Gene expression programming for modeling and predicting of leaching of chalcopyrite concentrates using 1-hexyl-3-methyl-imidazolium hydrogen sulfate ionic liquid aqueous solution

Abstract

This study aims to develop a precise predictive model for leaching chalcopyrite concentrates. It employs a leaching system comprising 1-hexyl-3-methylimidazolium hydrogen sulfate ([Hmim][HSO₄]) and hydrogen peroxide (H₂O₂), offering a more efficient alternative to conventional hydrometallurgical approaches. Gene expression programming (GEP) was used to develop this model. To construct these GEP models, 120 experimental data points were collected initially. Input variables included time, acid concentration, temperature, particle size, oxidant concentration, stirring speed, and solid/liquid ratio, while output variables included copper extraction percentage. For modeling purposes, the experimental dataset was randomly partitioned into a training set (84 data points) and a testing set (36 data points). A correlation analysis (BCA) revealed weak linear correlations between input variables, justifying the use of advanced methods such as GEP. Using criteria such as coefficient of determination (R²), mean absolute error (MAE), and root relative square error (RRSE), we proposed the optimal model (GEP-3). As a new model with simplified mathematical expressions for accurate prediction of copper extraction from chalcopyrite concentrate, this model achieves R² = 0.976, MAE = 2.80, and RRSE = 0.152 in the training set. Sensitivity analysis revealed that temperature, oxidant concentration, and particle size were the most influential parameters on the copper extraction percentage. By taking into account practical or economic constraints, the proposed model enables the optimization of the leaching process to maximize copper extraction and minimize material consumption.

Supplementary Information

The online version contains supplementary material available at 10.1038/s41598-026-51142-5.

Introduction

Over the past two decades, rising global copper demand has rendered the extraction of copper from low-grade sulfide ores, particularly chalcopyrite (CuFeS₂) which is the world’s most abundant copper mineral, economically viable^1,2. Over 80% of global copper production comes from pyrometallurgical routes, but they are hindered by concerns about SO₂ emissions, operational expenses, and performance when handling fine particles^3,4. Despite these challenges, hydrometallurgical techniques have emerged as viable substitutes, promising greener operations, simpler setup, reduced energy demands, and lower costs⁵. However, existing hydrometallurgical approaches, including chloride leaching⁶, ammonia leaching⁷, and sulfate leaching under pressure^8,9, all present inherent limitations. In spite of the fact that chloride systems accelerate copper dissolution and yield elemental sulfur rather than sulfate⁶, they are severely corrosive, making subsequent electrowinning of copper difficult¹⁰. Copper can be selectively extracted by complexation using ammoniacal solvents, but their use is limited by toxicity risks and costs of ammonia consumption and loss^11,12. The sulfate leaching method, on the other hand, while environmentally friendly, is sluggish and requires high temperatures and pressures in autoclaves because passivating films form on chalcopyrite surfaces¹³, which results in substantial energy and investment expenditures¹⁰.

Research into alternative leaching systems has intensified due to the need for more sustainable metallurgical practices. Due to their properties such as low volatility, high thermal stability, and tunable physicochemical characteristics, ionic liquids (ILs) have attracted considerable attention as potential alternatives to conventional aqueous acids^14–18. It is essential, however, to assess their overall sustainability and environmental footprint holistically. This assessment must extend beyond the leaching step itself to include critical factors such as the energy and resource intensity of IL synthesis, their long-term stability and potential for degradation during use, efficient recycling and recovery schemes, compatibility with downstream metal recovery operations (e.g., electrowinning, solvent extraction), and management of any secondary waste streams. ILs present a promising and innovative pathway for hydrometallurgy, but making definitive claims of superior environmental performance is premature. A comprehensive life-cycle analysis is necessary to address the multifaceted challenges outlined above. Compared to conventional acids, these compounds have shown to be effective as leaching media in hydrometallurgical processes for chalcopyrite leaching. According to studies, this enhancement may be due to a combination of factors, including their unique physico-chemical properties that alter the reaction interface and mitigate passivation effects^14,19. Their effectiveness has been demonstrated in numerous studies. With 1-butyl-3-methylimidazolium tetrafluoroborate ([Bmim]BF₄) at 100 °C over 8 h, copper was extracted to a maximum of 87%²⁰, copper extraction increased to 87% with increasing 1-butyl-3-methyl-imidazolium hydrogen sulfate (BmimHSO₄) concentration to 100% at 70 °C¹⁹, copper dissolution reached 88% after 24 h with BmimHSO₄ and oxygen (O₂)²¹, copper extraction of 80% with BmimHSO₄ and chloride at 90 °C²², 70% dissolution of copper with 1-hexyl-3-methyl-imidazolium hydrogen sulfate (HmimHSO₄) and sulfuric acid¹⁴, and a copper extraction of 90.2% using 1-ethyl-3-methyl-imidazolium hydrogen sulfate (EmimHSO₄) and potassium dichromate (K₂Cr₂O₇)²³. These results indicate that ILs have great potential in improving hydrometallurgical processes for copper extraction from chalcopyrite, especially when combined with strong oxidizing agents.

The combination of ILs with oxidizing agents significantly enhances the leaching efficiency of chalcopyrite. This is due to the resistance of chalcopyrite, which requires strong oxidation conditions. These conditions are achieved by using high temperature and pressure, along with powerful oxidizers such as hydrogen peroxide (H₂O₂), potassium dichromate (K₂Cr₂O₇), ammonium disulfate ((NH₄)₂S₂O₈), hypochlorous acid (HClO), sodium chlorate (NaClO₃), potassium nitrate (KNO₃), and ferric ions (Fe³⁺) in acidic media^5,24–33. In particular, H₂O₂ has found a pivotal application due to its high oxidation potential and the production of water as its only byproduct^5,24,25,32. Recent research in purification using H₂O₂ and ILs has shown very promising results, including the ability to achieve a leaching efficiency of 91% with 1,2-Ethane diammonium hydrogen sulfate ([EDA][HSO₄])³⁴, copper dissolution of approximately 97% with BmimHSO₄³⁵, and copper extraction yield greater than 98.3% with HmimHSO₄³⁶. Although ILs are more expensive and more viscous compared to sulfuric acid³⁷, the strategic selection of solvents, the development of synthesis methods that can be scaled up, and the implementation of efficient recycling systems can significantly reduce the overall process costs.

Copper extraction from chalcopyrite remains challenging due to the mineral’s refractory nature and process complexity^20,38. Machine learning techniques are now being employed to overcome these limitations, facilitating enhanced parameter analysis and process optimization in mineral processing. The adoption of artificial intelligence has grown significantly in this sector over the past decade. A variety of intelligent machine learning algorithms have been applied in the mineral processing industry. These include the multivariate adaptive regression splines (MARS) algorithm for predicting optimal iron recovery, which achieved a coefficient of determination (R²) of 0.978³⁹, and the genetic algorithm optimized neural network (GAANN) for predicting industrial magnetic separation efficiency⁴⁰. In a study⁴¹, a combination of variational mode decomposition (VMD) and random forest (RF) predicted the total iron content (Fet) with R² = 0.94, while a Generalized Regression Neural Network (GRNN) predicted iron recovery with a less than 11% absolute relative error⁴². As part of hydrometallurgical applications, partial least squares (PLS) and artificial neural networks (ANNs) were used to predict ferric iron precipitation in uranium bioleaching⁴³, and a recurrent neural network based on Long Short-Term Memory (LSTM) was applied to dynamic analysis and gold leaching rate prediction⁴⁴. Modeling and optimization of lead recovery were also carried out using ANN and genetic algorithms (GA), with the multi-layer perceptron (MLP) model from ANN providing the best results⁴⁵. In addition, ANN alone has been used to predict gold recovery results⁴⁶ and the support vector regression (SVR) model has been applied to evaluate pyrolusite leaching variables and predict manganese leaching rate with R² = 0.92⁴⁷. To predict the optimal conditions for leaching copper oxide ore, ANN and GA were combined, with GANN (ANN+GA) proving to be more effective in this context than ANN⁴⁸. The ANN model has also been shown to be superior for modeling and predicting copper recovery in heap leaching⁴⁹, where studies have shown that ANN-based models, when compared with SVR and RF, are suitable for capturing the nonlinear representation of the recovery process⁵⁰.

Although the machine learning algorithms mentioned above are capable of modeling complex processes, each possesses certain limitations and characteristics. Many ANNs operate as “black boxes,” providing accurate predictions, but lacking explicit mathematical formulae that restrict physical interpretation and generalizability. Alternatively, simple regression models may not capture the nonlinear relationships inherent in hydrometallurgical systems. GEP presents significant advantages as an advanced evolutionary computation method in this context. In addition to facilitating quantitative prediction and interpretable analysis of complex processes without the need for initial assumptions, one of its key strengths is the automatic construction of transparent, explicit mathematical equations from data. GEP exhibits high adaptability to limited datasets and effectively reduces experimental costs and time by combining genetic algorithm structures with symbolic programming. Due to its functional output, it is particularly useful for optimizing and predicting engineering process behavior, as it facilitates understanding of fundamental mechanisms unlike ANNs, which are opaque^51,52.

GEP has not been widely adopted in mineral processing and hydrometallurgy, despite showing considerable promise in applications such as predicting manganese and cobalt recovery from batteries^52,53, zinc bioleaching⁵⁴, and lead/zinc leaching⁵⁵. A significant research gap exists in applying GEP to model the leaching of refractory sulfide concentrates like chalcopyrite, as existing literature on copper hydrometallurgy has predominantly focused on oxide ores or employed black-box modeling approaches. This study utilizes the GEP algorithm to construct a robust and interpretable model for the atmospheric leaching of chalcopyrite concentrate with HmimHSO₄. Beyond delivering high predictive accuracy, this methodology generates an explicit mathematical function that offers deeper process understanding and supports optimization efforts^56,57.

Data collection and analysis methodology

Data set and input/output selection

Existing records demonstrate the importance of identifying the variables that influence copper dissolution from chalcopyrite during the leaching process for effective optimization and modeling. For instance, Hu et al.³⁶ investigated the leaching of chalcopyrite concentrates using HmimHSO₄ and H₂O₂. Their research highlighted that copper dissolution is significantly influenced by various operational parameters, such as reaction time (t), ionic liquid concentration (C_ionic), temperature (T), particle size (d), oxidant concentration (C_ox), stirring speed (St), and solid/liquid ratio (S/L). Table 1 details the specific values for these leaching parameters. (Despite incorporating elemental assay data (Cu, Fe, S) for the concentrate in the source study³⁶, the present GEP model utilizes only the operational parameters listed in Table 1).

Table 1.

Experimental conditions for leaching chalcopyrite concentrate.

Reaction parameter	Values
Time-t	15, 30, 60, 90, 120 Min
BmimHSO4 concentration-Cionic	0.45, 0.9, 1.36, 1.81, 2.27 Mol/L
Temperature-T	25, 30, 35, 40, 45 °C
Size-d	30, 60, 112.5, 225 μm
H2O2 concentration-Cox	2.94, 4.41, 5.88, 7.35, 8.82 Mol/L
Stirring speed-St	0, 100, 200, 400, 600 rpm
Solid/liquid ratio-S/L	10 g/L

The particle size (d) values are the mean diameters calculated from the sieve fractions (− 300 + 150, − 150 + 75, − 75 + 45, − 45 μm) reported in the source study³⁶, providing a single numerical input for the GEP model. For GEP modeling, a dataset of 120 data points was constructed based on the experimental conditions and results reported in reference³⁶. The copper extraction (output) values corresponding to the combinations of input parameters (Table 1) were digitized from the graphical results in³⁶ using GetData Graph Digitizer software (version 2.24). It is acknowledged that manual digitization of data from graphical plots can introduce a minor degree of uncertainty. To estimate this error, a subset of data points was re-digitized multiple times, yielding a standard deviation in the extracted copper values of less than ± 0.5%. This digitization-induced error is considered negligible compared to the overall range of the output variable (approximately 25–100%) and the magnitude of prediction errors reported by the final models (e.g., MAE ≈ 2.8–4.6%). Therefore, the digitized dataset is deemed suitable for the purpose of this study, which focuses on developing a robust predictive modeling framework rather than precisely replicating experimental measurements. The model’s performance on the independent test set further demonstrates its ability to learn the fundamental process relationships beyond any minor data noise. To develop GEP models, the data were divided into two sets: a training set (84 points) and a testing set (36 points).

Gene expression programming (GEP)

Developed by Ferreira^58–60, GEP is a population-based algorithm that evolved from Koza’s genetic algorithm (GA)⁶¹. To overcome the limitations inherent in both GA and genetic programming (GP), GEP was created. Predictive equations can be formulated using independent operational parameters as inputs^62–65. In GEP, there are five fundamental elements: the terminal set, the termination condition, the fitness function, the control parameters, and the function set¹⁰. Figure 1 presents a flowchart of the GEP evolutionary algorithm, highlighting the vital role of genetic operators^66–68.

Fig. 1.

The flowchart of the GEP evolutionary algorithm.

By affecting chromosomes, genetic operators increase genetic diversity. The GEP has a multi-gene structure, where every chromosome represents a mathematical function. There are several ways to display GEP output, including Karva language (the gene language), expression tree (ET), and mathematical functions^69–71. An example of GEP output is shown below: a linear string representing a chromosome encoded by two genes. Figure 2 shows the conversion of the Karva language to ET (left to right) and the transformation of ET to a mathematical function (right to left)⁷².

Fig. 2.

The genes structure of a chromosome and translation process in GEP.

Randomly selected genomes are validated using the head–tail method. The head of each gene contains terminals and functions, while the tail consists exclusively of terminal symbols^65,73. Equation (1) calculates the tail length (t) based on the head length (h) set by the GEP algorithm designer⁷⁴:

1

where is the maximum number of arguments. GEP is performed in five main steps. First, the fitness function, represented by Eq. (2), is selected⁷⁵.

2

In this equation, defines the selection range, represents the return value of chromosome using fitness function , and is the target value. If for all , then . Notably, when is set to 1, becomes 10, allowing the system to autonomously identify the optimal solution^76,77. The second step involves selecting the terminals (S), a function set (F) defined as for chromosome generation, arithmetic operators , and a range of mathematical functions including Exp, Ln, sqrt, X², X³, X⁴, X⁵, 3Rt, 4Rt, 5Rt, Atan, Tanh, Sin, and Cos. Additionally, the chromosomal architecture is determined by determining the number of genes and chromosomes, with the head length increasing sequentially. Training and testing processes are monitored based on performance, as shown in Table 2 for various GEP modeling trials. The fourth step involves selecting the linking function. Then, genetic operators are selected that are crucial for manipulating chromosomes as described in^61,76. The primary genetic operators include mutation, which alters elements within a chromosome; inversion, which reverses short fragments in the head; and transposition, which relocates genetic material through encompassing Insertion Sequence (IS), Root Insertion Sequence (RIS), or gene transposition^65,78,79.

Table 2.

Optimized GEP model parameters used in this study.

Model	Number of chromosomes	Head size	Number of genes	Linking function	Number of functions	Type of function
GEP-1	40	12	8	Addition	4	+, , , /
GEP-2	30	6	4	Multiplication	8	+, , , /, Exp, ln, X2, Atan
GEP-3	40	10	5	Multiplication	4	+, , , /
GEP-4	40	8	4	Multiplication	10	+, , , Exp, ln, X2, X3, 3RT, Atan, Tanh
GEP-5	30	8	3	Addition	4	+, , , /
GEP-6	40	9	4	Multiplication	14	+, , , /, sqrt, Exp, ln, X2, X3, X4, X5, 3RT, 4RT, 5RT
GEP-7	30	12	6	Addition	7	+, , , Exp, Sin, Cos, Atan
GEP-8	35	10	5	Subtraction	6	+, , , /, ln, 3RT
GEP-9	50	20	10	Division	12	+, , , /, sqrt, ln, X2, X3, 3RT, 4RT, Atan, Tanh
GEP-10	45	15	8	Multiplication	4	+, , , /

In this study, the GEP method and HmimHSO₄ were used to predict the leaching process of chalcopyrite concentrate. The operational parameters t, C_ionic, T, d, C_ox, St, and S/L were used as input variables, while copper extraction percentage was used as output. A total of 120 experimental data points were collected for developing GEP predictive models, of which 84 points were randomly selected for training and 36 points for testing. For modeling and determining the relationship between these operational parameters of the leaching process and copper extraction rates from chalcopyrite concentrate, GeneXproTools 5.0 software was used. The collected data, which served as input variables, were analyzed using SPSS software (version 27). For the proposed models in this study, Table 3 specifies the range of parameters used in the GEP method. GEP parameters (such as chromosome number, head size, and number of genes) are internal software settings that control the evolutionary search, not directly derived from leaching chemistry. Parameter tuning is an exploratory, trial-and-error process to determine their optimal values. The final configuration was chosen based on the combination of values with the highest predictive accuracy and the best balance between complexity and simplicity (Table 3). When applying evolutionary computation methods to complex engineering problems, parameter tuning is a standard procedure.

Table 3.

GEP configuration details.

Definition of GEP parameter	Values
Number of chromosomes	30–35–40–45–50
Head size	6–8–9–10–12–15–20
Number of genes	3–4–5–6–8–10
Linking function	Addition–Multiplication–Subtraction–Division
Fitness function error type	RRSE
Constant per gene	1
Mutation rate	0.00138–0.0044*
Inversion rate	0.00546–0.15*
One-point recombination rate	0.00277–0.2*
Two-point recombination rate	0.00277–0.2*
Gene recombination rate	0.00277–0.1*
Gene transposition rate	0.00277–0.1*
IS transposition	0.00546
RIS transposition	0.00546

*These values were used only for Model 7.

Results and discussion

Data analysis by SPSS

The Bivariate Correlation Analysis (BCA) method was used to determine the magnitude and direction of the relationship between operational parameters. Using this method, operational data is analyzed carefully, and then pairs with strong correlations are identified. A high negative or positive correlation coefficient among these pairs can significantly reduce the accuracy of the GEP model and make it difficult to explain how operational parameters affect chalcopyrite leaching. There is a possibility that the effect of each input parameter is overestimated if there is a significant interdependence between variables⁸⁰. In the correlation matrix (Table 4), the linear correlation between key process variables indicates that t exhibits negligible correlations with other input parameters such as C_ionic, T, d, C_ox, and St. As a result, t has a non-linear or cumulative effect on the process. In addition, there were weak correlations between some of these parameters; for example, C_ionic showed a weak correlation with T (0.110) and d (− 0.090), while T showed a weak positive correlation with d (0.061) and St (0.034). A lack of variance in the data for C_ox or the non-linear nature of its influence may explain its zero correlation with all other variables. Correlation calculations for this parameter were not possible due to the constant S/L throughout the experiments.

Table 4.

Pearson correlation matrix of operational parameters for leaching of chalcopyrite concentrate with HmimHSO₄.

Parameters	t	Cionic	T	d	Cox	St
T	1	0.000	0.000	0.000	0.000	0.000
Cionic	0.000	1	0.110	− 0.090	0.000	− 0.050
T	0.000	0.110	1	0.061	0.000	0.034
d	0.000	− 0.090	0.061	1	0.000	− 0.028
Cox	0.000	0.000	0.000	0.000	1	0.000
St	0.000	− 0.050	0.034	− 0.028	0.000	1

Principal component analysis (PCA) can be used to examine multicollinearity among practical parameters. As a dimensionality reduction technique, PCA converts correlated variables from a high-dimensional space into a space with fewer dimensions, effectively removing their correlation. The newly derived, uncorrelated variables are called principal components^81–83. As a key criterion for determining whether PCA is appropriate, the Kaiser–Meyer–Olkin (KMO) factor⁸⁴ can be calculated using Eq. (3).

3

The correlation coefficient r_ij and the practical correlation coefficient a_ij represent variables i and j, respectively. A KMO factor under 0.7 indicates that the dependencies among the practical parameters are not substantial, making PCA unsuitable⁸⁵. According to this study, the KMO factor was 0.444. As a result, PCA was deemed unnecessary for addressing potential interactions among practical parameters. A low KMO value indicates that the correlations among the variables are too weak to support the extraction of meaningful common factors. Variables exhibiting significant inter-correlations are typically used in factor analysis, and a low KMO value indicates deficiencies in these correlations.

Overall, the multivariate analysis (correlation matrix and PCA) served as a preliminary diagnostic step rather than a validation step for the GEP model. The purpose of the analysis was to determine whether the input variables were linearly interdependent. Based on the observed weak linear correlations and low KMO values, linear modeling approaches are insufficient for this system. Due to the inherently nonlinear and interactive nature of the leaching process, driven by complex physicochemical mechanisms, GEP was chosen. Our choice of GEP was based on its ability to capture these nonlinear relationships while producing an interpretable mathematical function.

Evaluation of GEP model prediction performance

Each developed model was validated using the coefficient of determination (R²), mean absolute error (MAE), and root relative square error (RRSE), as defined in Eqs. (4)–(6)⁸⁶.

4

5

6

Assume that t_i is the target value, o_i is the predicted value, and n is the number of data in the testing and training phases. R² values close to 1 and MAE and RRSE values close to zero indicate a good model fit⁸⁷. For both the training and testing datasets, Table 5 presents the statistical characteristics of ten appropriate models. The performance of model GEP-9 warrants closer examination. With R² = 0.986, MAE = 2.20, and RRSE = 0.118, this model demonstrated excellent fit to the data in the training phase. In real-world problems, however, the results of the testing phase, including R² = 0, MAE = 0, and RRSE = 0, are unusual. There is strong evidence of severe overfitting; that is, the model has retained the specific details and noise of the training data rather than generalizable patterns, and hence is unable to predict new, unseen data (the test data). Because of the high probability of overfitting and the lack of confidence in predicting new data, the GEP-9 model is not considered an optimal model despite the seemingly ideal testing results. In GEP-based models, repeated k-fold cross-validation may result in structurally different symbolic equations, which reduce interpretability. Therefore, model robustness was evaluated through consistency between training and testing results, explicit rejection of overfitted models, and evaluation of physical plausibility.

Table 5.

R², MAE, and RRSE amounts for six appropriate GEP models.

	R2		Error
			Training		Testing
No	Training	Testing	MAE	RRSE	MAE	RRSE
GEP-1	0.964	0.897	3.25	0.188	5.15	0.390
GEP-2	0.949	0.962	4.12	0.226	3.59	0.194
GEP-3	0.976	0.978	2.80	0.152	2.81	0.151
GEP-4	0.960	0.961	3.52	0.204	3.37	0.197
GEP-5	0.953	0.856	3.75	0.215	6.12	0.44
GEP-6	0.961	0.979	3.42	0.2	3.39	0.160
GEP-7	0.936	0.963	4.60	0.251	3.49	0.196
GEP-8	0.955	0.90	3.57	0.211	4.95	0.363
GEP-9	0.986	0	2.20	0.118	0	0
GEP-10	0.976	0.964	2.39	0.152	3.19	0.2

Figure 3 compares R², MAE, and RRSE values for ten superior GEP models. Based on the explanations provided (which indicate a significant difference between training and testing data for model GEP-9), models GEP-3, GEP-6, and GEP-10 demonstrated higher accuracy in predicting chalcopyrite concentrate leaching behavior.

Fig. 3.

Statistical indices, including R², MAE, and RRSE, for the training and testing data series within the GEP models.

The formulations for GEP-1 through GEP-10 are shown in Table 6. In terms of the leaching of chalcopyrite concentrates, these GEP models suggest a complex parameter space.

Table 6.

Mathematical expressions for various optimized GEP models.

Model	The equation of selected models
GEP-1	(Cox−(0.468*d)/(Cox−(Cox/((d−t)−Cionic))))  + (Cox−((((Cox−0.691)*S/L)*(Cox/0.691))/((T+Cionic)−Cox))) + ((S/L/((− 0.846*(− 0.846*− 0.846)) + ((− 0.846/d)*(St*Cox)))) − (− 0.846)) + ((d/− 0.043)/(((t*Cionic) + (− 0.043))*(T−S/L) + ((d/Cionic) − (Cox*t)))) + (((S/L/S/L)/(− 0.722 + (d/T) + (Cionic*Cox*− 0.722)))/− 0.722) + (T/((((t/d)−(S/L−d))*Cionic)/(Cox−((− 1.082*− 1.082)*− 1.082)))) + (((((Cox−((d/t)/0.618))/0.618) + Cox)/ 0.618)+t) + (((((Cox/Cionic)+Cox)*− 0.693)/− 0.693) − (((Cox−(− 0.693))−t)*− 0.693))
GEP-2	(atan((((t−Cionic)/(d+t))/atan(0.634))))*((atan(log(((St+1.042)*Cox)))*Cox))*(log((((d/Cox)/((− 0.796)/Cionic))+T)))*(log((T+(log(0.706)*pow(Cox,2)))))
GEP-3	((Cox/(((d/((S/L+St)*Cionic)) + 1.228)+T)))*((((T/d)+S/L+T) − (((Cox*0.011)*Cox)*(Cox+Cox))))*(((((T−0.919–0.919+(T/0.919))/(d+(t+t)))/0.919)))*(((Cionic/(((0.378−S/L)*(Cionic+Cox)) − ((Cox−d)+T))) + 0.378))*(((t/((t−(T/S/L)) + ((Cionic/Cox)*d)))*t))
GEP-4	(gep3Rt((t−gep3Rt((pow(log(t),3)*(d−T)))))) * (log(pow((((Cox−T)*(− 0.653−T)) − (gep3Rt(d)*d)),2))) * (gep3Rt(gep3Rt(((pow(Cionic,3)−tanh(− 0.049)) + gep3Rt(gep3Rt(St)))))) * (log((gep3Rt(((Cox+(−1.589))*(T+Cionic))) − exp(gep3Rt(Cionic)))))
GEP-5	(((d/((Cox−S/L)−St)) + ((T/S/L) + (t*0.314)))) + ((T−(((d/t) + (S/L/Cox))/((0.335*Cox)/S/L)))) + (((((T+T)*Cox)−(t/Cox))/(Cionic*(d−0.394))))
GEP-6	((Ln((0.01971673*((Ln(Cox) + Cionic)*(0.01971673^(1/5))))^(1/4)))^2) * (T+((((T−d)*(d+58.57453882))−(S/L*t))^(1/3)))^(1/3)* (Cox−(− 0.06213019+(T/((St+S/L)*Cox)))^(1/5)) * ((((S/L/Cionic)/(d−T))^4 + (t−S/L))^(1/3))
GEP-7	(((Cos(Cox)*Cos(− 0.92505326)) − (Sin((t−(− 0.92505326))) + ((− 0.92505326) + (− 0.92505326))))*S/L) + (t*(Cox*(0.09784995−Exp((atan(Cionic+0.09784995))*((S/L–T)*Cionic)))))) + (atan(((Exp(Cox)*0.02502634)*(Sin(St)+Cionic)))*(atan(St)−Cox)) + (atan(E1) + (Sin((T*Cionic)) − ((Sin(d) + Cionic)*Cox)))) + (((atan(0.33406096)*((Sin(Cox)*(0.33406096*0.33406096))*((T+T)+(T+St)))))+S/L) + (((− 0.30325216)*d) + Sin((((Cionic+T)*Cox) − (Sin((− 0.303252216))*(d+(− 0.30325216))))))
GEP-8	(((T−(((d+Cox) + T)/(t^(1/3)))) + 6.13839270)+T)−(((t^(1/3))^(1/3))/(− 0.34969491))− (Ln((Cox/((T*T)/d+(t+Cionic))))*(St^(1/3))) − (((222.17927959+St)*(St+222.17927959) + (T+(S/L*S/L)))/Cox) − ((Cionic+(((St/T) − (t+Cionic))/0.68411361)^(1/3))*Cox)
GEP-9	(pow(pow((log(sqrt(t))−gep3Rt(log((pow(pow(((sqrt(St)/(t−0.770))+sqrt(Cox)),2),2)/d)))),2),2)+pow(T,2))/pow(pow(0.674,(1.0/4.0)),2)/atan(pow((pow(pow((T*(pow(((Cionic/tanh(((St−1.046)/d)))−tanh(Cionic)),2)/pow(Cionic,3))),3),3)+t),(1.0/4.0)))/(pow(pow(pow((pow(pow((St/(− 0.967*(pow(((T/d)*T),3)+(atan(Cionic)−d)))),2),2)+S/L),2),(1.0/4.0)),2)/Cox)/T/((gep3Rt(sqrt(sqrt(pow((St*pow(Cionic,2)),2))))/pow((S/L+pow(((d−pow(Cox,2))/t),3)),2))− 0.318)/pow(((Cox+((log(atan(sqrt((d/Cox))))/((((Cox+0.266)*0.266)−gep3Rt(Cionic))−((d/t)/t)))/0.266)) + 0.266),(1.0/4.0))/log(gep3Rt((d−(atan(pow(((((t/Cionic)−pow(T,2))−(pow(− 0.168,3)*pow(d,3))) + (S/L+((t+T)*S/L))),3))/− 0.168))))/(pow((Cionic*log(pow(sqrt(atan(atan(((− 0.468+S/L)−(Cox+pow(pow(atan(− 0.468),2),2)))))),3))),3)− 0.468)/(gep3Rt((S/L*pow(((gep3Rt(atan((pow(((T−d)−Cionic),3) + (pow(Cox,2)−(t*t)))))−S/L)/t),2)))−tanh(− 0.4))
GEP-10	(((t−(((t−S/L) + (T−Cionic)) − (1.30259928*S/L))/(Cox+(S/L−d)))− 1.30259928)/Cionic) * ((− 0.71221335)/(((T*(− 0.71221335))*(d+S/L))+t) * ((d+(S/L+t)) * (((− 0.71221335)*St) − (St+T))))) * (t+((((− 0.61497582)*St)/S/L)/((Cox+(− 0.61497582))−Cionic)) − (St/((S/L/Cox)*(t/(− 0.61497582)))))) * ((T+(((St−(Cionic*Cionic)) − (S/L/Cionic)) + (d/(T/S/L)))) − ((− 0.32302798)−St))) * ((0.20374091+((t/S/L)−(0.20374091+(Cionic*((t/T) + (0.20374091+T))))))−T) * (T * (d+((0.22734040−((t/S/L)/((S/L−Cox) + (0.22734040+0.22734040))))− 0.22734040))) * (0.02089252/(t−(((T/(0.02089252 *Cionic))−St)+t)/(d*d)))) * (S/L * (S/L−(0.91475245/((Cox*((Cionic−0.91475245)+S/L))/((S/L/Cionic)*S/L))))))

ETs can be used to represent the output of models built by GEP. By using the ET, the user is able to recognize the sequence of genes. The ET of the GEP-3 model, which includes 5 genes (sub-ETs), is shown in Fig. 4. According to the presented figure, the complexity of the model increases with the number of genes and mathematical functions. As indicated in Tables 2 and 6, both the number and type of genes and mathematical functions play a crucial role in determining the performance of the GEP models.

Fig. 4.

ET of the GEP-3 model.

In Fig. 5, the actual and predicted copper extraction percentages for chalcopyrite concentrate using GEP‑3, GEP‑6, and GEP‑10 models are illustrated. The predicted values from all three models show close agreement with the experimental data, particularly during the training phase. During training for GEP-3, GEP-6, and GEP-10, the best fitness values were 867.68, 833.09, and 867.82, while during testing they were 868.27, 861.70, and 832.87.

Fig. 5.

Comparison of actual and predicted values for copper extraction percentages in the training and testing phases.

The comparison of the “actual extracted copper (%)” with the values “predicted” by the GEP-3, GEP-6, and GEP-10 models reveals the varying capabilities of these models in estimating copper extraction (Table 7). All three models predict values that agree closely with the actual data, indicating the applicability of GEP models to modeling this complex process.

Table 7.

Comparison of actual extracted copper percentage with values predicted by GEP models.

No	Actual extracted copper (%)	Predicted extracted copper (%)
		GEP-3	GEP-6	GEP-10
1	34.1	34.09	33.94	34.47
2	51.3	54.53	51.08	52.66
3	24.5	20.71	25.88	24.72
4	94.1	95.28	92.56	100.43
5	73.5	74.32	69.97	74.90
6	34	34.09	33.94	34.47
7	82.2	80.01	78.20	80.67
8	95.9	98.99	100.45	98.08
9	72.2	65.32	69.95	71.48
10	40.8	43.53	45.61	43.50
…
38	71.6	63.56	69.39	64.14
39	33.9	38.22	40.57	36.49
40	67.1	59.41	56.51	55.68
41	51.8	53.85	52.20	52.66
42	48.1	52.13	50.36	49.18
43	33.6	34.09	33.94	34.47
44	81.3	79.21	77.52	82.42
45	39.3	45.07	45.63	42.78
46	25.7	29.82	30.19	36.44
47	61.4	60.78	60.45	61.11
…
75	53.4	53.51	54.73	56.22
76	69.4	70.18	66.93	70.78
77	60.4	59.33	62.98	60.61
78	93.4	89.30	84.13	93.44
79	56.6	61.16	61.11	61.59
80	94.9	95.91	96.22	96.01
81	85.6	85.48	86.30	87
82	43	44.68	44.69	42.27
83	99	100.02	114.91	97.56
84	81.5	83.48	81.53	86.70

Figure 6 provides a graphical interpretation of the data presented in Table 7. Using this radar chart, the comparative performance of the models across different metrics can be observed. A few candidate models (e.g., GEP-6) generated predictions exceeding the theoretical maximum of 100% copper extraction during the evolutionary process. The initial approach was to normalize all parameters within a [0, 1] range. However, this method introduces a practical limitation for interpretable models such as GEP, since the final equation outputs a normalized value, which needs to be denormalized to obtain the actual extraction percentage. To preserve the direct interpretability and usability of the final model, normalization was not applied. By selecting rigorous models, physical boundaries were enforced instead. A well-selected GEP model can effectively respect the physical constraints of the system with its maximum prediction of 100.02% for an actual value of 99%.

Fig. 6.

Radar chart for models (a) GEP-3, (b) GEP-6, and (c) GEP-10.

Among the three top-performing models identified, GEP-3, GEP-6, and GEP-10, the GEP-3 model has the best accuracy in predicting copper extraction rate from chalcopyrite concentrate since it balances both training and testing. A balance must be struck between predictive accuracy and model simplicity when selecting a final model in GEP. The number of genes or the use of advanced mathematical operators can lead to lengthy equations that are difficult to interpret directly (as illustrated by the GEP-10 model in Table 6), but overly restricting the model structure can also result in poor accuracy and a failure to capture complex nonlinear relationships.

In this study, model selection was designed to ensure that the final model was not only highly predictive and generalizable, but also suitable for practical use. The GEP-3 model was selected as the optimal model. In both training and testing phases, R² was very close to one, indicating a strong and stable predictive performance. Second, the model uses only five genes and four basic arithmetic operators (+ , , , /). As a result of its relative simplicity, its corresponding ET in Fig. 4 is easier to comprehend, and more importantly, it facilitates its practical application.

Explicit equations of the GEP-3 model can be easily programmed into common software, such as Microsoft Excel, or integrated into plant control systems. By simply entering current operational parameters (e.g., temperature, concentration, and etc.) into a familiar computational environment, engineers or operators can perform rapid copper recovery predictions, sensitivity analyses, and optimizations without the need for complex manual calculations. As a result, the presented model (GEP-3) provides a bridge between advanced data-driven modeling and operational decision-making.

The optimized GEP-3 model exhibits exceptional predictive performance (R² = 0.976 for the training dataset). Its accuracy not only exceeds that of conventional “black-box” models such as ANNs and SVR used in copper hydrometallurgy^48–50, but also remains highly competitive when compared with other advanced GEP models reported in the literature. For instance, the model developed in this study achieves accuracy levels comparable to those of GEP models predicting manganese (R² = 0.988) and cobalt (R² = 0.979) recovery from spent Li-ion batteries^52,53. Its performance is clearly superior to GEP models reported for zinc solvent extraction from bioleaching solutions (R² = 0.955)⁵⁴, as well as for lead/zinc recovery from cerussite concentrates (R² = 0.82–0.90)⁵⁵. A major distinction of this study is its focus on a more complex and less-investigated system, namely the atmospheric leaching of a primary sulfide concentrate (chalcopyrite). The refractory behavior of chalcopyrite creates a far greater modeling challenge than those involving recycled materials, solvent extraction systems, or oxide ores used in prior GEP studies. Therefore, achieving such a high level of predictive accuracy for this complex sulfide system, along with deriving an explicit and interpretable mathematical function, represents a major advancement. This clearly highlights the robustness and broad applicability of the GEP methodology developed in this work.

Sensitivity analysis

The sensitivity analysis was first performed using the most appropriate model, GEP‑3, to determine the effect of operating parameters on the copper extraction percentage. In this analysis, the values of 15 min, 0.45 mol/L, 40 ℃, 60 μm, 5.88 mol/L, 200 rpm, and 10 g/L were considered for the parameters t, C_ionic, T, d, C_ox, St, and S/L ratio, respectively. The sensitivity analysis was done by changing each parameter by ± 10% of its original value, while keeping all parameters constant, and then measuring the copper extraction percentage.

Figure 7 provides critical visual insight into the chalcopyrite leaching process, showing the amount of change in copper extraction (%) in response to a ± 10% variation in each input parameter. Based on the magnitude of this change, parameters are categorized as having high (change < 5%), moderate (change = 1–5%), or low (change > 1%) sensitivity. The results reveal that T, d, and C_ox are the most influential parameters. The pronounced sensitivity of temperature is attributed to its fundamental role in enhancing particle kinetic energy, reducing solution viscosity, and increasing the reaction rate constant, thereby facilitating mass transfer and overcoming the activation energy barrier for breaking down the refractory chalcopyrite lattice⁸⁸. The significant influence of particle size is mechanistically linked to the increased specific surface area and shorter diffusion paths in finer particles, which exponentially accelerate the interfacial reaction kinetics and diminish the shielding effect of potential passivating products⁸⁹. The high sensitivity to H₂O₂ concentration underscores its function as a powerful oxidant; it generates hydroxyl radicals in the ILs medium, which aggressively oxidize sulfide species to sulfate and prevent the formation of a passive elemental sulfur layer^5,24. Furthermore, the presence of the HmimHSO₄ enhances the oxidative process, as its unique ionic structure can increase oxygen solubility, thereby potentiating the effectiveness of H₂O₂⁹⁰. In contrast, parameters such as t and C_ionic exhibit moderate sensitivity. Although their individual ± 10% variation leads to a measurable shift in extraction, their influence is secondary compared to the high-sensitivity parameters within the studied ranges. Within the investigated range, parameters St and S/L have low sensitivity on dissolution kinetics and, thus, the output of the model.

Fig. 7.

Sensitivity analysis of practical parameters.

The critical influence of T, C_ox, and d, as revealed by the sensitivity analysis in Fig. 7, is further corroborated by their individual effects on the average copper extraction percentage, detailed in Fig. 8. The results demonstrate that increasing T from 25 ℃ to 45 ℃ (Chart a) enhanced copper extraction from 40.02% to 77.9%, consistent with its role in accelerating kinetic rates. Likewise, elevating C_ox from 2.94 to 8.82 mol/L (Chart b) improved extraction from 25.1% to 78.1%, highlighting its pivotal function in maintaining a high oxidation potential. Furthermore, reducing the particle size to 30 μm (Chart c) maximized the extraction percentage to 85.7%, whereas a larger size (225 μm) curtailed it to just 24.8%, emphasizing the necessity of mechanical preparation. Thus, Fig. 8 provides the experimental justification for the sensitivity ranking, confirming that T, C_ox, and d are the most critical levers for process optimization.

Fig. 8.

The average copper extraction percentage from the experimental dataset³⁶ is affected by three factors: (a) T, (b) C_ox, and (c) d. The average extraction across all data points indicates the predominant trends that corroborate the results of the sensitivity analysis.

Conclusion

This study successfully developed a robust and interpretable predictive model for the atmospheric leaching of chalcopyrite concentrate using HmimHSO₄ and H₂O₂ through GEP. The optimal GEP-3 model demonstrated exceptional predictive performance, achieving R² = 0.976, MAE = 2.80, and RRSE = 0.152 on the training set, showcasing the capability of GEP to accurately model complex hydrometallurgical systems, achieving performance competitive with or superior to other advanced machine learning techniques. Unlike “black-box” models, the GEP framework provides the distinct advantage of generating an explicit, interpretable mathematical function, offering unparalleled transparency for deeper process insight and direct application in optimization and scenario analysis. Sensitivity analysis identified T, C_ox, and d as the most influential parameters, substantiated by their fundamental physicochemical roles in governing reaction kinetics and activation energy, controlling oxidation potential while preventing passivation, and determining available surface area, respectively. Analysis of parameter effects revealed clear optimization pathways, where increasing temperature from 25 ℃ to 45 ℃ enhanced copper extraction from 40.0% to 77.9%, elevating H₂O₂ concentration from 2.94 mol/L to 8.82 mol/L improved extraction from 25.1% to 78.1%, and reducing particle size to 30 μm maximized extraction to 85.7%, thereby providing practical guidance for maximizing copper yield under realistic economic constraints related to energy, reagents, and grinding. GEP is an empirical, data-driven tool. Its internal parameters (e.g., genes, chromosomes) are not physicochemical properties of the leaching system, but rather attributes of the evolutionary algorithm. As a predictive model, the model provides a function that can be interpreted for process optimization within the studied operational window. This research establishes GEP as a transformative tool that combines high predictive accuracy with exceptional transparency through its explicit functional form, paving the way for more efficient, interpretable, and sustainable hydrometallurgical processes while enabling the industry to optimize operations and enhance resource efficiency.

Supplementary Information

Below is the link to the electronic supplementary material.

Author contributions

A.M.: Investigation, Data collection, Software, Formula analysis, Validation, and Writing—Original draft; G.K.: Validation, Investigation, and Conceptualization; M.S.: Validation, Investigation, and Conceptualization; M.R.: Validation, Investigation, and Conceptualization.

Funding

The author(s) did not receive any specific funding for this research.

Data availability

The datasets used during the current study available from the corresponding author on reasonable request.

Declarations

Competing interests

The authors declare no competing interests.