Abstract
A novel approach was introduced to predict the solubility of hydrogen in Athabasca bitumen sample by leveraging a hybrid approach based on Harmony Search Algorithm (HS) and AdaBoost. The solubility of H2 in the samples is of great importance for treatment of heavy hydrocarbon in petroleum processing and can help optimize processes. In the correlation of data, pressure and temperature were used as the inputs, while the hydrogen solubility was assigned the sole response for the modeling. This approach was applied to three popular regression models: K-Nearest Neighbors (KNN), Theil-Sen, and Lasso, resulting in hybrid models named HSA-KNN, HSA-TS, and HSA-LAS, respectively. The HS algorithm is used to optimize the hyperparameters of the Adaboost and base models, and then AdaBoost is applied to enhance the performance of the base models. The HSA-KNN model achieved an R2 score of 0.96466, MSE of 6.2790E-03, and a maximum error of 1.80485E-01, while the HSA-TS model achieved an R2 score of 0.96433, MSE of 6.3994E-03, and a maximum error of 1.40763E-01. The HSA-LAS model, on the other hand, achieved an R2 score of 0.89249.
1 Introduction
In petroleum industry, hydrogen is an important gas which is used for hydrogeneration process by which heavy hydrocarbons are treated to meet the standards by removing their impurities. It is basically described as liquid-phase chemical reaction where the separation takes place in liquid feed by reactions between hydrogen and the hydrocarbon components (Bilto and Uyar, 2026; Gao et al., 2026). For instance, to remove organosulfur compounds from liquid heavy hydrocarbons, hydrogen reacts with sulfur compounds to form gaseous products and treat hydrocarbons (Han et al., 2022; Morales-Valencia et al., 2021; Rodríguez Castillo et al., 2023). The modeling of hydrogenation in liquid phase is complex and the concentration of H2 in liquid phase is needed to be determined. The concentration of H2 in hydrocarbon is governed by its solubility in the feed, so solubility is one of the important parameters which should be accurately obtained for modeling of hydrogenation process in petroleum industry such as hydrodesulfurization process (HDS) for diesel treatment (Kalam et al., 2023; Safari and Vesali-Naseh, 2019; Wu et al., 2018). As such, for H2 related processes, knowing accurate values of H2 solubility at different temperatures and pressures is valuable which can help predict and optimize the process (Li et al., 2024).
The solubility of gases such as H2 in liquid can be modeled by various modeling approaches such as thermodynamics, molecular modeling, and data-driven models. Given that various components are available in heavy hydrocarbon as the liquid phase, the system is considered to be complicated, therefore, data-driven models such as machine learning can be used for obtaining and predicting hydrogen solubility in liquid hydrocarbons (Hadavimoghaddam et al., 2022). The experimental data is needed for data-driven models to be built, and also a proper method should be selected to be well suited for the dataset. The model can also investigate the changes in H2 solubility versus other input parameters such as temperature, pressure, etc. Methods of machine learning (ML) are versatile in process modeling and can be used for predicting hydrogen solubility in different hydrocarbons.
The realm of machine learning (ML) has become a focal point of interest in recent years owing to its broad applicability in diverse domains. ML employs statistical and computational techniques to enable machines to assimilate data, and make decisions or predictions based on that data. One of the main challenges in machine learning is to build accurate and reliable models that can generalize well to new data (Alpaydin, 2020; Lashari et al., 2021; Shinde and Shah, 2018).
The ensemble methods have become a potent tool for enhancing ML models’ effectiveness. Ensemble methods combine the predictions of multiple base models to obtain a final prediction that is often more accurate and robust than any individual model. AdaBoost is one such popular ensemble method that has been shown to be effective in improving the performance of several base models.
Another approach to improving the performance of machine learning models is tuning models through hyper-parameters optimization techniques. Harmony Search (HS) is a metaheuristic optimization algorithm that is inspired by the process of musical improvisation (Yang, 2009). HS has been utilized successfully in a variety of optimization problems, including feature selection, hyperparameter tuning, and model selection.
K-Nearest Neighbors (KNN) is a non-parametric method that has garnered substantial attention in the field of machine learning for its effectiveness in regression and classification tasks. KNN predicts the target value of a new sample by finding the k-nearest neighbors in the training set and using their average or weighted average as the prediction. KNN has been shown to perform well on many datasets, but its performance can be sensitive to the choice of the number of neighbors and the distance metric used (Cover, 1968; Cover and Hart, 1967).
For feature selection and regularization, Lasso Regression is a linear regression technique. Lasso adds an L1 penalty component to the usual least squares objective function, which favors sparsity and promotes the selection of only the most important characteristics. Lasso has been shown to be effective in handling high-dimensional datasets and reducing overfitting (Ranstam and Cook, 2018).
Theil-Sen Regression is a highly robust and effective linear regression method that surpasses the standard linear regression method in handling noisy datasets and outliers. This method is based on the median of pairwise slopes between data points, which provides a robust estimate of the underlying trend. The Theil-Sen Regression can be applied to datasets with high variability, as it is less affected by outliers than standard linear regression, thus making it highly reliable and accurate. Theil-Sen Regression has gained widespread attention and adoption in recent years owing to its capacity to handle challenging datasets that standard linear regression methods struggle with (Ohlson and Kim, 2015; Wilcox, 1998).
In this paper, for prediction of hydrogen solubility we propose a novel hybrid method that combines AdaBoost and HS optimization to improve the performance of three base models: K-Nearest Neighbors (KNN), Theil-Sen Regression (TS), and Lasso Regression (LAS). We refer to these hybrid models as HSA-KNN, HSA-TS, and HSA-LAS, respectively. These models are developed for hydrogen solubility in heavy hydrocarbon (bitumen) for the first time, and the results will be used for evaluation of solubility changes versus input features, i.e., T and P.
2 Problem statement
The dataset used in this study contains the Solubility of hydrogen in Athabasca bitumen as reported in (Lal et al., 1999). Three parameters were taken into account including: temperature (T), pressure (P), and solubility (S), with a sole output which is S (Jin et al., 2023a). Other sources have reported the use of these data for machine learning analysis (Jin et al., 2023a; Jin et al., 2023b). The dataset contains 43 data points in total and are shown in Table 1 entirely. The solubility of hydrogen has been obtained after the thermodynamic equilibrium has reached in the system with the normal unit. Figure 1 presents a scatter matrix (pairplot) of the dataset variables, where the off-diagonal elements show pairwise relationships between temperature, pressure, and hydrogen solubility. The diagonal elements represent the marginal distribution of each variable in the form of histograms. These histograms share identical x- and y-axis scaling in each diagonal subplot because they depict frequency distributions of a single variable, rather than relationships between two different variables.
TABLE 1
| T (°C) | P (MPa) | S (g H2/kg solvent) |
|---|---|---|
| 50 | 3.678 | 0.114 |
| 50 | 8.898 | 0.24 |
| 50 | 0.26 | 0.0737 |
| 50 | 13.16 | 0.414 |
| 100 | 4.064 | 0.178 |
| 100 | 9.331 | 0.375 |
| 100 | 14.84 | 0.615 |
| 150 | 4.899 | 0.225 |
| 150 | 8.932 | 0.447 |
| 150 | 14.352 | 0.689 |
| 200 | 0.783 | 0.057 |
| 200 | 2.024 | 0.128 |
| 200 | 3.127 | 0.193 |
| 200 | 3.955 | 0.259 |
| 200 | 5.334 | 0.33 |
| 200 | 6.127 | 0.386 |
| 200 | 7.506 | 0.478 |
| 200 | 8.884 | 0.526 |
| 200 | 9.195 | 0.498 |
| 200 | 11.607 | 0.694 |
| 200 | 13.883 | 0.918 |
| 200 | 16.71 | 1.081 |
| 200 | 18.779 | 1.09 |
| 200 | 21.158 | 1.243 |
| 200 | 24.845 | 1.517 |
| 250 | 4.695 | 0.321 |
| 250 | 8.713 | 0.59 |
| 250 | 12.087 | 0.82 |
| 250 | 14.794 | 1.014 |
| 250 | 18.104 | 1.244 |
| 300 | 1.187 | 0.131 |
| 300 | 2.024 | 0.217 |
| 300 | 2.852 | 0.241 |
| 300 | 3.954 | 0.38 |
| 300 | 4.402 | 0.397 |
| 300 | 5.815 | 0.494 |
| 300 | 7.539 | 0.62 |
| 300 | 7.953 | 0.729 |
| 300 | 8.469 | 0.777 |
| 300 | 11.47 | 1.008 |
| 300 | 16.71 | 1.392 |
| 300 | 19.295 | 1.693 |
| 300 | 24.018 | 2.075 |
The values of gas solubility in bitumen (Lal et al., 1999).
FIGURE 1
3 Modeling of dataset
3.1 Base models
A ML algorithm that can be used for regression tasks is called K-Nearest Neighbors (KNN) regression. The algorithm is non-parametric, meaning it does not presuppose anything about the distribution of the data and only consults the training set when it is actually being used. KNN regression relies on the idea that similar data points tend to have similar outcomes (Bishop, 2006; Trevor et al., 2009). To perform the modeling in this study, the dataset was randomly partitioned into training (70%) and testing (30%). The split was performed at the sample level to ensure that both subsets preserve the overall distribution of temperature, pressure, and solubility values. The training set was used for model fitting and hyperparameter optimization, while the testing set was reserved exclusively for final performance evaluation and was not accessed during training or tuning. Moreover, external dataset was used to validate the robustness of model and prove the lack of overfitting.
Finding the K nearest neighbors in the training set allows the KNN algorithm to predict the output value for a new data point. Before an algorithm can be trained, a hyperparameter—the value of K—must be selected (Kramer et al., 2013). The distance metric used in KNN regression can be the Euclidean distance, Manhattan distance, or cosine distance, among others. Euclidean distance is the most common metric used in KNN, and it can be calculated as follows (Taunk et al., 2019):
Lasso regression, also referred to as L1 regularization, is a statistical method utilized in linear regression models to prevent overfitting. It achieves this by imposing a penalty on the magnitude of the coefficients, thus shrinking them towards zero (Tibshirani, 1996).
The Lasso regression technique aims to optimize the sum of squared differences between the predicted and actual values, while introducing a constraint on the sum of the absolute magnitudes of the regression coefficients. The constraint is imposed through the addition of a penalty term, which is proportional to the L1 norm of the coefficient vector (Sill et al., 2014):
Minimize:
Subject to:where Y represents the vector of observed values, X stands for the design matrix, β denotes the vector of coefficients, t is the penalty parameter, and ||.||1 denotes the L1 norm.
Lasso regression can be solved using various optimization algorithms, such as coordinate descent, least angle regression, or proximal gradient descent.
Theil-Sen regression is a nonparametric technique for estimating the slope of a linear relationship between two variables. The Theil-Sen regression algorithm works by first computing the slope among all possible pairs of data points and then taking the median of all the slopes (Ohlson and Kim, 2015; Wilcox, 1998). The median of the y-intercepts of the estimated intercept passing along the lines of the x and y values is then used to calculate the intercept. Mathematically, the Theil-Sen estimator for the slope can be written as (Ohlson and Kim, 2015):where i and j are indices for any two distinct data points and median [.] denotes the median of a set of values (Rousseeuw and Croux, 1993).
3.2 Adaboost
Adaboost regression, or adaptive boosting regression, is a powerful ML algorithm that is useful for solving regression problems. It is based on the idea of boosting weak learners, which are models that have slightly better performance than random guessing. Adaboost regression combines these weak learners to form a strong learner that can make accurate predictions (
Freund and Schapire, 1997;
Schapire, 1990). Adaboost regression works by iteratively training a sequence of weak regression models on the data and then combining their predictions using a weighted sum. In each iteration, the algorithm assigns higher weights to the data points that were incorrectly predicted in the previous iteration, which causes the subsequent models to focus on the hardest examples (
Solomatine and Shrestha, 2004). The algorithm can be summarized as follows (
Bereta, 2019;
Solomatine and Shrestha, 2004):
Initialize the weights of each data point to be equal.
For each iteration t = 1, 2, … T:
Develop a weak regression model ht(x) on the data using the current weights.
Determine the errors on the training samples as:
Determine the weight of the model as:
Update the weights of the data points as:
Where Zt is a normalization factor that ensures that the weights sum to 1.
Combine the weak models by taking a weighted sum:
The ultimate result of the algorithm is obtained as a weighted sum of the weak models, where the weights are assigned based on their efficacy in predicting the output values of the training data.
Adaboost regression has several advantages over other regression algorithms. Firstly, it is able to handle nonlinear relationships between the input variables and the target variable. Secondly, it is relatively insensitive to overfitting, which means that it can generalize well to new data. Finally, it is computationally efficient and can handle large datasets. In this study, AdaBoost serves as the ensemble wrapper around each base regression model, while hyperparameters are optimized using the Harmony Search algorithm.
3.3 Model construction framework
The proposed modeling framework follows a hybrid optimization–ensemble pipeline designed to enhance predictive accuracy and robustness of the employed models. First, the dataset consisting of temperature (T), pressure (P), and hydrogen solubility (S) is randomly divided into training (70%) and testing (30%) subsets. The training subset is used for model fitting and Harmony Search-based hyperparameter optimization, whereas the testing subset is strictly held out and used only for final model evaluation. Prior to model training, no explicit feature transformation is applied, ensuring consistency across all base models.
In the first stage, three base regression learners are defined: K-Nearest Neighbors, Theil–Sen regression, and Lasso regression. Each base learner is embedded within an AdaBoost regression framework to improve weak prediction performance through iterative reweighting of training samples.
In the second stage, the Harmony Search (HS) algorithm is employed as a global hyperparameter optimization strategy. HS tunes critical model parameters such as the number of estimators, learning rate, and base model-specific hyperparameters (e.g., number of neighbors for KNN and regularization strength for Lasso). The optimization objective is to minimize prediction error on validation data.
The final hybrid model structure is constructed by combining HS-optimized parameter selection with AdaBoost-based ensemble learning. This results in three hybrid models: HSA-KNN, HSA-TS, and HSA-LAS, where HS performs global search over the hyperparameter space and AdaBoost enhances model stability through weighted ensemble learning.
The overall workflow ensures sequential optimization (HS) followed by ensemble refinement (AdaBoost), enabling improved generalization performance for hydrogen solubility prediction by the combined methodology. No information from the testing set was used during model training or optimization, ensuring an unbiased evaluation of model performance. Finally, external data (unseen) was used to further validate the model performance and check the overfitting due to the small size of dataset for H2 solubility.
4 Results and discussions
Table 2 shows the hyperparameters and performance metrics of three hybrid models. This table clearly shows the higher accuracy of the HSA-KNN model compared to the other two models. In addition, Figures 2–4 also show the comparison of the values predicted by the models with the actual values, which confirm the same fact, and validity of the tuned models for solubility correlation. The blue points represent the training data predictions, while the orange points represent the validation (test) data predictions. Similar trend has been reported by Jin et al. (2023a), Jin et al. (2023b) for modeling H2 solubility data. Therefore, in the following, we use the HSA-KNN model as the main and superior model of this research to evaluate and analyze the process and solubility variations with the input parameters.
TABLE 2
| Model | Base model hyperparameters | Hybrid model hyperparameters | R2 score | MSE | Max error |
|---|---|---|---|---|---|
| HSA-KNN | algorithm = 'ball_tree’, n_neighbors = 4 weights = 'distance' | learning_rate = 0.91211 loss = 'square' n_estimators = 82 | 0.9646 | 6.279E-03 | 1.804E-01 |
| HSA-TS | max_iter = 5 tol = 1.0706 | learning_rate = 2.2227 loss = 'linear' n_estimators = 438 | 0.9643 | 6.399E-03 | 1.407E-01 |
| HSA-LAS | alpha = 0.0355 tol = 0.007034 | learning_rate = 0.7498 loss = 'square' n_estimators = 385 | 0.8924 | 2.344E-02 | 2.221E-01 |
Hyperparameters and performance metrics of the base models and hybrid models.
FIGURE 2
FIGURE 3
FIGURE 4
The Learning Curve of HSA-KNN is illustrated in Figure 5 in which the training and cross-validation scores is shown as a function of the training set size. Based on this figure this model is not over-fitted or under-fitted on the dataset of this study (Jin et al., 2023b). Figure 6 displays the 3D surface of the model, which depicts the relationship between the pressure and temperature inputs and the corresponding output values. The figure highlights that as the pressure and temperature inputs increase, there is an increase in the output value (Jin et al., 2023a; Jin et al., 2023b). The same figures for two additional models can be found in the appendix section (See Supplementary Appendix Figures A1, A2).
FIGURE 5
FIGURE 6
The impact of both features on the output is further demonstrated in Figures 7, 8. The 2D figures are in agreement with the aforementioned observations derived from the 3D figure, indicating that an increase in both inputs leads to an increase in the output value (Jin et al., 2023a). The pressure enhancement can compress hydrogen and increases the solubility of hydrogen in the hydrocarbon sample which could be attributed to the enhanced intermolecular interactions between H2 and solvent molecules. Also, the temperature can enhance the reaction rate of H2 with the hydrocarbon sample which finally will increase the consumption of hydrogen and its mass transfer from gas towards the liquid phase. Also, the mass transfer rate and molecular diffusivity increase with rising temperature which leads to enhanced gas dissolution in the solvent phase (Li et al., 2024).
FIGURE 7
FIGURE 8
Finally, the optimized ML model is tested for estimation of H2 solubility for external dataset, and the results are illustrated in Figure 9 by comparing theoretical and measured data. The results are fitted using HSA-KNN model, which is the best model determined in this work to show the robustness and validity of the model for unseen dataset. The data have been collected from different sources for solubility of H2 in organic solvent (Ji et al., 2013). Given the small dataset size in this study, which is a limitation of current study, the external validation reveals great accuracy of the model, while no overfitting has been observed.
FIGURE 9
5 Conclusion
To summarize, this study has presented a novel approach that combines the Harmony Search Algorithm (HS) with AdaBoost to optimize and enhance the performance of three popular regression models: K-Nearest Neighbors (KNN), Theil-Sen, and Lasso. The models were optimized and applied for physical case study which is solubility of hydrogen in bitumen as the heavy hydrocarbon. The results demonstrate that the hybrid models, named HSA-KNN, HSA-TS, and HSA-LAS, outperform the base models in terms of accuracy, as evidenced by higher R2 scores, lower mean squared errors (MSE), and smaller maximum errors. The HSA-KNN model achieved the highest accuracy with an R2 score of 0.96466, MSE of 6.2790E-03, and maximum error of 1.80485E-01. The HSA-TS model achieved an R2 score of 0.96433, MSE of 6.3994E-03, and maximum error of 1.40763E-01. The HSA-LAS model achieved an R2 score of 0.89249, MSE of 2.3449E-02, and maximum error of 2.22157E-01. Both pressure and temperature were indicated to have direct relationship with the solubility of H2 in the hydrocarbon sample.
Statements
Data availability statement
The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding author.
Author contributions
ZZ: Methodology, Validation, Conceptualization, Investigation, Software, Writing – review and editing, Resources, Visualization, Formal Analysis, Writing – original draft.
Funding
The author(s) declared that financial support was not received for this work and/or its publication.
Conflict of interest
Author ZZ was employed by China National Aviation Fuel Group Corporation Limited, CNAF.
Generative AI statement
The author(s) declared that generative AI was not used in the creation of this manuscript.
Any alternative text (alt text) provided alongside figures in this article has been generated by Frontiers with the support of artificial intelligence and reasonable efforts have been made to ensure accuracy, including review by the authors wherever possible. If you identify any issues, please contact us.
Publisher’s note
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.
Supplementary material
The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fchem.2026.1876389/full#supplementary-material
References
1
AlpaydinE. (2020). Introduction to Machine Learning. MIT press.
2
BeretaM. (2019). Regularization of boosted decision stumps using tabu search. Appl. Soft Comput.79, 424–438. 10.1016/j.asoc.2019.04.003
3
BiltoM.UyarT. S. (2026). Hydrogen as a viable decarbonization enabler for critical sectors: a comprehensive review. Int. J. Hydrogen Energy251, 156025. 10.1016/j.ijhydene.2026.156025
4
BishopC. M. (2006). Pattern recognition. Mach. Learning62, 1–8.
5
CoverT. (1968). “Estimation by the nearest neighbor rule,” IEEE Trans. Inf. Theory, 14150–55. 10.1109/tit.1968.1054098
6
CoverT.HartP. (1967). “Nearest neighbor pattern classification,” IEEE Trans. Inf. Theory13121–27. 10.1109/tit.1967.1053964
7
FreundY.SchapireR. E. (1997). A decision-theoretic generalization of on-line learning and an application to boosting. J. Computer System Sciences55 (1), 119–139. 10.1006/jcss.1997.1504
8
GaoS.BoC.WangZ.YuG.GaoF.YangG.et al (2026). Dynamic multi-objective operation optimization of dimethyl oxalate hydrogenation process under green hydrogen flow fluctuation. Appl. Energy421, 128097. 10.1016/j.apenergy.2026.128097
9
HadavimoghaddamF.MohammadiM. R.AtashrouzS.NedeljkovicD.Hemmati-SarapardehA.MohaddespourA. (2022). Data-driven modeling of H2 solubility in hydrocarbons using white-box approaches. Int. J. Hydrogen Energy47 (78), 33224–33238. 10.1016/j.ijhydene.2022.07.238
10
HanD.LiQ.WangE.XieW.ChenG.ZhangQ.et al (2022). The evolution of NiMo unsupported catalysts with 3DOM structure for thiophene hydrodesulfurization. Catal. Today405-406, 329–336. 10.1016/j.cattod.2022.03.010
11
JiS.WangZ.GuoA.ZhouY.ChenK. (2013). Determination of hydrogen solubility in heavy fractions of crude oils by a modified direct method. J. Chem. and Eng. Data58 (12), 3453–3457. 10.1021/je400729v
12
JinH.JinZ.KimY. G.FanC. (2023a). Intelligence-based simulation of solubility of hydrogen in bitumen at elevated pressure and temperature: models optimization using fireworks algorithm. J. Mol. Liq.390, 122948. 10.1016/j.molliq.2023.122948
13
JinH.JinZ.KimY. G.FanC. (2023b). Development of machine learning-based solubility models for estimation of hydrogen solubility in oil: models assessment and validation. Case Stud. Therm. Eng.51, 103622. 10.1016/j.csite.2023.103622
14
KalamS.ArifM.RazaA.LashariN.MahmoudM. (2023). Data-driven modeling to predict adsorption of hydrogen on shale kerogen: implication for underground hydrogen storage. Int. J. Coal Geol.280, 104386. 10.1016/j.coal.2023.104386
15
KramerO.and K-nearest neighbors (2013). Dimensionality Reduction with Unsupervised Nearest Neighbors, 13–23.
16
LalD.OttoF.MatherA. (1999). Solubility of hydrogen in Athabasca bitumen. Fuel78 (12), 1437–1441. 10.1016/s0016-2361(99)00071-x
17
LashariN.GanatT.OtchereD.KalamS.AliI. (2021). Navigating viscosity of GO-SiO2/HPAM composite using response surface methodology and supervised machine learning models. J. Petroleum Sci. Eng.205, 108800. 10.1016/j.petrol.2021.108800
18
LiD.YadavA.AliE.AlbaghdadiM. F.BainsP. S.NaeemY. A.et al (2024). Advanced machine learning computations for estimation of hydrogen solubility in oil samples: model comparisons and validation. Case Stud. Therm. Eng.60, 104684. 10.1016/j.csite.2024.104684
19
Morales-ValenciaE. M.Vargas-MontañezO. J.Monroy-GarcíaP. A.Avendaño-BarónL. G.Quintero-QuinteroE. A.Elder-BuenoC.et al (2021). Conditions for increasing the hydrodesulfurization of dibenzothiophene when co-feeding naphthalene, quinoline, and indole. J. Catal.404, 204–209. 10.1016/j.jcat.2021.09.021
20
OhlsonJ. A.KimS. (2015). Linear valuation without OLS: the Theil-Sen estimation approach. Rev. Account. Stud.20 (1), 395–435. 10.1007/s11142-014-9300-0
21
RanstamJ.CookJ. (2018). LASSO regression. J. Br. Surg.105 (10), 1348. 10.1002/bjs.10895
22
Rodríguez CastilloL. J.Bello BarreraJ. I.Morales-OrtuñoJ. C.KlimovaE. I.de Ita LozadaV. J.Puente LeeI.et al (2023). Hybrid alumina-1D titania nanotube materials as supports for NiMo catalysts with improved activity in hydrodesulfurization. Chem. Phys. Lett.813, 140300. 10.1016/j.cplett.2023.140300
23
RousseeuwP. J.CrouxC. (1993). Alternatives to the median absolute deviation. J. Am. Stat. Association88 (424), 1273–1283. 10.1080/01621459.1993.10476408
24
SafariA.Vesali-NasehM. (2019). Design and optimization of hydrodesulfurization process for liquefied petroleum gases. J. Clean. Prod.220, 1255–1264. 10.1016/j.jclepro.2019.02.226
25
SchapireR. E. (1990). The strength of weak learnability. Mach. Learning5 (2), 197–227. 10.1007/bf00116037
26
ShindeP. P.ShahS. (2018). “A review of machine learning and deep learning applications,” in 2018 Fourth International Conference on Computing Communication Control and Automation (ICCUBEA) (IEEE).
27
SillM.HielscherT.BeckerN.ZucknickM. (2014). c060: Extended Inference for Lasso and Elastic-Net Regularized Cox and Generalized Linear Models; R Package Version 0.2-4.
28
SolomatineD. P.ShresthaD. L. (2004). “AdaBoost. RT: a boosting algorithm for regression problems,” in 2004 IEEE International Joint Conference on Neural Networks (IEEE Cat. No. 04CH37541) (IEEE).
29
TaunkK.DeS.VermaS.SwetapadmaA. (2019). “A brief review of nearest neighbor algorithm for learning and classification,” in 2019 International Conference on Intelligent Computing and Control Systems (ICCS) (IEEE).
30
TibshiraniR. (1996). Regression shrinkage and selection via the lasso. J. R. Stat. Soc. Ser. B Methodol.58 (1), 267–288. 10.1111/j.2517-6161.1996.tb02080.x
31
TrevorH.RobertT.JeromeF. (2009). The Elements of Statistical Learning: Data Mining, Inference, and Prediction. Spinger.
32
WilcoxR. (1998). A note on the theil-sen regression estimator when the regressor is random and the error term is heteroscedastic. Biometrical J. J. Math. Methods Biosci.40 (3), 261–268. 10.1002/(sici)1521-4036(199807)40:3<261::aid-bimj261>3.0.co;2-v
33
WuQ.LiY.HouZ.XinJ.MengQ.HanL.et al (2018). Synthesis and characterization of Beta-FDU-12 and the hydrodesulfurization performance of FCC gasoline and diesel. Fuel Processing Technology172, 55–64. 10.1016/j.fuproc.2017.12.003
34
YangX.-S. (2009). “Harmony search as a metaheuristic algorithm,” in Music-Inspired Harmony Search Algorithm: Theory and Applications, 1–14.
Summary
Keywords
harmony search algorithm, hydrogenation, hyper-parameter tuning, machine learning, modeling
Citation
Zhao Z (2026) Modeling and optimization of hydrogenation for crude oil by estimating hydrogen solubility in the solvent at different temperatures. Front. Chem. 14:1876389. doi: 10.3389/fchem.2026.1876389
Received
08 May 2026
Revised
04 July 2026
Accepted
06 July 2026
Published
11 August 2026
Volume
14 - 2026
Edited by
Utpal Sarkar, Assam University, India
Reviewed by
Najeeb Ullah, University of Technology Petronas, Malaysia
Ahmed Salah Al-Shati, Kut University College, Iraq
Updates
Copyright
© 2026 Zhao.
This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Zhongtian Zhao, m352qs1231538@163.com
Disclaimer
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.