Original: Molecular Energy Learning Using Alternative Blackbox Matrix-Matrix Multiplication Algorithm for Exact Gaussian Process
Revised: Molecular Energy Learning Using Alternative Blackbox Matrix-Matrix Multiplication Algorithm for Exact Gaussian Process

Original: Anonymous Author(s)

Original: AfﬁliationAddress email

Original: Abstract
Revised: Abstract

Original: We present an application of the blackbox matrix-matrix multiplication (BBMM)algorithm to scale up the Gaussian Process (GP) training of molecular energiesin the molecular-orbital based machine learning (MOB-ML) framework.
Revised: We present an application of the blackbox matrix-matrix multiplication (BBMM) algorithm to scale up the Gaussian Process (GP) training of molecular energies in the molecular-orbital based machine learning (MOB-ML) framework.

Original: An alternative implementation of BBMM (AltBBMM) is also proposed to train moreefﬁciently (over four-fold speedup) with the same accuracy and transferability asthe original BBMM implementation.
Revised: An alternative implementation of BBMM (AltBBMM) is also proposed to train more efﬁciently (over four-fold speedup) with the same accuracy and transferability as the original BBMM implementation.

Original: The training of MOB-ML was limited to220 molecules, and BBMM and AltBBMM scale the training of MOB-ML up byover 30 times to 6500 molecules (more than a million pair energies).
Revised: The training of MOB-ML was limited to 220 molecules, and BBMM and AltBBMM scale the training of MOB-ML up by over 30 times to 6500 molecules (more than a million pair energies).

Original: The accuracyand transferability of both algorithms are examined on the benchmark datasets oforganic molecules with 7 and 13 heavy atoms.
Revised: The accuracy and transferability of both algorithms are examined on the benchmark datasets of organic molecules with 7 and 13 heavy atoms.

Original: These lower-scaling implementations of the GP preserve the state-of-the-art learning efﬁciency in the low-dataregime while extending it to the large-data regime with better accuracy than otheravailable machine learning works on molecular energies.
Revised: These lower-scaling implementations of the GP preserve the state-of-the-art learning efﬁciency in the low-data regime while extending it to the large-data regime with better accuracy than other available machine learning works on molecular energies.

Original:  Introduction
Revised:  Introduction

Original: Machine-learning (ML) for quantum chemistry has emerged as a versatile method in chemicalsciences in recent years, facilitating innovation in a variety of domains such as molecular modeling [4, 17, 20], drug discovery [6, 14, 18] and material design [9, 12, 15, 19] by delivering accuratepredictions at a low computational cost.
Revised: Machine-learning (ML) for quantum chemistry has emerged as a versatile method in chemical sciences in recent years, facilitating innovation in a variety of domains such as molecular modeling [4, 17, 20], drug discovery [6, 14, 18] and material design [9, 12, 15, 19] by delivering accurate predictions at a low computational cost.

Original: One successful example of such methods is the recently developed molecular-orbital-based machine learning (MOB-ML)
Revised: One successful example of such methods is the recently developed molecular-orbital-based machine learning (MOB-ML)

Original: [3, 13, 24] which can predict molecular energies with a high degree of accuracy while requiring relatively low amounts of training data.
Revised: [3, 13, 24] which can predict molecular energies with a high degree of accuracy while requiring relatively low amounts of training data.

Original: Nevertheless, extending MOB-ML to the large data regime has remained challenging due to thesteep O ( N 3 ) complexity scaling associated with the use of Gaussian Process (GP) regression.
Revised: Nevertheless, extending MOB-ML to the large data regime has remained challenging due to the steep O ( N 3 ) complexity scaling associated with the use of Gaussian Process (GP) regression.

Original: Astrategy to reduce the complexity of GP is to introduce a low-rank kernel approximation, whichhas been exploited in the Sparse Gaussian Process Regression [21] and Stochastic Variational Gaussian Processes [11] methods.
Revised: A strategy to reduce the complexity of GP is to introduce a low-rank kernel approximation, which has been exploited in the Sparse Gaussian Process Regression [21] and Stochastic Variational Gaussian Processes [11] methods.

Original: However, such treatments of GP sometimes result in signiﬁcant lossof accuracy.
Revised: However, such treatments of GP sometimes result in signiﬁcant loss of accuracy.

Original: In contrast, Gardner et al.
Revised: In contrast, Gardner et al.

Original: [7, 23] recently proposed the blackbox matrix-matrix multiplication (BBMM) method, which provides exact GP inference while reducing the training timecomplexity to O ( N 2 ) and allowing for multi-GPU usage.
Revised: [7, 23] recently proposed the blackbox matrix-matrix multiplication (BBMM) method, which provides exact GP inference while reducing the training time complexity to O ( N 2 ) and allowing for multi-GPU usage.

Original: In this work, we employ BBMM and a novel alternative implementation for BBMM (AltBBMM)to speedup and scale the GP training in MOB-ML for molecular energies.
Revised: In this work, we employ BBMM and a novel alternative implementation for BBMM (AltBBMM) to speedup and scale the GP training in MOB-ML for molecular energies.

Original: We show that AltBBMMdelivers more efﬁcient training on over 1 million pair energies without sacriﬁcing transferabilityacross chemical systems of different molecular sizes.
Revised: We show that AltBBMM delivers more efﬁcient training on over 1 million pair energies without sacriﬁcing transferability across chemical systems of different molecular sizes.

Original: The accuracy and efﬁciency of BBMM and
Revised: The accuracy and efﬁciency of BBMM and

Original: AltBBMM in modeling physical problems are demonstrated by comparisons with literature resultson the same datasets.
Revised: AltBBMM in modeling physical problems are demonstrated by comparisons with literature results on the same datasets.

Original:  Background2.1 Molecular orbital based machine learning (MOB-ML)
Revised:  Background 2.1 Molecular orbital based machine learning (MOB-ML)

Original: The total energy of a given chemical system can be written as the sum of the Hartree-Fock andcorrelation energies.
Revised: The total energy of a given chemical system can be written as the sum of the Hartree-Fock and correlation energies.

Original: The correlation energy, E corr , can be further decomposed into pair energies,ϵ ij , associated with occupied molecular orbitals (MO) i and j , such that E corr =
Revised: The correlation energy, E corr , can be further decomposed into pair energies, ϵ ij , associated with occupied molecular orbitals (MO) i and j , such that E corr =

Original: P ij ϵ ij .
Revised: P ij ϵ ij .

Original: MOB-MLlearns pair energies by ϵ ij ≈ ϵ ML ( f ij ) , where the features f ij describe the interactions betweenthe molecular orbitals.[13]
Revised: MOB-ML learns pair energies by ϵ ij ≈ ϵ ML ( f ij ) , where the features f ij describe the interactions between the molecular orbitals.[13]

Original: Due to the different scales of the values, the diagonal pair energies ϵ d =41 { ϵ ij (cid:12)(cid:12) i = j } and the offdiagonal pair energies
Revised: Due to the different scales of the values, the diagonal pair energies ϵ d = { ϵ ij (cid:12)(cid:12) i = j } and the offdiagonal pair energies

Original: ϵ o = { ϵ ij (cid:12)(cid:12) i ̸
Revised: ϵ o = { ϵ ij (cid:12)(cid:12) i ̸

Original: = j } are trained separately, effectivelyproducing two different models.
Revised: = j } are trained separately, effectively producing two different models.

Original: 2.2 Gaussian Processes (GP)
Revised: 2.2 Gaussian Processes (GP)

Original: Gaussian Processes (GP) are non-parametric kernel-based machine learning methods that predict theprobabilistic distribution of unobserved data.
Revised: Gaussian Processes (GP) are non-parametric kernel-based machine learning methods that predict the probabilistic distribution of unobserved data.

Original: Given the observed data ( X, y ) , X ∈ R N × d , y ∈ R Nwith a Gaussian noise σ 2 ∈ R and a prior covariance function or kernel K : R d × R d → R , theprediction f ( X ′ ) test points X
Revised: Given the observed data ( X, y ) , X ∈ R N × d , y ∈ R N with a Gaussian noise σ 2 ∈ R and a prior covariance function or kernel K : R d × R d → R , the prediction f ( X ′ ) test points X

Original: ′ ∈ R M × d is a joint Gaussian distribution such thatE
Revised: ′ ∈ R M × d is a joint Gaussian distribution such that

Original: [Equation]
Revised: [Equation]

Original: where ˆ K = K ( X, X ) + σ 2 I .
Revised: where ˆ K = K ( X, X ) + σ 2 I .

Original: The hyperparameters, which include the Gaussian noise and kernelparameters, are learned in GP training by maximizing the log marginal likelihood
Revised: The hyperparameters, which include the Gaussian noise and kernel parameters, are learned in GP training by maximizing the log marginal likelihood

Original: [Equation]
Revised: [Equation]

Original: The typical way to calculate the above quantities is the Cholesky decomposition, which has a timecomplexity of O ( N 3 ) and a memory complexity of O ( N 2 ) .
Revised: The typical way to calculate the above quantities is the Cholesky decomposition, which has a time complexity of O ( N 3 ) and a memory complexity of O ( N 2 ) .

Original: Such complexities limit the trainingsize of GP-based models, such as MOB-ML, to around 50000 data points.
Revised: Such complexities limit the training size of GP-based models, such as MOB-ML, to around 50000 data points.

Original:  Method 
Revised:  Method 3.1 Conjugate gradient (CG)

Original: 3.1 Conjugate gradient (CG)

Original: The conjugate gradient (CG) [22] algorithm offers another way to obtain the predictive mean in Eq.1by iteratively solving ω
Revised: The conjugate gradient (CG) [22] algorithm offers another way to obtain the predictive mean in Eq. by iteratively solving ω

Original: = ˆ K − 1 y , or equivalently ˆ Kω = y with an O ( N 2 ) cost in each iteration.
Revised: = ˆ K − 1 y , or equivalently ˆ Kω = y with an O ( N 2 ) cost in each iteration.

Original: CG requires only the matrix-vector multiplications (MVMs) with the kernel matrix ˆ K , which isamenable to multi-GPU acceleration.
Revised: CG requires only the matrix-vector multiplications (MVMs) with the kernel matrix ˆ K , which is amenable to multi-GPU acceleration.

Original: In the iteration k , the solution is found in the order- k Krylovspace
Revised: In the iteration k , the solution is found in the order- k Krylov space

Original: [Equation]
Revised: [Equation]

Original: The solution ω k of CG at iteration k converges to the exact solution ω ∗ exponentially measured bythe relative residual || ˆ Kω k − y
Revised: The solution ω k of CG at iteration k converges to the exact solution ω ∗ exponentially measured by the relative residual || ˆ Kω k − y

Original: || ||
Revised: || ||

Original: y || .
Revised: y || .

Original: However, the total number of iterations k c to converge is usuallyvery large for a kernel ˆ K with high singularity.
Revised: However, the total number of iterations k c to converge is usually very large for a kernel ˆ K with high singularity.

Original: A common way to reduce k c is to construct apreconditioner P and then solve the equivalent equation P − 1 ˆ Kω =
Revised: A common way to reduce k c is to construct a preconditioner P and then solve the equivalent equation P − 1 ˆ Kω =

Original: P − 1 y such that P − 1 ˆ K is lesssingular than ˆ K .
Revised: P − 1 y such that P − 1 ˆ K is less singular than ˆ K .

Original: Block conjugate gradient (BCG) [16], as a variant of CG, can also be used to further reduce k c .
Revised: Block conjugate gradient (BCG) [16], as a variant of CG, can also be used to further reduce k c .

Original: Itextends CG to solve s linear equations ˆ Kω i
Revised: It extends CG to solve s linear equations ˆ Kω i

Original: =
Revised: =

Original: y i , i = 0 , 1 ...s − 1 simultaneously.
Revised: y i , i = 0 , 1 ...s − 1 simultaneously.

Original: The number oflinear equations s is also known as block size.
Revised: The number of linear equations s is also known as block size.

Original: In the iteration k of BCG, the solution is found in
Revised: In the iteration k of BCG, the solution is found in

Original: [Equation]
Revised: [Equation]

Original: block = span

Original: [Equation]

Original: By setting y 

Revised: By setting y

Original: = y , and y i ∼ N (0 , I ) for i > 0 , BCG can converge to the same exact solutionω ∗ 0 = ω ∗ with fewer iterations since K k ⊂ K block k .
Revised: = y , and y i ∼ N (0 , I ) for i > 0 , BCG can converge to the same exact solution ω ∗ 0 = ω ∗ with fewer iterations since K k ⊂ K block k .

Original: 3.2 Blackbox matrix-matrix multiplication (BBMM) and Alternative BBMM (AltBBMM)
Revised: 3.2 Blackbox matrix-matrix multiplication (BBMM) and Alternative BBMM (AltBBMM)

Original: BBMM [7, 23] calculates the GP inference by utilizing CG combined with the pivoted Cholesky decomposition preconditioner [1, 10].
Revised: BBMM [7, 23] calculates the GP inference by utilizing CG combined with the pivoted Cholesky decomposition preconditioner [1, 10].

Original: Furthermore, a modiﬁed batched version of conjugate gradients(mBCG 1 ) [7] is also proposed to estimate the marginal likelihood and its derivatives, which are required in the GP hyperparameter optimization.
Revised: Furthermore, a modiﬁed batched version of conjugate gradients (mBCG 1 ) [7] is also proposed to estimate the marginal likelihood and its derivatives, which are required in the GP hyperparameter optimization.

Original: These enhancements reduce the training complexityto O ( N 2 ) in time, and O ( N ) in memory and therefore enable the training of a million data points.
Revised: These enhancements reduce the training complexity to O ( N 2 ) in time, and O ( N ) in memory and therefore enable the training of a million data points.

Original: In this work, we propose an alternative realization of BBMM (AltBBMM) to achieve similar accuracy as BBMM with a lower cost in molecular energy prediction applications, where a low Gaussiannoise ( 10 − 5 ∼ 10 − 8 ) is required to reach the desired accuracy.
Revised: In this work, we propose an alternative realization of BBMM (AltBBMM) to achieve similar accuracy as BBMM with a lower cost in molecular energy prediction applications, where a low Gaussian noise ( 10 − 5 ∼ 10 − 8 ) is required to reach the desired accuracy.

Original: However, sine the low Gaussian noisesigniﬁcantly increases the singularity of ˆ K , CG would converge slowly or even fail to convergewhen the rounding errors exceed the Gaussian noise.[8]
Revised: However, since the low Gaussian noise signiﬁcantly increases the singularity of ˆ K , CG would converge slowly or even fail to converge when the rounding errors exceed the Gaussian noise.[8]

Original: In order to speedup the CG convergence,we employ the BCG algorithm described in the previous section.
Revised: In order to speedup the CG convergence, we employ the BCG algorithm described in the previous section.

Original: The additional computational costof BCG in each iteration is negligible compared with the kernel matrix calculations.
Revised: The additional computational cost of BCG in each iteration is negligible compared with the kernel matrix calculations.

Original: To furtherimprove the robustness of the convergence, we use the double-precision ﬂoating numbers in theimplementation and employ the symmetric preconditioning P − 1 / 2 ˆ KP − 1 / 2 .
Revised: To further improve the robustness of the convergence, we use the double-precision ﬂoating numbers in the implementation and employ the symmetric preconditioning P − 1 / 2 ˆ KP − 1 / 2 .

Original: The Nystreoem preconditioner [5] is used as an example, but we note that better preconditioners could exist.
Revised: The Nystreoem preconditioner [5] is used as an example, but we note that better preconditioners could exist.

Original: Finally,the hyperparameters are optimized on a random subset of the entire training set in AltBBMM sincethe optimized hyperparameters remain similar across various training sizes for MOB-ML.
Revised: Finally, the hyperparameters are optimized on a random subset of the entire training set in AltBBMM since the optimized hyperparameters remain similar across various training sizes for MOB-ML.

Original:  Computational details
Revised:  Computational details

Original: We train all the models on random subsets of the QM7b-T dataset [3, 13, 24], which contains 7211organic molecules with up to 7 heavy atoms.
Revised: We train all the models on random subsets of the QM7b-T dataset [3, 13, 24], which containsorganic molecules with up to 7 heavy atoms.

Original: The test sets are the remaining QM7b-T molecules andthe whole GDB-13-T dataset [3, 13, 24] containing 1000 organic molecules with 13 heavy atoms.
Revised: The test sets are the remaining QM7b-T molecules and the whole GDB-13-T dataset [3, 13, 24] containing 1000 organic molecules with 13 heavy atoms.

Original: The Matérn 5/2 kernel is used in all the GP trainings.
Revised: The Matérn 5/2 kernel is used in all the GP trainings.

Original: We independently implement the BBMM according to the description of the mBCG and hyperparameter optimization in Ref. 7.
Revised: We independently implement the BBMM according to the description of the mBCG and hyperparameter optimization in Ref. 7.

Original: The symmetric
Revised: The symmetric

Original: Nystroem preconditioner and the block CG are used in this work.
Revised: Nystroem preconditioner and the block CG are used in this work.

Original: In both BBMM and AltBBMM,the rank r of the preconditioner is chosen as 10000, the BCG block size s is ﬁxed as 50 , and the
Revised: In both BBMM and AltBBMM, the rank r of the preconditioner is chosen as 10000, the BCG block size s is ﬁxed as 50 , and the

Original: BCG iterations stop when all the s relative residuals are smaller than 10 − 6 .
Revised: BCG iterations stop when all the s relative residuals are smaller than 10 − 6 .

Original: The hyperparametersare optimized from a full GP trained on 50 random molecules.
Revised: The hyperparameters are optimized from a full GP trained on 50 random molecules.

Original: To overcome the memory limit andmaximize the multi-GPU efﬁciencies, the kernel computations in CG are performed in 4096 × 4096batches, and such computations are dynamically distributed to all the available GPUs.
Revised: To overcome the memory limit and maximize the multi-GPU efﬁciencies, the kernel computations in CG are performed in 4096 ×batches, and such computations are dynamically distributed to all the available GPUs.

Original: Additionally,we add a Gaussian noise regularization σ 2 add = 10 − 5 to the optimized Gaussian noise reduce thesingularity of ˆ K .
Revised: Additionally, we add a Gaussian noise regularization σ 2 add = 10 − 5 to the optimized Gaussian noise reduce the singularity of ˆ K .

Original:  Results 
Revised:  Results 5.1 Low noise regularization for accurate GP

Original: 5.1 Low noise regularization for accurate GP

Original: We ﬁrst demonstrate the necessity of utilizing a low noise regularization to achieve accurate predictions.
Revised: We ﬁrst demonstrate the necessity of utilizing a low noise regularization to achieve accurate predictions.

Original: We train all the offdiagonal energies ( ϵ o ) pairs from 1000 QM7b-T molecules with differentσ 2 add and test on the ϵ o of the rest QM7b-T molecules.
Revised: We train all the offdiagonal energies ( ϵ o ) pairs from 1000 QM7b-T molecules with different σ 2 add and test on the ϵ o of the rest QM7b-T molecules.

Original: The training time and the prediction meanabsolute error (MAE) are displayed in Table 1.
Revised: The training time and the prediction mean absolute error (MAE) are displayed in Table 1.

Original: For both BBMM and AltBBMM, regularizing withσ 2 add = 10 − 1 results in a less singular ˆ K and saves half the training time, but its prediction MAEdoubles when compared to the results of σ 2 add = 10 − 5 .
Revised: For both BBMM and AltBBMM, regularizing with σ 2 add = 10 − 1 results in a less singular ˆ K and saves half the training time, but its prediction MAE doubles when compared to the results of σ 2 add = 10 − 5 .

Original: Since the MOB-ML data generation is signiﬁcantly more expensive than model training, we ﬁx σ 2 add = 10 − 5 for all of the following BBMMand AltBBMM experiments to achieve the most accurate model with the least amount of data.5.2 BBMM and AltBBMM for energies of organic molecules
Revised: Since the MOB-ML data generation is signiﬁcantly more expensive than model training, we ﬁx σ 2 add = 10 − 5 for all of the following BBMM and AltBBMM experiments to achieve the most accurate model with the least amount of data.

Original: We now examine the accuracy and transferability of BBMM and AltBBMM in learning QM7b-Tand GDB-13-T molecular energies.

Original: The transferability of MOB-ML is assessed by the MAEs per 7heavy atoms (MAE/7HA) of test GDB-13-T molecules predicted by the models trained on QM7b-Tmolecules.

Original: Table 2 lists the wall-clock time of training on 6500 QM7b-T molecules by BBMM

Original: [Table]
Revised: [Table]

Original: [Table]
Revised: [Table]

Original: a Training size of ϵ o is 1,
Revised: a Training size of ϵ o is 1,152,157 and training size of ϵ d is 124,

Original: [Table]
Revised: [Figure]

Original: and training size of ϵ d is

Original: [Table]
Revised: [Figure]

Original:  and AltBBMM and the corresponding prediction MAEs on test QM7b-T and GDB-13-T molecules.
Revised: We now examine the accuracy and transferability of BBMM and AltBBMM in learning QM7b-T and GDB-13-T molecular energies. The transferability of MOB-ML is assessed by the MAEs perheavy atoms (MAE/7HA) of test GDB-13-T molecules predicted by the models trained on QM7b-T molecules. Table 2 lists the wall-clock time of training on 6500 QM7b-T molecules by BBMM and AltBBMM and the corresponding prediction MAEs on test QM7b-T and GDB-13-T molecules.

Original: Similar to the results in Table 1, by utilizing our AltBBMM approach, we gain a four-fold speedup inthe training timings while only introducing 4% and 1% additional MAE in the prediction of QM7b-Tand GDB-13-T, respectively, compared with BBMM.
Revised: Similar to the results in Table 1, by utilizing our AltBBMM approach, we gain a four-fold speedup in the training timings while only introducing 4% and 1% additional MAE in the prediction of QM7b-T and GDB-13-T, respectively, compared with BBMM.

Original: In addition, we compare the performance of BBMM and AltBBMM with the results of the current most accurate literature methods, i.e., MOB-ML with full GP (MOB-ML/Full GP) [13] and
Revised: In addition, we compare the performance of BBMM and AltBBMM with the results of the current most accurate literature methods, i.e., MOB-ML with full GP (MOB-ML/Full GP) [13] and

Original: DeePHF with extended neural network regressor (DeePHF/extended NN) [2].
Revised: DeePHF with extended neural network regressor (DeePHF/extended NN) [2].

Original: The literature resultsof MOB-ML/Full GP are only available with up to 220 training molecules due to the limited memory resources.
Revised: The literature results of MOB-ML/Full GP are only available with up to 220 training molecules due to the limited memory resources.

Original: The introduction of BBMM and AltBBMM allows MOB-ML to scale up the trainingto 6500 molecules (over 1 million training pair energies) while retaining the accuracy and transferability compared with MOB-ML/full GP in Figure 1.
Revised: The introduction of BBMM and AltBBMM allows MOB-ML to scale up the training to 6500 molecules (over 1 million training pair energies) while retaining the accuracy and transferability compared with MOB-ML/full GP in Figure 1.

Original: By training on 6500 molecules, BBMM and
Revised: By training on 6500 molecules, BBMM and

Original: AltBBMM reach the current best MAE/7HA for GDB-13-T as 0.490 kcal/mol and 0.493 kcal/mol,respectively.
Revised: AltBBMM reach the current best MAE/7HA for GDB-13-T as 0.490 kcal/mol and 0.493 kcal/mol, respectively.

Original: In all the cases we tested, BBMM and AltBBMM provide a better accuracy on QM7b-Tand a better transferability on GDB-13-T than DeePHF/extended NN.[Figure]
Revised: In all the cases we tested, BBMM and AltBBMM provide a better accuracy on QM7b-T and a better transferability on GDB-13-T than DeePHF/extended NN.

Original:  Conclusion
Revised:  Conclusion

Original: In this work we successfully apply the BBMM algorithm and a new alternative implementation,
Revised: In this work, we successfully apply the BBMM algorithm and a new alternative implementation,

Original: AltBBMM, to Gaussian process-training for the MOB-ML method on over a million pair energies.
Revised: AltBBMM, to Gaussian process-training for the MOB-ML method on over a million pair energies.

Original: Even though the use of BBMM alone increases our previously attainable training-set size limitover 30 times, our newly introduced AltBBMM implementation improves this further by offering afour-fold speed-up while maintaining high accuracy.
Revised: Even though the use of BBMM alone increases our previously attainable training-set size limit over 30 times, our newly introduced AltBBMM implementation improves this further by offering a four-fold speed-up while maintaining high accuracy.

Original: With the BBMM and AltBBMM approaches,
Revised: With the BBMM and AltBBMM approaches,

Original: MOB-ML models can be trained using datasets with over 6500 QM7b-T molecules, yielding thebest accuracy to date for the QM7b-T and GDB-13-T datasets.
Revised: MOB-ML models can be trained using datasets with over 6500 QM7b-T molecules, yielding the best accuracy to date for the QM7b-T and GDB-13-T datasets.

Original: Future work will include extensionof the approach for the prediction of other molecular properties within the MOB-ML framework
Revised: Future work will include extension of the approach for the prediction of other molecular properties within the MOB-ML framework. Acknowledgments and Disclosure of Funding

Revised: We thank Dr. J. Emiliano Deustua for helpful discussions.

Revised: This work is supported in part by the U.S. Army Research Laboratory (W911NF-12-2-0023), the U.S. Department of Energy (DESC0019390), the Caltech DeLogi Fund, and the Camille and Henry Dreyfus Foundation (Award

Revised: ML-20-196).

Revised: Computational resources were provided by the National Energy Research Scientiﬁc

Revised: Computing Center (NERSC), a DOE Ofﬁce of Science User Facility supported by the DOE Ofﬁce of Science under contract DE-AC02-05CH11231.