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Predicting hydrogen storage in MOFs via machine learning

Summary The H2 capacities of a diverse set of 918,734 metal-organic frameworks (MOFs) sourced from 19 databases is predicted via machine learning (ML). Using only 7 structural features as input, ML identifies 8,282 MOFs with the potential to exceed the capacities of state-of-the-art materials. The identified MOFs are predominantly hypothetical compounds having low densities (<0.31 g cm−3) in combination with high surface areas (>5,300 m2 g−1), void fractions (∼0.90), and pore volumes (>3.3 cm3 g−1). The relative importance of the input features are characterized, and dependencies on the ML algorithm and training set size are quantified. The most important features for predicting H2 uptake are pore volume (for gravimetric capacity) and void fraction (for volumetric capacity). The ML models are available on the web, allowing for rapid and accurate predictions of the hydrogen capacities of MOFs from limited structural data; the simplest models require only a single crystallographic feature.


In brief
The adoption of hydrogen as a lowcarbon fuel has been slowed by the low energy density of H 2 gas. Hydrogen adsorption in MOFs presents a pathway for storing hydrogen at the densities desired for mobile applications, such as fuel cell vehicles. Nevertheless, identifying a suitable MOF remains a challenge because the number of MOFs is essentially limitless. To accelerate the discovery of high-capacity hydrogen adsorbents, machine learning models are developed to predict hydrogen uptake across a diverse set of MOFs.

INTRODUCTION
Hydrogen (H 2 ) is considered to be a future automotive fuel. [1][2][3][4][5][6] This potential reflects its high specific energy compared with competing fuels, such as natural gas and gasoline, and the ability of H 2 to be produced renewably and consumed without CO 2 emissions. 2,7 Nevertheless, the adoption of hydrogen in mobile applications, such as fuel cell (FC) vehicles has been limited by its low volumetric energy density. 2,6,7 Consequently, the design of low-cost H 2 storage systems that overcome these volumetric limitations has been the focus of recent research. 4,[8][9][10][11][12] At present, FC vehicles employ storage systems based on gaseous H 2 compressed to pressures up to 700 bar. 13 This approach is costly and can incur limitations in driving range. 7,11,13,14 Storage based on adsorption in porous hosts is an alternative to high-pressure compression. 15 Due to their high gravimetric densities, fast kinetics, and reversibility, metal-organic frameworks (MOFs) have emerged as one of the most promising THE BIGGER PICTURE The efficient storage of hydrogen fuel remains a barrier to the adoption of fuel cell vehicles. Although many storage technologies have been proposed, adsorptive storage in metal-organic frameworks (MOFs) holds promise due to the low operating pressures, fast kinetics, reversibility, and high gravimetric densities typical of MOFs. Nevertheless, the volumetric storage densities of known MOFs are generally low; hence, new MOFs with improved volumetric performance are desired. Identifying optimal MOFs remains a challenge, however, because relatively few MOFs have been characterized experimentally, and the building-block structure of MOFs suggests that the number of possible materials is limitless. To accelerate the discovery process, this study develops machine learning models that predict the hydrogen capacity of MOFs. The models identify promising materials, clarify structure-property relations, and can be used-on the web or through an API-to predict the performance of new MOFs.
Machine learning (ML) could provide a path forward. [62][63][64][65] For ML to be helpful, access to high-quality training data is essential.
Unfortunately, training on experimental H 2 storage data in MOFs is non-trivial 1,2,6,66-68 : experimental uptake data are generally restricted to a relatively small number of MOFs, and can depend sensitively upon the experimental conditions and the purity of the sample. 2,67,69 Employing a dataset based on a consistent set of computational predictions may be a better choice. 62,63 Earlier work has demonstrated that accurate isotherms for H 2 uptake in MOFs can be predicted using the pseudo-Feynman-Hibbs potential (to describe H 2 ) combined with general interatomic potentials to describe the MOF. 1,2,6,68 This approach was used to screen a database of 5,309 real MOFs, from which IRMOF-20 was identified and experimentally demonstrated to have a favorable balance of high gravimetric and volumetric H 2 density. 2 In a follow-on study, a larger database of 495,305 MOFs was compiled from several publicly available databases (see Table S1 for details). 1,29,31,33,[36][37][38][39][40]45 Following a pre-screen based on crystallographic properties and empirical correlations, the H 2 capacities of a subset of 43,777 MOFs were evaluated using GCMC. Three additional MOFs-SNU-70, UMCM-9, and PCN-610/NU-100-were identified and shown experimentally to out-perform the leading MOF candidate, IRMOF-20. 1 The database of MOF properties 70 generated in these previous studies presents an opportunity to develop ML models that can predict H 2 uptake across even larger MOF datasets. 1,70 43 Expanding upon these previous reports, this study applies ML to explore a large database of 918,734 known and proposed MOFs. The database was assembled from a diverse collection of publicly available MOF repositories, 1,29,31,33,34,[36][37][38][39][40][41][42][43][44][45]81,82 and allows for a wide-ranging and consistent assessment of H 2 uptake in MOFs.
Here, the extremely randomized trees (ERT) 76,83 algorithm was identified as the most accurate ML model for predicting H 2 up-take. A training set comprising 24,674 MOFs was sufficient to enable accurate predictions of usable capacities across 820,039 unseen compounds. 70 These predictions were made using a small set of seven crystallographic features as input: single-crystal density, pore volume, gravimetric and volumetric surface area, void fraction, largest cavity diameter, and pore limiting diameter. Importantly, ML identified 8,282 MOFs-8,187 appropriate for pressure swing (PS) operation and 95 for temperature-PS (TPS) use-with the potential to exceed both the gravimetric and volumetric capacities of state-of-the-art materials. These compounds are comprised predominantly of hypothetical MOFs, and exhibit low densities (<0.31 g cm À3 ) in combination with high surface areas (>5,300 m 2 g À1 ), void fractions ($0.90), and pore volumes (>3.3 cm 3 g À1 ). In addition to identifying high-capacity MOFs, the relative importance of the input features is quantified; dependencies on the ML algorithm and training set size and are also assessed. The most important features for predicting H 2 uptake are pore volume (for gravimetric capacity) and void fraction (for volumetric capacity). A simplified model using only two input features is demonstrated to predict capacities with high accuracy-within 0.2 wt % and 1.4 g-H 2 L À1 of more expensive Monte Carlo calculations. The ML models are available for use via the web, 84 allowing for rapid and accurate predictions of hydrogen capacities with only a small amount of structural data required as input.
A previous study examined a subset of the present database, wherein the hydrogen uptake in 495,305 MOFs was estimated using the Chahine rule. 1,2,70 Subsequently, usable uptake in a portion of this subset comprising 43,777 MOFs predicted to be promising based on the Chahine rule was evaluated using GCMC. This GCMC-evaluated dataset contained a mix of real and hypothetical MOFs: 15,235 real MOFs were sourced from the UM, 31 CoRE, 33 and CSD, 29,30 and 28,542 hypothetical MOFs were extracted from the mail-order, 38 in silico deliverable, 46 in silico surface, 39 MOF-74 analogs, 40 ToBaCCo, 59 Zr-MOFs, 45 Northwestern, 36 University of Ottawa, 37,85,86 and in-house 1 hypothetical MOF databases (see Ahmed et al. 1 or Table S1 for details). 1,29,31,33,[36][37][38][39][40] Hydrogen uptake isotherms for two operating conditions were predicted: for an isothermal PS at T = 77 K between 5 and 100 bar, and for a combined Martin et al. 38 mail-order 38 112 Bao et al. 46 in silico deliverable 46 2,816 Bao et al. 39 in silico surface 39 8,885 Witman et al. 40 MOF-74 analogs 40 61 Coló n et al. 59 ToBaCCo 59 13,512 Gomez-Gualdron et al. 45 Zr-MOFs 45 204 Wilmer et al. 36 Northwestern 36 137,000 Aghaji et al., 37 Boyd et al. 85 Anderson et al. 42 CSM-2018-I 42 117 Anderson et al. 43 CSM-2018-II 43 32 Anderson et al. 44 CSM-2019-I 44 99 Ahmed et al. 1 in-house 1 1 and Table S1 for further details). These additional MOFs were selected at random from the 495,305-entry HyMARC database and therefore represent a more diverse sampling of the MOF property space. To this dataset, 423,429 additional compounds were added from 7 additional datasets: BJT (Beijing, Jiangsu, Tianjin) MOFs, 81 R-WLLFHS, 41,87 MTV, 82 CSM-2018-I, 42 CSM-2018-II, 43 and CSM-2019-I, 44 and selected MOFs from the CSD 2017 dataset. 29,30 Subsequently, the capacities of the MOFs from these additional datasets were predicted by the ML models without retraining (i.e., no MOFs from these datasets were used for training or testing, and none of their isotherms were evaluated in advance with GCMC). In total, the dataset employed in this study contains H 2 uptake data for 98,695 MOFs 70 and crystallographic property data for 918,734 MOFs.
The present dataset includes approximately 74,000 MOFs having open metal sites (OMS), comprising roughly 8% of the total dataset. As the interatomic potential used in our GCMC calculations is not tuned to capture the unique aspects of the H 2 -OMS interaction, it is possible that the calculated capacities for this class of MOFs will be less accurate. Figure S1 and Table S3 compare experiments and the present GCMC calculations of H 2 capacities across a benchmark set of OMS MOFs discussed by Garcia-Holley et al. 88 and in our previous work. 1 These data show that GCMC calculations using the pseudo-Feynman-Hibbs potential are in good agreement with experimental data for these OMS MOFs. The good agreement between theory and experiments is a consequence of the low temperature operating conditions used in our study, combined with the relatively low density of OMS in these MOFs.
The goal of the ML models is to predict four output properties: UG and UV for each of PS and TPS operating conditions. This was accomplished by developing separate ML models for each of the four targeted capacities. Figure S3 illustrates the overall work flow.
The existing dataset of 98,695 MOFs (for which both crystallographic and capacity data are available) 70 was initially split into training and test sets of 74,201 and 24,674 MOFs, respectively, after shuffling the entire dataset. 104 ML algorithms 90,93,94,104 (Table 3) were implemented using the Scikit-learn library. 104 Both scaled and unscaled features were used in training ML models. Ten-fold cross-validation was used to optimize the hyperparameters of each model. The performance of the ML algorithms was assessed by comparing the predicted H 2 capacities with the capacity predicted by GCMC for the MOFs in the test set. The metrics used for the performance assessment of ML models were the R 2 , AUE, RMSE, MAE, and t. Additional details regarding these calculations can be found in supplemental note S2 of the supplemental information. Dataset size An obstacle to wider adoption of ML in materials science is the availability of sufficient quantities of high-quality training data. 110,111 Unfortunately, it is not yet clear how much data are needed to construct a useful ML model for a given system. Fernandez et al. 72 found that a reasonable balance between accuracy (R 2 $ 0.85-0.93) and computational expense for predicting methane storage in MOFs was achieved for a training set containing data on 10,000 MOFs with 3 features. In contrast, Fanourgakis et al. 112 showed that a much smaller training set of $1,000 MOFs was sufficient to predict methane uptake when using six crystallographic features and four fictitious features. The Table 3. Machine learning regression algorithms employed in this work different training set sizes required in these previous studies arise from the differing numbers and types of features used.
This study explores this issue further by systematically examining the effect of training set size, and the training set to test set ratio, on ML accuracy. For each of the four targeted capacity outputs, 100 independent ML models were developed by varying the size of the training set between 100 and 74,000 MOFs (see Table S5 for a list of the training set sizes). The four best-performing ERT ML algorithms identified earlier were used with 10-fold cross-validation. The resulting models were assessed using a common test set of 24,674 MOFs. Feature importance/selection The well-known Chahine rule proposes a linear correlation between gravimetric surface area and excess gravimetric H 2 capacity in adsorbents. 113,114 Nevertheless, the Chahine rule overpredicts H 2 capacities for MOFs with high surface areas, 114 and has not been extended to predict usable capacities. 1,2,6 Hence, a model for predicting H 2 uptake that is more general than the Chahine rule, yet requires limited input data, would be very helpful. In principle, ML could be used to generate such a predictive model if the features that are the most important for predicting H 2 uptake could be identified. Along these lines, Pardakhti et al. reported improved accuracy in predicting CH 4 adsorption when using a combination of (7) crystallographic and (19) chemical features. 71 Recently, Moosavi et al. explored feature importance in predicting the synthesis of MOFs. 115 This study determines the minimum number and optimal combination of crystallographic features necessary to achieve a specified accuracy in predicting H 2 uptake. The relative importance of the input features was assessed for all possible univariate and multivariate feature combinations using ERT ML models. The number of multivariate feature combinations, M, is given by: Mðn tot ; n sub Þ = ntot ! nsub!ðntot ÀnsubÞ! , where n tot = 7 is the total number of available features, and 1 % n sub % 7 is the number  Table 3.
of features used as input to a given ML model. A total of 127 feature combinations are possible. ML models were developed for each of these feature combinations for each of the 4 output capacities, resulting in a total of 508 distinct ML models. All models were trained using a dataset of 74,021 MOFs and tested on a common set of 24,674 MOFs. Ten-fold cross-validation was used for tuning and validating the models using only the training set. Univariate feature importance was further assessed using (1) Pearson's correlation coefficient (r), [116][117][118] (2), Breiman and Friedman's tree-based algorithm as implemented in Scikit-learn, 90,104 and (3) the permutation importance method as implemented in rfpimp package. 119 Additional details regarding these methods can be found in Figure S7.

Evaluating ML algorithms
Tables S6-S9 illustrate the effect of several feature scaling methods on the performance of the ML algorithms examined here. Only the SVM family of models (SVM/L-K, SVM/RBF-K, and Nu-SVM/RBF-K) 76,90,96,98,99,104 were impacted by the choice of scaling method. Figure 1 compares the accuracy of the ML algorithms for predicting hydrogen uptake in MOFs. Coefficient of determination (R 2 ) and average unsigned error (AUE) were used as performance metrics. SVM variants were trained using min-max feature scaling; unscaled features were used in training the remaining models. The performance of the algorithms as measured by four additional metrics-root-mean-square error (RMSE), explained variance (EV), median absolute error (MAE), and Kendall rank correlation coefficient (t)-is reported in Tables S6-S9. Overall, these data indicate that the tree-based ensemble methods are superior to the other methods examined. In particular, the ERT 76,83,104 algorithm exhibited the best performance overall. Boosted decision trees, 76,90-92,102,104 random forest, 76,94,104 and Bagging algorithm variants 76,93,104,120,121 (with tree-based base estimators) are nearly as accurate. The R 2 values for ERT predictions exceed 0.997 for gravimetric capacities, which are equivalent to errors of $0.14 wt %. Volumetrically, the accuracy of the ERT algorithm is slightly worse than its gravimetric performance: R 2 = 0.967-0.984, equivalent to errors of $1.1 g-H 2 L À1 on average. In general, the worst-performing algorithms were linear regression, ridge regression, and SVM with linear kernel. For these algorithms R 2 varies between 0.913 and 0.992 depending on the conditions (i.e., gravimetric/volumetric and PS/TPS). As expected, the linear nature of these algorithms fails to fully capture the nonlinear dependence of output capacities on the multiple input features. Figure 1 also shows that all the algorithms tested yield more accurate predictions of usable gravimetric (UG) capacities compared with those for usable volumetric (UV) capacities. Likewise, all algorithms more accurately predict usable capacities under PS conditions than under TPS conditions. This reflects the fact that the functional relationships between output capacities (UG/UV) and input features under PS and TPS conditions are likely different, as was observed in previously reported structure(feature)-property(capacity) relationships. 1,6,122 Table 4 summarizes the performance of the ERT algorithm in further detail. A comparison of Tables 1 and 4 indicates that the accuracy of the present ML models surpass previously reported models for H 2 uptake. Furthermore, the present models also appear to be an improvement over earlier models that aim to predict the adsorption capacities of MOFs for any gas species, Table S2. This improved performance can be attributed to the exploration and optimization of multiple ML algorithms, use of an appropriate feature set, and the relatively large size of the present training set. Figure 2 illustrates the degree of agreement between ERT ML predictions and GCMC calculations of usable H 2 capacities under PS conditions as a function of MOF source database (Figure S4 shows similar data for TPS conditions; see also Table  4). As mentioned above, the present ML models more accurately predict UG capacities than UV capacities. The largest differences between ML and GCMC capacities ( Figures 2C, 2F, S4C, and S4F) primarily occur for the real MOF dataset. In principle, these differences may arise either from ML overfitting or from inaccurate GCMC predictions caused by non-ideal/incomplete MOF crystal structure data (i.e., missing atoms, disorder, etc.), as mentioned in previous studies. 1,32,35,[123][124][125] ERT algorithms are fairly robust against overfitting. 83 To examine the possibility for overfitting, test set errors were compared with training set errors, as shown in Figure S5 and Table 4. These data suggest that the outliers are not a consequence of over fitting; hence, inaccuracies in the crystal structure data are proposed as the most likely source of this disagreement. 1,32,35,[123][124][125] Effect of training set size Figure 3 illustrates the impact of training set size on the accuracy of the ERT ML models, as quantified using R 2 and AUE (Table S5 summarizes the dataset sizes used in these plots). For training sets containing more than 5,000 MOFs, R 2 and AUE vary slowly and in a monotonic fashion, with AUE decreasing and R 2 increasing. The accuracy of the models is more sensitive to the size of the training set for smaller training sets containing roughly 5,000 or fewer MOFs. Figure S6 highlights the variation in performance for these smaller training sets.
The trends AUE as a function of training set size can be fit to a power law expression of the form AUE(m) = am b + g, wherem represents the size of the training set and b is the power law exponent. Fitting this model to the data shown in Figure 3 reveals that the AUE for UG converges faster with training set size (b = À0.37 and À0.43) than it does for UV (b = À0. 16 and À0.23). A full tabulation of the power law parameters is given in Table S10. Based on these power law expressions, one can determine the necessary size of the training set to achieve a desired level of accuracy. For example, assuming PS operation, to achieve an AUE of approximately 0.25 wt % and 1.5 g-H 2 L À1 requires training set sizes (for UG and UV) of less than 300 MOFs randomly selected from the diverse datasets used here.
Univariate feature importance Figure 4 illustrates the relative importance of the seven crystallographic features in predicting usable hydrogen uptake in MOFs. Feature importance was determined by developing ERT models for each single feature individually. Additional details for these models are provided in the supplemental information. Based on these models, it is evident that pore volume (pv) and void fraction (vf) are the dominant features in predicting H 2 capacity; these two properties appear as the first-or second-most important single features regardless of operating condition or capacity type. The importance of these features can be rationalized by two factors. First, based on the empirical Chahine rule, the pore volume of an MOF correlates with its excess uptake. 113 Second, pore volume and void fraction are related (since pv = vf d À1 )-MOFs with larger pv have larger vf, and vice versa. 1 Conversely, the largest cavity diameter (lcd) and volumetric surface area (vsa) are the single features whose ML models yield the lowest accuracy. The relative importance of the individual features for predicting UG capacities is: pv > d > vf > gsa > pld > lcd > vsa. This ordering is the same for PS and TPS conditions. In contrast, the importance ordering for UV capacities differs based on the operating condition. Nevertheless, vf and pv remain the two most important single features for both UV conditions, in that order (Figure 4).
Despite their limited input, the single-feature ML models illustrated in Figure 4 achieve high accuracy. For example, any of the

OPEN ACCESS
Article three independent models for UG-PS based only on pv, d, or vf can predict capacities with R 2 > 0.95 and with AUE of less than 0.5 wt %. The accuracy and simplicity of the univariate ML models suggest that they can be used to quickly screen new MOFs for their utility in hydrogen storage. To that end, optimized single-feature ML models for the four categories of usable capacities considered here have been made available for use on the web with an interactive web form or with a python API. 84 Furthermore, the ML models can be downloaded via figshare. 126 These models take as input either pv (for UG predictions) or vf (for UV predictions) of a given MOF. These input data can be quickly calculated from a MOF's crystal structure using modern structure analysis codes. 25,47,[127][128][129][130] As shown in Figure 4, these models can predict UG with an average error of less than 0.4 wt %, and UV with errors less than 2.2 g-H 2 L À1 . Figure S7 compares the single-feature importance assessments based on ERT ML models (as reported in Figure 4) with three popular methods for determining feature importance: Pearson's correlation coefficient (r), [116][117][118] Breiman and Friedman's tree-based algorithm as implemented in Scikit-learn, 90,104 and the permutation importance method as implemented in the rfpimp package. 119 It is clear that the feature importance methods do not reproduce in detail the rank ordering of feature importance that is suggested by our ERT ML models. Nevertheless, good agreement is evident more broadly. For example, in the case of UG ( Figures S7A and S7C), the three feature importance methods suggest that in aggregate pv is the most important feature, while vsa is the least, in agreement with the ERT models ( Figures 4A and 4B). Similarly, for UV, the importance methods suggest that vf and lcd are among the most and least important features, respectively. This is the same trend found in the univariate ERT models (Figures 4C and 4D). Figure 5 illustrates how the accuracy of the ML models varies with the number and combination of features. Assuming 7 features, 2 7 -1 = 127 possible combinations exist. For a given number of features, Figure 5 plots the combination of features resulting in the highest accuracy model. (The supplementary  file [Table S11] summarizes the performance for all 508 possible feature combinations and capacity/operating condition types.) As expected, Figure 5 shows that ML accuracy generally increases as the number of input features increases. As previously discussed, when limited to a single feature, vf yields the best accuracy for predicting UV, while pv is the best choice for UG. When the feature set is extended to 2 features, the combination of d and pv is the optimal choice among the 7 2 = 21 possible pairs regardless of the capacity (UG versus UV) or operating condition (PS versus TPS). For larger numbers of features, the optimal feature combination depends upon the operating condition and the capacity type. Based on the AUE, whose value tends to plateau as more features are added, highly accurate ML models can be generated using only 5 input features (Table 5). These data lend further support to the notion that the accuracy of a given ML model depends on both the number and identity of the input features. As a slightly more accurate alternative to the univariate web models described above, a subset of the present multivariate ML models that use 4, 5, and 7 input features are also available on the web using an interactive web form and via a python API. 84 The ML models can also be downloaded via figshare. 126 H 2 uptake in unseen MOFs Figure 6 illustrates the H 2 storage capacities of 820,039 MOFs as predicted by the 7-feature ERT ML models developed here. (This dataset is publicly accessible via HyMARC data hub. 70 ) These MOFs are referred to as ''unseen'', in that they have not been included in the training or test sets used to develop the models. Figures 6A and 6B show UV capacities as functions of UG capacities under PS and TPS conditions, respectively. Both plots exhibit a rapid increase in UV at low values of UG, and reach a maximum in UV at UG values of approximately 9 wt %. Beyond the maximum, UV decreases relatively slowly with increasing UG. These trends are consistent with our earlier findings derived from GCMC calculations on smaller datasets. 1,2,6 In the case of PS operation, the maximum UV across the MOFs in the dataset is 37.4 g-H 2 L À1 ; for TPS operation the maximum UV is 48.5 g-H 2 L À1 . In the case of UG, the maximum value predicted is 39 wt % for PS operation and 42 wt % for TPS. These values can be placed in context by comparing against the Department of Environment hydrogen storage targets, which stipulate system-level hydrogen densities of 5.5 wt % and 40 g-H 2 L À1 by 2025 and 6.5 wt %/50 g-H 2 L À1 longer-term (''Ultimate target''). 6 Given that the tank and balance-of-plant for the storage system have non-zero mass and volume, the MOFs examined here cannot meet the Ultimate target for UV, regardless of operating condition. 12 More optimism exists, however, for meeting the gravimetric targets given the high UG exhibited by these systems on a MOF-only basis. Of course, an additional challenge is to identify MOFs that excel both gravimetrically and volumetrically. 1,2,6,31,131 It is also helpful to compare the performance predictions in Figures 6A and 6B with that of state-of-the-art materials. In the case of PS operation, our previous study demonstrated that PCN-610 (NU-100) exhibits a hydrogen capacity of 10.1 wt % and 35.5 g-H 2 L À1 , 1 which, to our knowledge, is the best combination of gravimetric and volumetric capacities  Figure 6B), MOF-5 remains the benchmark, which a measured capacity of 7.8 wt % and 51.9 g-H 2 L À1 . 2 Figure  6D shows that only 21 MOFs out-perform MOF-5 under these conditions. Regarding the accuracy of the present ML predictions, Table 4 shows that the AUE of these models are on the order of 0.15 wt % and 1.3 g-H 2 L À1 . Although these errors are small, a more rigorous validation of the ML can be achieved with GCMC calculations. Thus, GCMC calculations were performed on a subset of MOFs that ML predicted to exhibit high UV and UG capacities. These MOFs fall within the rectangular regions shown in Figures 6A  and 6B, and exhibit capacities that meet or exceed 36 g-H 2 L À1 and 7.5 wt % for PS conditions and 48 g-H 2 L À1 and 7.5 wt % under TPS conditions. In total, 21,700 compounds were re-examined with GCMC based on their ML-predicted PS capacities, and another 7,901 were re-examined for TPS. Figure 6C compares ML and GCMC predictions for usable capacities for 21,700 high-capacity MOFs under PS conditions. The strong overlap in the two datasets further highlights the accuracy of the ML models. A total of 8,187 MOFs were predicted by GCMC to out-perform PCN-610/NU-100 under these conditions. A summary of the 10 highest-capacity MOFs, sorted based on their GCMC capacities, is provided in Table 6 (a more extensive listing is provided in Table S12). The highest-capacity MOFs are all hypothetical compounds: five originate from the ToBaCCo database, 59 two are from the University of Ottawa database, 37

capacities in MOFs
The accuracy of ERT ML models, as determined by R 2 and AUE, was determined as a function of the number and combination of input features. Each data point represents the most accurate feature combination for a given number of features. ERT models were trained on a dataset of 74,201 MOFs. R 2 and AUE were calculated using a test of 24,674 MOFs. Feature abbreviations are defined in Figure 4.
0.89, on average. The range of these property values are consistent with those reported in an earlier study, 1,132,133 and suggest that maximizing the surface area is an important design guideline for PS operation. The highest-capacity MOF, mof_7642, 59 is predicted to exhibit capacities of 11.1 wt % and 40.5 g-H 2 L À1 , surpassing that of PCN-610/NU-100, the record-holder under PS conditions. The crystal structure of mof_7642 is shown in Figure 7A.
A search in the CCDC 134 was performed to identify MOFs that have been synthesized that are similar to the high-capacity compounds identified here. The existence of similar MOFs may suggest synthetic procedures that could be adapted to the present systems. The top 5 MOFs under PS conditions contain relatively long tritopic linkers. In the case of mof_7642, this search identified the interpenetrated MOF RANCEQ 135 as having a similar index of 0.82. Interpenetration is fairly common in MOFs (such as mof_7642) with longer linkers, and is generally undesirable for achieving high uptake. Nevertheless, several examples of successful synthesis of MOFs with long, multi-topic linkers that do not undergo interpenetration, have been reported. These include MOF-180 and MOF-200, 136 the PCN-6X series, 137 and NOTT-112. 138 The next four PS candidates in Table 6 exhibit pillared Zn paddlewheel clusters with long ditopic linkers. Karagiaridi et al. 139 demonstrated the feasibility of synthesizing pillared paddlewheel MOFs with long linkers; the SALEM-X series are examples. 139 Finally, str_m3_o5_o20_f0_nbo.sym.1.out is based on a Zn paddlewheel cluster and a ditopic linker. HOFSUS (CSD Refcode) is an example of such a MOF. 140 Figure 6D provides a similar comparison between ML predictions and GCMC calculations for MOFs expected to exhibit high capacities under TPS conditions. Under these conditions, only 95 MOFs were predicted by GCMC to out-perform MOF-5. A summary of the 10 highest-capacity MOFs, sorted by their GCMC capacities, is provided in Table 6 (see Table S13 for a more extensive tabulation). As found for PS operation, all of the top performing candidates are hypothetical compounds. One difference with the PS case is that all of these MOFs originate from the University of Ottawa database. 37 Furthermore, none of the highest-capacity MOFs identified for PS operation appear as top candidates for TPS. Comparing the highest-capacity MOFs for both operating conditions, it can be seen that the high-capacity TPS MOFs systematically exhibit lower surface areas (average = 4,073 m 2 g À1 ), smaller void fractions (average = 0.83), and higher densities. Hence, the categories of MOFs that maximize uptake under PS and TPS conditions exhibit distinct properties. These differences suggest that maximizing the surface area-which, as discussed above, is desirable for maximizing PS capacity-is not advantageous for TPS operation. This behavior can be explained by trends in total capacities, 6 which the TPS capacities reported here approximate. More specifically, it is known that total volumetric capacities are maximized for intermediate values of the surface area; for larger surface areas the volumetric capacity decreases.
Returning to the list of promising MOFs for TPS operation, Table  6 reports that the highest-capacity MOF, str_m1_o1_o11_f0_pcu.sym.102.out, has a GCMC-predicted capacity of 10.4 wt % and 53.1 g-H 2 L À1 . This capacity surpasses that of MOF-5, which, to our knowledge, holds the capacity record under these conditions. The crystal structure of this MOF is shown in Figure 7B.
The top 10 MOFs under TPS conditions contain the same Zn metal cluster and terephthalic acid linkers, where the linkers have been modified with varying functional groups. The slight differences in the capacities of these MOFs can be traced to differences in the functional groups. A similarity search based on str_m1_o1_o11_f0_pcu.sym.117.out identified 40 similar MOFs. Approximately 30 of these (for example, HIFTOG, MIB-QAR, UNIGEE, VUSJUP, and ZELROZ) contain Zn metal clusters and linkers based on variants of terephthalic acid. Figures S8 and S9 and Table S14 quantify the differences between ML and GCMC predictions on the subset of high-capacity MOFs shown in Figures 6C and 6D. For PS operation, the AUE of ML relative to GCMC is 0.24 wt % and 0.66 g-H 2 L À1 , while for TPS the AUE is 0.24 wt % and 1.28 g-H 2 L À1 . Both sets of errors are comparable with the errors reported in Table 4 for the original test set of MOFs. Figures  S8C and S8F and S9(c,f) plot the frequency distribution of the differences between GCMC and ML. These distribution plots suggest that the largest differences occur for predictions involving real MOFs and for hypothetical MOFs extracted from databases other than those from Northwestern, 36 University of Ottawa, 37 and BJT. 81 (These MOFs are referred to as ''other hypothetical MOFs'' in Figure 6). These MOFs, along with the real compounds, exhibit higher structural diversity than those contained in the other databases. For example, the diversity of the topologies used in the ToBaCCo 59 and Zr-MOFs 45 databases and in the linkers used in MTV-MOF 82 database are larger than what is found in the databases from Northwestern, 36 University of Ottawa, 37 and BJT. 81

DISCUSSION
Limitations of this study As described previously, some of the high-capacity MOFs identified here may prove difficult to synthesize. Although this limitation applies primarily to the hypothetical MOFs, in some cases real MOFs are also known to undergo framework collapse during activation, which would reduce capacity. 1,2 Nevertheless, future improvements to synthesis techniques may overcome these limitations-what is difficult to make today may be possible in the future. Secondly, our models do not distinguish between realistic MOFs having non-defective crystal structures and those for which the structures are defective/unrealistic. Unrealistic structures can result from incomplete or imperfect virtual solvent removal and the presence of partial occupancies or symmetry disorder in the crystal structure. 31 Consequently, a defective/unrealistic MOF could be erroneously predicted to be a promising candidate. Follow-up calculations using GCMC and visual inspection of the crystal structure are recommended for all promising candidates identified by ML. Finally, the ML models developed here are non-interpretable, ''black-box'' models. Although these models are demonstrated to be highly accurate, additional effort is required to assess the relative importance of their input data. (The approach demonstrated here for evaluating feature importance involved the development of multiple models with varying numbers and combinations of features.) Alternatively, interpretable white-box ML models could be developed to provide more insight into feature importance. However, our experience suggests that white-box models generate less accurate predictions.

Concluding remarks
The H 2 storage capacities of nearly a million MOFs have been predicted via ML. The predictions span a diverse collection of MOFs sourced from 19 databases and reveal performance under two operating conditions: PS and temperature + PS. More than a dozen ML algorithms were benchmarked, with the ERT method found to be the most accurate. The resulting ML models are accessible on the web at the HyMARC data hub. 84 These models allow for accurate, rapid screening of the hydrogen stor- age properties of new MOFs using minimal structural data as input; only a single feature is needed for the simplest models. The accuracy of the ML models was characterized as a function of training set size and the number/combination of input features. Regarding the dependence on the training set, the accuracy of the models can be well described using a simple power law function of the training set size. The dependence on the number and combination of input features was determined by evaluating 508 independent ML models generated from all possible combinations of the seven features. The most important features for predicting H 2 uptake are pore volume (for gravimetric capacity) and void fraction (for volumetric capacity).
Using these models, 8,282 MOFs are identified that have the potential to exceed the capacities of state-of-the-art materials under usable conditions. The identified MOFs are predominantly hypothetical compounds, which (for PS operation) exhibit low densities (<0.31 g cm À3 ) in combination with high surface areas (>5,300 m 2 g À1 ), void fractions ($0.90), and pore volumes (>3.3 cm 3 g À1 ). These MOFs are suggested as targets for experimental synthesis.