Prediction of Hydrogen Adsorption and Moduli of Metal–Organic Frameworks (MOFs) Using Machine Learning Strategies

Abstract

For hydrogen-powered vehicles, the efficiency cost brought about by the current industry choices of hydrogen storage methods greatly reduces the system’s overall efficiency. The physisorption of hydrogen fuel onto metal–organic frameworks (MOFs) is a promising alternative storage method due to their large surface areas and exceptional tunability. However, the massive selection of MOFs poses a challenge for the efficient screening of top-performing MOF structures that are capable of meeting target hydrogen uptakes. This study examined the performance of 13 machine learning (ML) models in the prediction of the gravimetric and volumetric hydrogen uptakes of real MOF structures for comparison with simulated and experimental results. Among the 13 models studied, 12 models gave an R² greater than 0.95 in the prediction of both the gravimetric and the volumetric uptakes in MOFs. In addition, this study introduces a 4-20-1 ANN model that predicts the bulk, shear, and Young’s moduli for the MOFs. The machine learning models with high R² can be used in choosing MOFs for hydrogen storage.

1. Introduction

Hydrogen (H₂) is of particular interest in research due to the rising interest in clean energy as a response to the progression of climate change. As vehicle fuel consumption constitutes a significant percentage of fossil fuel use and, in turn, global carbon emissions, considerable interest has been invested in developing hydrogen as an energy carrier for powering light-duty, fuel cell-powered vehicles with energy derived from cleaner sources [1,2]. However, while fuel cell technology has been around for some time, the major obstacle hampering its widespread use remains the lack of efficient and economically sound storage and transportation methods for hydrogen [3,4]. The widely adopted physically based storage methods require high pressures or cryogenic temperatures, making them expensive and difficult to maintain [5,6]. The US Department of Energy (DOE) sets technical targets for on-board hydrogen storage parameters to accelerate the development of a robust hydrogen storage method. The latest target set for 2025 is a usable energy density of 1.3 kWh L⁻¹, corresponding to 0.040 kg H₂ L⁻¹ [7]. As such, highly porous materials that readily facilitate physisorption, such as carbon materials, nanocomposites, and MOFs, are potential alternatives for storage that allow energy-efficient storage that is capable of achieving densities comparable to those of the liquid state. Furthermore, MOFs offer highly favorable kinetics that enable rapid adsorption and desorption, allowing the easy storage and use of H₂ [8,9].

Metal–organic frameworks (MOFs) are porous materials that are known for their exceptional tunability. They are composed of organo-metallic hybrid bonds with metal joints serving as the primary building units (PBUs) and organic ligands or linkers serving as secondary building units (SBUs). The precise selection of the constituents allows flexibility in the modification of the physical and chemical features, in addition to general properties such as large surface areas and high porosity [10]. As such, the development of MOFs and the enhancement of their properties has been a primary research focus for further applications in gas adsorption and separation [11,12,13], storage [14,15,16], and catalysis [17,18,19,20]. However, the infinite combinations resulting from MOF tuning make the efficient screening and performance evaluation for MOFs with appropriate properties that would fit each application challenging in terms of material resources, workforce, and time [21].

In line with this, machine learning (ML) strategies have been introduced to address high-throughput screening and performance evaluation efficiency. ML refers to a subset of artificial intelligence applications that utilize computing systems capable of working through historical data using pattern-based techniques to solve problems; subsequently, the solutions to the problems are applied to the judgment, association, and prediction of possible scenarios [21]. It is important to note that a critical factor in controlling the performance of these ML strategies and defining the exploration space of the approach is the judicious representation of compounds in the dataset [22,23,24,25,26]. These representations are referred to as features or descriptors. In the field of MOFs, ML strategies have been widely employed in the emerging fields of MOF adsorption applications for carbon dioxide (CO₂), methane (CH₄), and hydrogen (H₂), as shown in

Table 1. Summary of studied modeling gas adsorption in MOFs using ML.

Year	Application	Machine Learning Method	Best Performing Algorithm	Ref.
2016	CH4 storage	Gaussian Process Regression (GPR)	-	[1]
2016	CH4 storage	Principal Component Analysis (PCA), Gaussian Process (GP), Support Vector Regression (SVR), Neural Network (NN), Linear Regression (LR)	GP	[1]
2017	H2 storage	Neural Network (NN)	-	[2]
2017	CH4 storage	Decision Trees (DT) Poisson Support Vector Machines (SVMs), Random Forest (RF)	RF	[26]
2018	CO2 capture, CO2/N2 and CO2/H2 separation	Multilinear Regression (MLR), Gradient Boosting Machines (GBMs), NN, DT, RF	GBM	[3]
2018	CO2/N2/CH4 separation	PCA, MLR, DT	-	[4]
2019	H2 storage	Artificial Neural Networks (ANNs)	-	[7]
2019	CO2/H2 separation	Gradient Boosted Regression Trees (GBRTs)	-	[5]
2019	CH4 storage	RF, DT, SVM	RF	[6]
2020	CO2 capture from air	Back Propagation Neural Network (BPNN), DT, SVM, RF	RF	[27]
2021	H2 storage	Extremely Randomized Trees	-	[9]

.

Many research and development efforts have been made to identify and develop MOF structures capable of meeting DOE hydrogen storage efficiency targets. On this front, the application of ML strategies that enable high-throughput screening has become highly beneficial in the search for benchmark structures. In a 2021 study, Ahmed et al. utilized extremely randomized trees to screen for MOFs that met the 2021 DOE target uptake capacities across a combined database that comprised an unprecedented quantity of MOF structures, numbering around 820,000 [9]. In addition, many other studies have the same objective of predicting H₂ with a wide variety of ML algorithms. Studies also opt to utilize molecular simulation techniques, most commonly the grand canonical Monte Carlo simulations, to procure the uptakes of thousands of MOFs under similar conditions, which experimental studies cannot provide.

The main objective of this study is to examine and compare the predicted outputs of machine learning algorithms and experimental and simulation uptake values for a set of MOFs. In addition, the study aims to include the prediction of mechanical stability parameters like the bulk modulus. To achieve this, the researchers acquired the open-access database created by Ahmed and Siegel [9], which comprised MOF structures and their properties relevant to hydrogen uptake capacity. This database is a combination of 19 existing databases. The details can be found in [9]. Afterwards, the predictive performances of 13 different ML algorithms were evaluated relative to the simulated and experimental values.

Also, another publicly available database created by Moghadam et al. [28], comprising the bulk and shear moduli of 3384 MOFs containing 41 unique network topologies obtained through density functional theory (DFT) calculations, was used to predict the bulk modulus of MOFs using an artificial neural network.

The prediction of the hydrogen adsorption performance of MOFs and the subsequent prediction of mechanical stability moduli is relevant to the furthering of the efforts towards the identification of materials suitable for the physisorption and storage of H₂ in industrial applications as MOFs are often synthesized in powder form and require pressure-driven post-synthetic shaping before downstream applications [29]. Furthermore, this study’s findings may contribute to the further development of ML strategies in the cost-effective and time-efficient performance predictions of both existing and hypothetical MOFs.

2. Methods

2.1. Data Gathering

For the first part of the study, the researchers identified and retrieved 98,695 MOFs from a dataset made publicly available by Ahmed and Siegel [9]. This dataset contains real and hypothetical MOFs that were collated from 11 databases. Their physical properties, gravimetric uptake, and volumetric uptake under isothermal pressure swing (PS) conditions were calculated using the grand canonical Monte Carlo (GCMC) simulation. The PS condition was specified to be isothermal at 77 K, with the pressure ranging from 5 to 100 bar. From this dataset, only 29,608 data points from real MOFs were considered in this study. On the other hand, for the second part, the database for the prediction of the bulk, shear, and Young’s moduli that consists of 3384 MOFs was obtained from the pioneering study of Moghadam et al. on the structure and mechanical stability relations [28].

Lastly, to assess the accuracy of the predicted uptakes from ML, the researchers compared the results with the experimental data collected from a study by Garcia-Holley et al. [30], using MSE as the primary metric for accuracy. These experimental data were chosen because this is the only study that reported gravimetric and volumetric H₂ uptakes for the same PS specified by Ahmed and Siegel [9]. Only 6 out of the 14 MOFs examined by Garcia-Holley et al. were found in Ahmed and Siegel’s database.

2.2. Data Normalization

The data were normalized using the Min–Max method. This method serves as a pre-processing step to keep the values within a specific range and to avoid significant variations.

$$x_{i,normalized}=\frac{x_i-x_i^{min}}{x_i^{max}-x_i^{min}}$$

(1)

2.3. Machine Learning Models

2.3.1. Hydrogen Uptake Prediction

Using the data gathered, 13 machine learning methods with preset hyperparameters using MATLAB 2021b Regression Learner were tested. Five input variables were used in predicting the gravimetric uptake (UG) and volumetric uptake (UV) of hydrogen in 29,608 MOF structures. The schematic diagram of the prediction using different machine learning models is given in Figure 1.

The five input variables and their units are enumerated in

Table 2. Input variables used in predicting H₂ update in MOF.

Feature	Unit
Crystal density ($\rho$)	g cm−3
Pore volume (PV)	cm3 g−1
Void fraction (VF)	-
Pore Limiting Diameter (PLD)	Å
Largest Cavity Diameter (LCD)	Å

.

The presets used for each model are enumerated in

Table 3. Hyperparameters and preset values used in MATLAB regression learner.

ML Algorithm	Hyperparameters and Presets Used
DT (Fine Tree)	Minimum leaf size = 4, Surrogate decision splits = Off
Random Forest	Minimum leaf size = 8, Number of learners = 30
Boosted Trees	Minimum leaf size = 8, Number of learners = 30, Learning rate = 0.1
Quadratic SVM	Kernel function = Quadratic, Kernel scale = Automatic, Box constraint = Automatic, Epsilon = Automatic, Standardized data = True
Linear SVM	Kernel function = Linear, Kernel scale = Automatic, Box constraint = Automatic, Epsilon = Automatic, Standardized data = True
Exponential GPR	Basis function = Constant, Kernel function = Exponential, Use isotropic kernel = True, Kernel scale = Automatic, Signal standard deviation = Automatic, Sigma = Automatic, Standardize = True, Optimize numeric parameters = True
Matern 5/2 GPR	Basis function = Constant, Kernel function = Matern 5/2, Use isotropic kernel = True, Kernel scale = Automatic, Signal standard deviation = Automatic, Sigma = automatic, Standardize = True, Optimize numeric parameters = True
Rational Quadratic GPR	Basis Function = Constant, Kernel function = Rational Quadratics, Use isotropic kernel = True, Kernel scale = Automatic, Signal standard deviation = Automatic, Sigma = Automatic, Standardize = True, Optimize numeric parameters = True
Bi-layered Neural Network	Fully connected layer = 2, First layer size = 10, Second layer size = 10, Activation = ReLU, Iteration limit = 1000, Regularization strength (lambda) = 0, Standardized data = Yes
Tri-layered Neural Network	Fully connected layer = 3, First layer size = 10, Second layer size = 10, Third layer size = 10, Activation = ReLU, Iteration limit = 1000, Regularization strength (lambda) = 0, Standardized data = Yes
Narrow Neural Network	Fully connected layer = 1, First layer size = 10, Activation = ReLU, Iteration limit = 100, Regularization strength (lambda) = 0, Standardized data = Yes
Medium Neural Network	Fully connected layer = 1, First layer size = 25, Activation = ReLU, Iteration limit = 1000, Regularization strength (lambda) = 0, Standardized data = Yes
Wide Neural Network	Fully connected layer = 1, First layer size = 100, Activation = ReLU, Iteration limit = 1000, Regularization strength (lambda) = 0, Standardized data = Yes

. For all the models, the ratio of the training to the testing used was 7:3. To prevent overfitting, cross-validation folds of 5 were used in all the models. In MATLAB, the model referred to as bagged tree is also the random forest model. Ahmed and Siegel pointed out the shortage of examined algorithms for H₂ storage; they then addressed this shortage with the introduction of 14 different algorithms in their 2021 study. Similarly, this study opted to introduce and comparatively study algorithms not addressed in the study mentioned above. These algorithms include the quadratic support vector machine (QSVM), the Gaussian process regression (GPR) family of algorithms, and multi-layered neural networks (NNs).

2.3.2. Modulus Prediction

The bulk modulus was predicted using an artificial neural network that used four input variables, as enumerated in

Table 4. Input variables used in predicting modulus of MOF.

Feature	Unit
Crystal density ($\rho$)	g cm−3
Void fraction (VF)	-
Largest cavity diameter (LCD)	Å
Gravimetric surface area (GSA)	m2 g−1

. The schematic diagram for the prediction is given in Figure 2.

This ANN was trained using the Levenberg–Marquardt algorithm. The training to the testing ratio used was 7:3.

2.4. Model Performance Evaluation

The performances of the ML models were evaluated by comparing the predicted values to the GCMC simulated values and the experimental values. The following performance indicators were used: the mean squared error (MSE), coefficient of determination (R²), root mean square error (RMSE), and mean absolute error (MAE). Equation (2) shows the equation for the determination of the MSE. The MSE shows the average squared difference between the predictions and the observations that were fed to the algorithm.

Mean squared error (MSE)

$$MSE=\frac{1}{N} \sum _{t=1}^N{\left(x_t^{Actual}-x_t^{predicted}\right)}^2$$

(2)

Root mean square error (RMSE)

$$RMSE=\sqrt{\frac{1}{N} {\sum }_{t=1}^N{\left(x_t^{Actual}-x_t^{predicted}\right)}^2}$$

(3)

The RMSE defines the deviation between the simulated value and the actual value. It is a quantitative trade-off method. However, it becomes challenging to utilize the RMSE in the measurement of the model efficacy with differing dimensions. In this case, the coefficient of determination (R²) is also utilized to determine how well the predicted results match the actual values.

Coefficient of determination (R²)

$${\mathrm{R}}^2={\left(\frac{\mathrm{N}\sum x_t^{Actual}{x}_t^{Predicted}-(\sum x_t^{Actual}\left)\right(\sum x_t^{Predicted})}{\sqrt{\left[N\sum x_t^{{\left(Actual\right)}^2}-{(\sum x_t^{Actual}) }^2\right]\left[N\sum x_t^{{\left(Predicted\right)}^2}-{(\sum x_t^{Predicted}) }^2\right]}}\right)}^2$$

(4)

Mean absolute error (MAE)

$$MAE=\frac{1}{N}\sum _{t=1 }^N\left|x_t^{Actual}-x_t^{predicted}\right|$$

(5)

3. Results and Discussion

3.1. Gravimetric and Volumetric Uptakes at PS Data

The gravimetric and volumetric uptakes at the PS in the dataset used in the machine learning modeling are represented with respect to the input variables using scatter plot matrices, as shown in Figure 3 and Figure 4. The scatter plot matrices were generated using R Studio.

The most important plots in these scatterplots are the plots in the last column. These are the pairwise relationships of the input variables with UG and UV. Some pairwise relationships are almost linear with some variability, like PV vs. UG and PLD vs. UG. All the others are nonlinear.

3.2. Evaluation of ML Models

Table 5. Test RMSE for gravimetric (UG) and volumetric (UV) uptakes at PS.

ML Algorithm	UG				UV
	MS	RMSE	MAE	R2	MSE	RMSE	MAE	R2
DT (Fine Tree)	0.23736	0.4872	0.24973	0.98	4.2562	2.0631	1.4417	0.97
Random Forest (RF)	0.31249	0.5590	0.22022	0.98	3.1772	1.7825	1.2571	0.97
Boosted Trees (BT)	0.31756	0.5635	0.27857	0.98	4.3197	2.0784	1.6089	0.96
Quadratic SVM (QSVM)	0.23116	0.4808	0.25987	0.98	6.3876	2.5274	1.5874	0.95
Linear SVM (LSVM)	2.0368	1.4272	0.5023	0.84	25.894	5.0886	3.0743	0.79
Exponential GPR (GPR-E)	0.081152	0.2849	0.16338	0.99	2.8525	1.6889	1.15	0.98
Matern 5/2 GPR (GPR-M)	0.11929	0.3454	0.15802	0.99	2.8764	1.6960	1.1226	0.98
Rational Quadratic Kernel GPR (GPR-RQK)	0.11649	0.3413	0.15935	0.99	2.9242	1.7100	1.1298	0.98
Bi-layered NN (Bi-NN)	0.076928	0.2774	0.16577	0.99	2.9854	1.7278	1.1848	0.98
Tri-layered NN (Tri-NN)	0.056056	0.2368	0.16005	1.00	3.5125	1.8742	1.1972	0.97
Narrow NN (N-NN)	0.075871	0.2755	0.1635	0.99	2.9605	1.7206	1.1706	0.98
Medium NN (M-NN)	0.067059	0.2590	0.16678	0.99	3.0895	1.7577	1.1802	0.97
Wide NN (W-NN)	0.07361	0.2713	0.17357	0.99	1.9125	1.9125	3.6575	0.97

shows the comparison of the RMSE values for all the ML algorithms evaluated in this study. The RMSE is used as the primary performance metric for comparison due to the fact that the presentation of R² values in MATLAB is restricted to two decimal points, leading to rounded values that are too close for comparison and even to values that are the same. For predicting the gravimetric uptakes, the best performing algorithm was found to be the tri-NN across all the performance metrics, with an RMSE value of 0.23676, closely followed by the GPR family of algorithms with RMSE values ranging from 0.285 to 0.345 for the UG uptake prediction. As for the prediction of the volumetric uptakes, the tri-NN lagged behind the GPR family of algorithms, with an RMSE of 1.874 compared to the GPR algorithm results, which ranged from 1.688 to 1.71. This result suggests that while the tri-NN suits the data for UG uptake prediction, it fails to perform well for UV uptakes.

The same trend was shown across the board, with the RMSE values differing widely between the UV and UG uptake datasets per algorithm. However, this was shown to be a notable trend across all the algorithm performances between the UV and UG uptake predictions, with the RMSE values for UV being much higher. The algorithms used all exhibited a better fit for the UG data, implying that the features used were much more relevant for the gravimetric uptakes than the UV uptakes under PS conditions and that the relationship between the input features and the output UG and UV uptakes were largely different. In any case, the data shown further reinforce the fact that the optimal choice of ML is largely problem-specific.

In general, the worst performing algorithm is the LSVM, with an RMSE value of 1.427 for UG and 5.089 for UV. This result follows the findings of Ahmed and Siegel [9], wherein the same model performed the worst among the 14 algorithms they examined. They attributed the low performance to the linear nature of the algorithms failing to fully grasp the nonlinear dependence of the multiple inputs and outputs.

3.3. H₂ Uptake Prediction

To verify the reliability and accuracy of the predicted uptakes, the researchers compared the results to the experimental data taken from the benchmark study on H₂ storage for MOFs under the PS conditions conducted by Garcia-Holley et al. [30], the predicted uptakes of extremely randomized trees (ERTs) used by Ahmed and Siegel [9], and the GCMC simulated results from Ahmed and Siegel. Furthermore, the accuracy of the models used in this study was compared relative to the results of the published ERT and GCMC data and experimental values.

Table 6. Comparison of H₂ gravimetric uptake under PS conditions obtained through different methods.

MOF	Usable Gravimetric Uptake (UG)
	Exp. [30]	GCMC [9]	ERT [9]	Fine Tree	RF	GPR-E	LSVM	QSVM	Tri-NN	M-NN	N-NN
HKUST-1	2.00	2.19	2.37	1.99	5.13	1.71	2.09	1.93	1.73	1.78	1.63
NU-125	4.10	3.76	3.78	4.06	3.73	4.09	4.05	4.01	3.83	3.59	3.73
rht-MOF-7	1.80	2.30	3.00	2.50	2.63	2.80	3.05	2.81	2.57	2.51	2.64
PCN-250	1.80	1.81	1.53	1.99	1.62	1.60	1.85	1.65	1.61	1.62	1.54
CYCU-Al	5.50	4.4	5.1	5.13	5.47	5.14	5.87	5.47	4.81	4.83	4.72
Zn2(bdc)2 (dabco)	1.60	1.1	1.48	1.77	1.59	1.42	1.87	1.59	1.46	1.42	1.45

shows the gravimetric uptakes under PS conditions obtained through the models that showed improvement compared to the ERT and GCMC data reported by Ahmed and Siegel relative to the experimental values. Low-performance algorithms are omitted from the table.

Figure 5 shows a graphical representation of the MSE values calculated for the simulated GCMC data and ML predictions relative to the experimental UG values denoted in Table 6. The top-performing algorithm was found to be the fine tree DT variant, with an average MSE of 0.1105. To first address the discrepancy between the GCMC and predicted values, this is carried over from Ahmed and Siegel’s study [9], wherein inaccuracies in the MOF crystal structure data were cited as possible points for errors in the GCMC simulations, which directly contributed to the disagreement between the predicted and GCMC values. The remarkable results shown by the fine tree relative to the experimental values, despite the training conducted with the GCMC simulated values, are considerable and suggest that the model fit compensates for the errors incurred in the GCMC simulation.

From the data, it is evident that while the DT algorithm had a low score in the performance metrics for the prediction of the UG and UV uptakes in the previous section, it was able to yield values that were relatively close to the experimental benchmarks, with an MSE that was 64.5% lower than the ERT values and 63.2% lower than the GCMC values. This difference suggests that when given unseen data, this model may fare better at predicting values closer to the actual experimental values, despite the fact that the raw performance metrics were less than desirable.

Moreover, several algorithms were able to improve upon the ERT and GCMC performance in predicting the UG uptake values when compared to the experimental values. This result shows that the generated models in this study, which are easily reproducible as they require less rigorous programming, can compete with the published models in terms of accuracy.

As for UV uptake prediction, while several models improved upon the ERT performance relative to the experimental values, only the W-NN showed considerable improvement compared to both the ERT predicted values and GCMC values, as shown in

Table 7. Comparison of H₂ volumetric uptake under PS conditions obtained through different methods.

MOF	Usable Volumetric Uptake (UV)
	Exp. [30]	ERT [9]	GCMC [9]	W-NN [This Work]
HKUST-1	17	23.4	21.04	17.13
NU-125	24	25.93	25.6	19.21
rht-MOF-7	14	23.76	20	20.39
PCN-250	16	15.2	17.45	16.75
CYCU-Al	27	25.84	24.21	26.89
Zn2(bdc)2(dabco)	14	15.03	10.99	14.33

and

Table 8. MSE for volumetric uptakes (UV) under PS conditions relative to experimental values.

Method	MSE
ERT	23.8
GCMC	12.3
W-NN	10.7

. The W-NN predicted values with an average MSE that was less than that of the ERT values and GCMC values at 10.74. This improvement is shown in the 54% reduction in MSE relative to ERT and the 12.74% reduction relative to GCMC. Lastly, like the observations in the preceding section, the MSE values for the UV uptakes are greater across the board than the UG MSE values.

3.4. Univariate Feature Importance

The importance of each descriptor in the ML algorithm’s predictive ability was evaluated by simulating a tri-layered NN model for UG and an exponential GPR model for UV that was developed with a single descriptor. The same ratio of 70:30 was used, splitting the data for the training and testing of these models.

Based on Figure 6, it can be seen that the void fraction and pore volume are the two most important features that can effectively predict the MOFs’ H₂ adsorptive capacity. For UG, the most crucial feature was PV, followed by density and VF. On the other hand, VF was followed by PV in terms of importance for UV. These results mirror the results reported by Ahmed and Siegel [9], who credited the significant role of pore volume to the Chahine rule. In his previous study in 2017, Ahmed [31] found that both the PV and the VF are highly correlated with the excess uptake obtained through the Chahine rule. This study by Ahmed further verifies the significance of both descriptors to the prediction.

The hierarchy of importance for UG is as follows: PV > Density > VF > PLD > LCD. Meanwhile, the order of importance of each feature for UV prediction is as follows: VF > PV > Density > PLD > LCD. From this sequence, it can be determined that LCD was the most irrelevant feature to the prediction for both types of uptakes. Therefore, a model that was developed using only the LCD of an MOF may yield inaccurate predictions. Also, if the LCD values for the MOFs are not available within the database, one may still proceed with the conviction that the model will perform well if one possesses values for the other features, such as those used in this study or other more relevant available predictors.

On the other hand, with PV and VF being able to single-handedly produce the same accuracy as that of the model that uses all the features, it can be deduced that using these two features may prove to be enough to create a reliable and accurate model. This situation makes the model creation and data gathering procedure more straightforward and less intensive since the data for these two descriptors are usually readily available in the literature and the crystallographic data found in the databases.

3.5. Modulus Prediction

Among the ML algorithms used to predict the bulk modulus, only the created ANN was able to exceed the R² reported by Moghadam et al. [28]. This study utilized a 4-20-1 ANN, or 4 neurons for the input layer, 20 hidden neurons for the single hidden layer, and 1 for the output layer. For easier visualization, the flow of information for the simulated ANN is shown in Figure 2.

Table 9. Comparison between the performances of ANNs in predicting the bulk modulus.

ML Algorithm	Number of Hidden Neurons	R-Squared
ANN [this work]	20	0.761
ANN [28]	30	0.696

compares the R² values of the simulated ANN and the ANN used in the literature. The only difference between the construction of the two ANNs was the number of hidden neurons used. The results show that the algorithm’s performance exceeds the neural network (NN) used by Moghadam et al. to predict the bulk modulus. A 9.339% increase in the coefficient of determination (R²) was observed in the simulated ANN model. This increase may be attributed to the difference in the number of hidden neurons. Shibata and Ikeda explain that the greater the number of hidden neurons present, the greater the probability of similar hidden neurons, worsening the learning performance [32]. When the learning performance worsens, it is expected that the accuracy of the predicted value compared to the actual or calculated value will be negatively affected. This concept could explain why the fit of the predicted data for the ANN in [28] is lower than this work’s ANN.

The corresponding R² for the model indicates that the ANN could explain around 76% of the variance and that the variables used may still be quite inadequate in their ability to predict the modulus as accurately as possible [33]. However, one cannot simply add variables until the R² becomes one because it could result in a highly complex model that might produce an overfit to the data and possibly yield poor predictions when it encounters unseen data.

Among the other algorithms, it was also the 4-20-1 ANN that gave the most accurate prediction for the shear and Young’s moduli, respectively. Figure 7 contains the scatter plots for the predicted and calculated moduli. From the graphs, it can be noted that the ANN gave the highest R² for the bulk modulus, which indicates that among the three moduli, the predicted bulk moduli are quite close to the bulk moduli that were computed using DFT. However, the accuracy for the shear and Young’s moduli is not very far behind.

4. Future Work

As the use of hydrogen as a renewable fuel becomes increasingly viable in the future, it is important to continuously collect more data on the hydrogen uptake in different adsorbent materials, like other nanomaterials. It is also important to try different methods, like the fuzzy method [34], probabilistic methods [35,36], and deep learning methods, to increase the accuracy of the prediction of the bulk modulus of MOFs.

5. Conclusions

In this study, the researchers were able to successfully predict the gravimetric and volumetric hydrogen uptake of more than 20,000 MOFs that were obtained from several databases. Among the 13 algorithms comparatively studied, the tri-NN and GPR/E-K exhibited the best performance metrics for the UG and UV uptake prediction, respectively. The GCMC simulated and model predicted values’ reliability was verified by determining the MSE values relative to the experimental data. Afterwards, the MSE values calculated for the models utilized in this study were compared to the MSE values for GCMC and the ERT predicted values derived from Ahmed and Siegel’s study. From the comparisons made, it was found that the DT variant (fine tree) and W-NN models were able to compete with the GCMC simulated values and the ERT predicted values in terms of accuracy relative to the experimental values for the UG and UV uptakes, respectively, under PS conditions, despite their inherent simplicity. These results suggest that the top-performing models utilized in this study are viable for the prediction of the H₂ storage capacities for MOFs before their synthesis to check whether the MOF can store H₂ as needed by the situation.

Furthermore, the single feature importance of each of the five variables was analyzed by the generation of a tri-layered NN for UG and an exponential GPR for UV that only used a single descriptor. Through this, it was found that the pore volume and the void fraction of the MOF were indispensable yet simple features that could match the prediction fit provided by using all the features. This result simplifies the ML process and can be applied whenever there are insufficient readily available data to retrieve all the five features used in this study.

Lastly, the researchers were also able to predict the bulk, shear, and Young’s moduli of certain MOFs that were in the database of Moghadam et al. [28]. Upon using the same algorithms as those used in the uptake prediction, it was found that the 4-20-1 ANN performed the best in the prediction of the moduli. The prediction of such moduli will be relevant to the furthering of the efforts towards identifying MOFs suitable for the physisorption of hydrogen in industrial applications.