a highly effective technique for optimizing multiple indices in blasting operations, where several objectives can be improved simultaneously [26, 27].

This study uses Kolmogorov-Arnold networks to predict dilution, overbreak, and costs in underground mining. It combines geological, geomechanical, and operational data in a deep learning model to improve economic viability. This new approach integrates theory with deep learning, enhancing efficiency and optimization for sustainable mining.

This research follows the following structure: Section 2 presents the materials and methodology, including the fundamentals of the KAN model. Section 3 discusses the model's performance and results. Finally, Section 4 provides conclusions and recommendations for further research.

A total of 732 records were processed, with 19 critical variables selected for the prediction of dilution, overbreak, and operational costs. The models were evaluated on different datasets using k-fold, and their results are detailed below.

3.1 Training, validation and test by model

Figure 3 shows the loss function process during training and validation over 300 epochs for the KAN model. The training and validation lines are observed for each fold. Both lines decrease rapidly and then stabilize, converging towards constant values. This behavior indicates that the model is not experiencing overfitting, as the training and validation losses remain similar, suggesting an adequate generalization capability and accuracy on unseen data.

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Figure 3. Training and validation curve – KAN

Figure 4 presents three scatter plots comparing the KAN model's predictions with actual values across the k-fold phases. The results show high correlations, suggesting that the model is robust and generalizes effectively. For dilution, overbreak, and cost, the correlation is nearly perfect (0.99) across all phases, with data points closely following the ideal regression line, indicating precise learning. Furthermore, the results for each fold remain consistent without significant variance, reinforcing the conclusion that the model is reliable and does not exhibit overfitting, demonstrating its ability to generalize to new data.

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Figure 4. Evaluation of real vs. predicted values – KAN

Figure 5 presents the results of the XGBoost model. The results show high accuracy for each fold, with an overall R² of 0.97 for operational dilution, while for overbreak, the value is 0.99. For total operating cost, it demonstrates high accuracy with an overall R² of 0.94.

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Figure 5. Evaluation of real vs. predicted values – XGBoost

Figure 6 shows the results of the RF model. The overall R² is 0.91 for dilution, demonstrating its good performance in predicting this target. For overbreak, the model showed a slight decrease in performance for each fold, obtaining an R² of 0.94. Similarly, in cost prediction, its performance declined for each fold, resulting in an overall R² of 0.84.

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Figure 6. Evaluation of real vs. predicted values – RF

Figure 7 depicts the evolution of the loss function during the training and validation phases of the MLP model, using MSE. The loss function shows a rapid initial decrease, followed by stabilization, as the training and validation curves align closely. This behavior suggests the model is free from overfitting, since the training loss does not significantly exceed the validation loss.

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Figure 7. Training and validation curve – MLP

In Figure 8, the MLP model demonstrates strong performance across all variables. For operational dilution, the R² is 0.99 for all datasets, indicating excellent correlation during training and high accuracy. For overbreak, it achieved an overall R² of 0.93, reflecting solid overall performance with slight dispersion across folds, with the lowest values in folds 1 and 5, where R² is 0.91. Finally, for total operating cost, the R² values remain at 0.99 across all datasets, demonstrating robust performance.

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Figure 8. Evaluation of real vs. predicted values – MLP

3.2 Comparison and evaluation of the KAN model

Table 3 presents the analysis of the metrics. It is evident that the KAN model excels in all evaluations. In terms of dilution, KAN achieves the best values with MSE, RMSE, MAE, and R² of 0.001, 0.032, 0.055, and 0.999, respectively, indicating an exceptional fit. For the overbreak metric, KAN has an MSE of 0.002 and RMSE of 0.045, an MAE of 0.004, and an R² of 0.984. Regarding costs, KAN remains the leader with MSE, RMSE, MAE, and R² of 0.001, 0.032, 0.027, and 0.999, respectively, reflecting the best fit among all models.

To rigorously evaluate the effectiveness of the implemented KAN model, both non-parametric and parametric statistical tests were conducted based on the data distribution. The Wilcoxon signed-rank test (WSRT) was applied to OD and OB, as these data do not follow a normal distribution. On the other hand, for the TOC, which exhibits a normal distribution, the student's t-test was used.

Table 3. Performance metrics by model

Table 4. p-value result

The results of the WSRT test and student's t-test, presented in Table 4, reveal p-values (all below 0.05, except for XGBoost in overbreak), statistically validating significant distinctions between the KAN model and the alternative models evaluated. Regarding operational dilution, overbreak, and total operating cost, the KAN model demonstrated superior performance compared to MLP and RF. Conversely, XGBoost achieved better performance in predicting overbreak; however, KAN showed a significant difference in predicting dilution and cost. Therefore, the KAN model was chosen as the best-performing approach, outperforming MLP, XGBoost, and RF in the evaluated tasks.

As discussed previously, the KAN model outperformed other techniques based on various key metrics of performance. The superiority of the KAN model over other architectures lies in its use of learnable univariate functions. Unlike MLPs, which apply fixed activation functions such as ReLU or Tanh, KAN dynamically optimizes its activation functions through parameterized functions, such as splines, allowing for a more precise capture of complex relationships within the data. This leads to significantly more accurate models. Liu et al. [31] state that the primary strength of KAN architectures is their ability to increase the expressiveness of the approximation function by eliminating traditional fixed weights and inferring activation functions from univariate functions. This facilitates a better function fitting and helps solve partial differential equations with greater structural flexibility compared to MLP architectures [31, 32]. KAN also has the important ability to modify only the relevant regions without damaging prior knowledge and is less vulnerable to catastrophic forgetting during continuous learning; this latter feature is crucial in problems where stability and flexibility are essential. On the other hand, Ta [37] proposed the BSRBF-KAN architecture, which combines KAN, B-Splines, and radial basis functions (RBF); this architecture can achieve better fitting in image classification problems, converging more stably and rapidly than MLPs and other KAN architectures, optimizing flexibility and interpretability. Moreover, in time series prediction problems, KAN models have outperformed MLPs and Recurrent Neural Networks (RNNs), achieving greater interpretability, efficiency, and accuracy while using significantly fewer parameters [34]. This establishes KAN as a remarkable architecture for modeling complex problems. In one study, Spline-KAN was used and achieved the same accuracy as MLPs but in half the training rounds, which translated into an improvement, despite a slight transition in computational time, as it demonstrated faster and more efficient convergence [38]. Similarly, Shukla et al. [39] analyzed the effectiveness of KAN and MLP in solving partial differential equations trained with noisy data; the results showed that when KAN was properly adjusted, it outperformed MLP in terms of accuracy and stability, enhancing KAN's capability in environments where flexibility, robustness, and computational efficiency are essential.

In this study, KAN achieved similar and superior performance compared to MLP, XGBoost, and RF models, while also providing advantages in both stability and flexibility in data representation. In this case, although MLP and XGBoost achieved high accuracy, KAN was chosen for its adaptability and ability to reach high levels of accuracy in the tasks performed.

3.3 Sensitivity analysis

The Sobol analysis is a sensitivity technique based on the concept of variance decomposition, which quantifies the effect that each input variable has on the model's response.

This procedure was used in the present research to characterize the effect of certain geomechanically and operational parameters on two key variables in underground mining: dilution behavior and overbreak behavior. The variables used for the analysis were defined along with their respective ranges, and samples were generated using the Saltelli method (5000 samples), which allows for a robust estimation of sensitivity indices. The analysis was also complemented by the use of normalized values in the range [0,1], which express the relative influence of each variable within the response of the model.

The first-order Sobol indices (S₁) were calculated to determine the effect of each input variable on the output, while the second-order indices (S₂) were computed to characterize interaction effects between pairs of input variables on the model’s response. The total-order indices (ST) were calculated, including interactions with other input variables, even those corresponding to first- and second-order effects. The results were complemented with 95% confidence intervals, allowing for a more rigorous interpretation of the influence each factor has on the variability of the obtained results.

Table 5 presents the S₁, ST and half-width (HW) of the 95% confidence interval. MW emerges as the most relevant parameter in both scenarios, with S₁ values of 0.643 in dilution and 0.533 in overbreak. This suggests that adjusting the MW can significantly reduce mineral losses and unwanted excavations. On the other hand, both PF and EQ exert a notable influence, especially in overbreak situations, with S₁ values of 0.129 and 0.102, respectively. This highlights that an inadequate configuration of explosive energy significantly contributes to overbreak. Therefore, optimizing fragmentation through a proper selection of specific energy can positively impact stope stability. Regarding RMR, its impact is more significant in dilution (S₁=0.115) than in overbreak (S₁=0.057), reinforcing the idea that weaker rocks experience higher mineral losses. The discrepancy between S₁ and ST in variables such as PF, EQ, and AL suggests that their effect is not only direct but also depends on interactions with other factors. Hence, controlling these parameters must be approached holistically rather than in isolation. Finally, the minimal impact of CL and CF indicates that managing explosive charge design should prioritize the total amount of explosive and its distribution, rather than focusing solely on CL.

The S₂ interaction analysis between key variables is presented in Table 6. The results indicate that RMR has a significant influence on the MW, as the coefficients remain stable at 0.054 for overbreak and 0.052 for dilution, suggesting that lower rock quality amplifies this effect. The powder factor (PF) also affects MW, with coefficients of 0.043 and 0.033, indicating that excessive explosive energy results in unnecessary fracturing, while an insufficient charge compromises rock fragmentation. The EQ must also be adjusted, as coefficients of 0.051 for overbreak and 0.055 for dilution indicate that an overload leads to excessive excavation, whereas an insufficient charge leaves unblasted rock. Finally, the EQ-RMR interaction confirms that rock mass strength determines the required explosive charge, as harder rocks demand more energy, whereas in weaker rock masses, excessive explosives increase dilution and overbreak.

Table 5. S₁ and ST values

Table 6. S₂ values

Therefore, to reduce dilution and overbreak, optimization should focus on ensuring that the MW is adjusted to optimal operational limits, guaranteeing both stability and safety. The controllability of explosive energy must also be considered, as adjusting the powder factor and the amount of explosive should help reduce unwanted excavation and operational risks. Proper characterization of the rock mass before designing drilling and blasting is essential to prevent structural failures and improve efficiency. Finally, optimizing PF, EQ, and considering RMR will lead to a more precise, efficient, and safer excavation.

3.4 Application in an underground mine

The KAN model was applied to an underground mine located in the province of Pataz, northern Peru. The mining operation shares similar characteristics with the variables used to train the model, ensuring its applicability. The dataset used for testing consisted of 246 records.

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Figure 9. Prediction of OD (%)-OB(m³) – TOC ($/ton)

Figure 9 indicates that the model demonstrates strong predictive potential for dilution, with a coefficient of determination of R²=0.99. Similarly, its prediction accuracy for overbreak is also high, with R²=0.97. Additionally, the model exhibits good predictive performance in estimating operational costs, reinforcing its effectiveness in optimizing the mining process.

Once its validity was verified, its effectiveness was compared with traditional methods (TM) used in this mine to estimate dilution and overbreak. In this mine, the operational dilution estimation method considers the impact on the footwall and hanging wall due to overbreak caused by explosive detonation, taking into account the mining section, blast effects, geomechanically conditions, and the cleaning of extracted ore. This combination of factors determines the designed operational dilution percentage, which is calculated using the formulas established for this case by Pakalnis. To improve overbreak estimation, numerical models such as the Finite Element Method (FEM) are used.

A total of 30 blasts were evaluated, comparing the performance of the KAN model against traditional methods. Figure 10 shows that KAN achieved a 97% accuracy in predicting dilution, significantly outperforming the traditional method, which obtained 77%.

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Figure 10. Comparison real vs. TM vs. KAN

Regarding overbreak, the KAN model reached an accuracy of 94%, whereas the FEM achieved 86%, demonstrating the superiority of KAN in estimating these key parameters for optimizing mine design.

3.5 Parameter optimization with MOPSO

MOPSO was used exclusively to determine the optimal controllable parameters, such as MW (m), CL (m), AL (m), and EQ (kg). This approach focuses on optimization on directly adjustable variables to minimize dilution, overbreak, and costs.

Table 7 presents the optimal values identified by MOPSO. The MW shows a difference of 3.38%, which remains within an acceptable margin, ensuring consistency in mining practices. The CL deviates by 3.94%, indicating potential opportunities to optimize material loading precision. Similarly, the AL exhibits a variation of 2.76%, which, while within operational limits, could be further refined to enhance efficiency. Finally, the quantity of explosive used exceeds the optimal value by 5.07%.

Table 7. Main optimized parameters

Although these variables allow for a high degree of control, their adjustment is constrained by technical limitations, operating costs, safety regulations, and local geomechanically conditions. In this case, the RMR averages 65 (IIB), classifying the rock as good quality.

The optimal values recommended for use in drilling and blasting design were analyzed according to the geomechanically factors of the operation, achieving changes that improve rock mass stability by reducing overbreak and minimizing dilution. These adjustments can lead to increased operational efficiency, better material fragmentation, and optimized explosive consumption, thereby ensuring the safety and profitability of the mining process.

Figure 11 presents a comparison between the average values from the database and the improvements achieved by the model after applying the new parameters in the simulation. The initial average dilution was 79.5%, whereas the model successfully reduced it to 68.2%, decreasing by 11.3%. This represents a significant improvement in the accuracy of dilution control. Similarly, for overbreak, which initially averaged 0.32 m³, the model optimized it to 0.24 m³, decreasing by 25%, indicating a notable reduction in unwanted material volumes.

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Figure 11. OD (%) - OB(m³) before vs. after

Figure 12 shows that for drilling and blasting, the cost decreased from 48.65 to $45.30 \mathrm{US} \$ / \mathrm{Tn}$, representing a $6.89 \%$ reduction. In clean-up, the cost dropped from 22.35 to 19.82 US $\$ / \mathrm{Tn}$, equating to an $11.33 \%$ saving. Haulage costs declined from $15.85 \mathrm{US} \$ / \mathrm{Tn}$ to $14.62 \mathrm{US} \$ / \mathrm{Tn}$, showing a $7.77 \%$ improvement, while loading costs decreased from 16.42 to $15.34 \mathrm{US} \$ / \mathrm{Tn}$, resulting in a $6.58 \%$ saving. Finally, the support cost was reduced from 5.60 to $5.52 \mathrm{US} \$ / \mathrm{Tn}$, achieving a $1.43 \%$ reduction.

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Figure 12. Operational costs before vs. after

Overall, the total operating cost decreased from 108.87 to 100.60 US$/Tn, representing a global optimization of 7.60%. These improvements support the synergy between KAN and MOPSO, highlighting how their combination enhances modelling capabilities by providing more accurate and applicable solutions in the mining sector. This approach establishes itself as a solid and efficient framework for addressing optimization in complex underground mining scenarios.

Chimunhu et al. [15] utilized geological, geotechnical, and design data accessible early in the mine planning phase to forecast dilution, attaining a remarkable accuracy of 93% with the PCA-CART hybrid model. Their investigation also observed a minor underestimation of dilution by 13% relative to the stopes in the initial design. In a similar vein, Jang et al. [17] applied a neuro-fuzzy system to predict unplanned dilution and ore loss, achieving an R-value of 0.72. Their model also offered practical recommendations to reduce dilution and ore loss, emphasizing the importance of factors like powder factor and ground support, highlights the critical role of selecting the right variables.

This study included significant geometric aspects, such as MW, and geomechanically characteristics like RMR, in addition to drilling and blasting variables, to guarantee precise forecasts of dilution and overbreak. Hong et al. [22] determined that tunnel diameter was the predominant factor in calculating overbreak, accounting for 90.5%, with RMR contributing 1.80%. Their BO-XGBoost model attained a robust R² score of 0.94. Jang and Topal [18] utilized RMR and overbreak as input and output parameters, respectively, attaining a remarkable R² of 0.95 with an optimized ANN. Mottahedi et al. [24] conducted a sensitivity analysis utilizing the cosine amplitude method (CAM) and found that the S/B and S/D parameters exerted the most significant influence on their model, whereas RMR demonstrated an effectiveness of 0.79 in the ANFIS-PSO model, achieving an R² of 0.96. Similarly, this study showed that MW is the most influential variable responsible for overbreak and dilution. It contributes 64.3% to dilution and 53.3% to overbreak, also highlighting the importance of RMR and RQD in the prediction of these targets, achieving an R² greater than 0.95.

Guo et al. [27] integrated the MOPSO algorithm into their model, thereby identifying the optimal set of blasting parameters. In practical operational environments, these parameters demonstrated both efficacy and reliability, resulting in a 3% decrease in total costs across the mining process, encompassing drilling, blasting, loading, hauling, and crushing. This research demonstrated that MOPSO facilitated the KAN model in precisely determining the parameters to prioritize for minimizing dilution and overbreak, achieving a 7.60% reduction in total operating expenses.