Improving accuracy of physics-based hydration free energy predictions by machine learning the remaining error relative to the experiment

Abstract

The accuracy of computational models of water is key to atomistic simulations of biomolecules. We propose a computationally efficient way to improve the accuracy of the prediction of hydration free energies (HFEs) of small molecules: the remaining errors of the physics-based models relative to the experiment are predicted and mitigated by machine learning (ML) as a postprocessing step. Specifically, the trained graph convolutional neural network attempts to identify the “blind spots” in the physics-based model predictions, where the complex physics of aqueous solvation is poorly accounted for, and partially corrects for them.

The strategy is explored for five classical solvent models representing various accuracy/speed trade-offs, from the fast analytical generalized Born (GB) to the popular TIP3P explicit solvent model; experimental HFEs of small neutral molecules from the FreeSolv set are used for the training and testing. For all of the models, the ML correction reduces the resulting root-mean-square error relative to the experiment for HFEs of small molecules, without significant overfitting and with negligible computational overhead. For example, on the test set, the relative accuracy improvement is 47% for the fast analytical GB, making it, after the ML correction, almost as accurate as uncorrected TIP3P. For the TIP3P model, the accuracy improvement is about 39%, bringing the ML-corrected model’s accuracy below the 1 kcal/mol threshold. In general, the relative benefit of the ML corrections is smaller for more accurate physics-based models, reaching the lower limit of about 20% relative accuracy gain compared with that of the physics-based treatment alone. The proposed strategy of using ML to learn the remaining error of physics-based models offers a distinct advantage over training ML alone directly on reference HFEs: it preserves the correct overall trend, even well outside of the training set.

Graphical Abstract

1. Introduction

Atomistic modeling and simulation methods enable a modern molecular approach to biological research,^1–4 including structure-based drug design.⁵ The ability of these methods⁶ to address biologically relevant problems is largely determined by the accuracy and computational efficiency of the treatment of complex solvation and electrostatic effects in biomolecules surrounded by water. A large variety of water models have been developed, yet none of the current models commonly used in practice is perfect:^7–10 various compromises between speed and accuracy had to be made, see, e.g., Ref.¹¹ for a recent review.

Broadly speaking, there are two different approaches to modeling aqueous solvation that are widely used in classical biomolecular simulations at atomic resolution: explicit and implicit solvation.¹¹ The former treats each water molecule individually, at the same resolution as the target biomolecule. One of the simplest, most efficient, and still widely used models of this class is TIP3P,¹² with rigid geometry and fixed partial charges on its 3 atoms.

A computationally attractive alternative to the explicit solvation is the implicit solvation, in which all of the solvent is approximated as (infinite) continuum with dielectric and non-polar properties of water.^13–23 The relative computational efficiency of the approach is considered to be its main advantage over the explicit solvation. The history of the development of the implicit solvation framework is longer than 100 years: Max Born introduced his model²⁴ for the solvation (hydration) free energy of a single ion in 1920. The generalization of this very simple, yet powerful model – generalized Born (GB) approximation^18,25–57 – is the implicit solvation model often used in atomistic molecular dynamics (MD) simulations,⁵⁸ where it can provide up to several orders of magnitude speed-up of conformational sampling relative to the explicit solvation.⁵⁹ Needless to say, the GB model is separated from reality by a multitude of approximations,⁶⁰ each with its own speed-accuracy trade-offs.

The implicit solvation framework is particularly well suited to adding individual physical effects to the base model. For example, the linear response Poisson-Boltzmann (PB) or GB models are manifestly invariant under charge inversion, but real water is not – as a result, the so-called charge hydration asymmetry (CHA)^61–68 is missing from these models, but can be re-introduced explicitly.^69–76 Likewise, efforts to introduce the “non-polar” effects of real water into the implicit solvation framework have a long history, starting from an empirical, single parameter surface energy term to more sophisticated, multi-parameter models^45,77 aimed at capturing the more nuanced physics of solute-solvent interactions, including the hydrophobic effect. Approaches based directly on variational principles^78–85 attempt to account for multiple physical effects all at once. Generally, the accuracy gains of implicit solvent models come at a price of reduced computational efficiency, and, often, introduce additional parameters to account for more subtle and diverse physics of hydration. Yet, even multi-parameter models are not perfect, while the lofty goal of capturing it all in a single formula, remains elusive. Breaking the stubborn “accuracy barrier”, while keeping the models computationally tractable, is still a major problem in the field of atomistic modeling and simulations.

The development of the explicit solvent models has followed a similar path in several respects. Over the decades since TIP3P was introduced, multiple accuracy improvements to classical explicit solvent models generally required either better accounting^86,87 for the same overall physics already present at the conceptual level of the older model, or an explicit addition of the missing physics into the model, such as the inclusion of electronic polarization effects,^88–90 often (but not always^91,92) at a substantial additional computational cost.

Still, none of these classical models, explicit or implicit, account for all of the complex physics of liquid water, including various quantum effects. Quantum modeling of tens of thousands of water molecules at room temperature remains out of reach for time-scales relevant for most biomolecular systems and processes. In practical simulations, stubborn errors remain.

It is then not surprising that recent explosive development of machine learning (ML) techniques,⁹³ including deep neural networks (DNNs),^94–96 is already making a noticeable impact on this field,^97–110 including works aimed directly at improving the accuracy of description of complex solvation effects.^111–117 It should be noted that the majority of these recent works combine QM-based methodology with ML, while our interest here is purely classical approaches. Among recent purely ML-based approaches, a featurization algorithm, based on functional class fingerprints, and implemented within the DeepChem ML framework,⁹⁵ was used in Ref.¹¹⁷ to predict hydration free energies (HFEs) of a diverse set of 642 neutral small molecules available in FreeSolv¹¹⁸ – arguably the largest public database of experimentally measured HFEs. The overall accuracy achieved with this approach was on par with, or even better than, several existing, computationally efficient classical models of solvation. Still, despite several novel features, the resulting accuracy did not reach that of the explicit models such as TIP3P or even some of the GB-based models.^56,73 Apparently, the pure neural network based approach struggles with the complex physics of hydration, at least when the training set is under 1000 data points, providing a motivation for combining ML with classical physics-based water models to improve the accuracy of computationally efficient HFE prediction. In fact, in the context of water models, it was recently demonstrated⁹⁷ that DNN potentials face accuracy limitations in predicting properties not included in the training process.

Physics-guided machine learning (PGML)^119–121 is an emerging area of research that aims to utilize physics knowledge in the design and training of ML models to achieve better generalization performance on samples outside of training data. One of the research directions in PGML that has received considerable attention is focused on incorporating various physics-based constraints directly in the process of training ML models to provide additional sources of supervision to the ML models beyond the empirical loss observed on labeled data. This direction has been explored in several scientific applications including lake modeling,^122,123 quantum mechanics,¹²⁴ and solving partial differential equations.^125,126 For prediction of HFEs in conjunction with physics-based modeling, ML is often used to train the parameters of the physics-based model, e.g., the force-field.¹¹³

Here we propose, and explore in detail, an alternative way to benefit from the Physics while harnessing the power of DNNs: in our approach the physics and ML parts of the prediction pipeline are completely decoupled. Namely, a physics-based model is applied first, while a “pure” DNN (ML) learns and attempts to correct the remaining errors relative to the experiment as a separate, post-processing step. The motivation for this approach is several-fold. First, we expect that it may automatically preserve the correct trends set by the underlying physics model outside of the ML training set. Second, assuming that the physics model is already reasonably accurate, the errors remaining after application of the physics-based treatment should be relatively small, so that the power of ML is not “wasted” on learning what the existing classical physics can already handle well. Note that within this approach, the larger fraction of the computed quantity automatically satisfies various physical constrains imposed by the physics-based model. The relatively small inaccuracies that remain are complex, hard to decompose into distinct independent components, and are overall difficult or even impossible to describe accurately within computationally facile models of classical physics. In contrast, DNNs may be just the right tool to handle these small, but complex and convoluted errors, difficult to attribute to specific bits of the missing physics. Finally, a potential practical benefit of the proposed strategy is that it should be easy to adopt for any physics-based model that predicts the quantity of interest, with no additional, and often non-trivial, effort to integrate the physics-based and ML parts.

We test the ability of the proposed approach to improve the accuracy of prediction of HFEs of small molecules. Accurate HFE prediction is important in its own right:¹²⁷ this single number incorporates many facets of the complex physics of hydration, being able to predict it correctly is key to many types of computations common to molecular biophysics and computational biology.¹²⁸ For example, the accuracy of predicting receptor-ligand binding energetics^129,130 depends critically on how well the hydration effects are described,^131–133 and these are encapsulated in HFEs. It is also important that a reasonably large data set of experimental HFEs is available,¹¹⁸ which makes it possible to train the DNN directly against experiment, as opposed to yet another model.

We apply our approach to several explicit and implicit physics models, with varying computational complexity and accuracy, to determine how much accuracy can be gained using this approach, and how this accuracy improvement depends on the accuracy of the physics model. If the small molecule test shows promise, further exploration of the approach may be warranted in the future.

The extensive “Methods” section begins with an introduction of the overall approach, then describes the choice of physics-based solvent (water) models used here; their details and parameters are given in the Supporting Information. The “Methods” section proceeds to define and describe the neural network and the data sets. Details of the parameter optimization and neural network training follow. The performance of “Physics + ML”, and “ML alone” approaches are presented in “Results and Discussion”, including model performance on molecules well outside of the training set. The overall findings are summarized and discussed further in “Conclusions”, along with limitations of the approach.

2. Methods

Every practical physics-based model for prediction of HFE of molecules has some residual error relative to the ground truth, that is experiment. Here we use this remaining error – the difference between the experimental values and the predictions of a given physics-based model – as the target for ML (DNN) models. The resulting prediction of the ML model is added to the physics-based prediction to reduce the remaining error, Fig. 1.

Figure 1:

Schematic showing our overall approach of using ML to reduce the remaining error between the hydration energies predicted by physics-based models and experiment. The HFE remaining error between physics-based and experimental predictions is fit via ML; this ML correction augments the HFE of the physics-based models, reducing the remaining error to experiment. The physics and the ML parts of the overall workflow are completely separate and independent: the output of the physics-based model becomes the input of the ML (DNN) part (the latter is not a physics-motivated ML potential).

Specifically, our ML models use the L₂ loss function to compare ML-predicted corrections to the target to evaluate its performance as it learns. The L₂ loss function is defined as:

$$L_2\text{loss}=\frac{{\sum }_{i=1}^N{\left(y_i-{\widehat{y}}_i\right)}^2}{N}={\text{RMSE}}^2,$$ (1)

where N is the number of data points, $y_i$ are the reference data points, ${\widehat{y}}_i$ are the predicted values, and RMSE is the root-mean-square error relative to the reference. A final trained model predicts the difference between the experiment and physics model predictions to reduce the overall error.

2.1. Physics-based Water Models

The five classical physics-based water models – one explicit and four implicit – used here as baselines for ML improvement are TIP3P, CHA-GB, GBNSR6, IGB5 (GB-OBC), and AASC, listed here in the approximate order of expected accuracy. The last model, AASC,¹³⁴ is a relatively new addition to the implicit solvation family: it is an approximation of the apparent surface charge at the solute dielectric boundary that provides a description of the electric field around the solute in aqueous solution, allowing for a straightforward estimate of the polar part of the HFE. More details about the models, and their parameters, are provided in the Supporting Information.

The main rationale for choosing this specific subset from the many available classical solvent models is that we aim to test our new approach on a wide spectrum of accuracy/speed trade-offs of solvation models currently used in practice, see Table 1 below. The implicit solvation models alone provide the wide range we seek, but we also need to make sure our conclusions are robust to the type of solvent model, this is why we add an explicit solvent model to the mix. Our choice of the implicit solvent models attempts to illustrate two aspects of the evolution of the framework: accuracy improvements made while keeping the same level of the underlying physics, as in the IGB5(GB-OBC) to GBNSR6 step, and improvements due to the addition of new physics, absent from the previous level of approximation, as in the GB to CHA-GB step. Here, CHA-GB⁷³ is the generalized Born model modified to take into account charge hydration asymmetry (CHA) – non-invariance of the polar part of the solvation energy upon solute charge inversion. Also, while we use IGB5 (GB-OBC) and AASC in conjunction with an early, single-parameter model for the non-polar part of the free energy, a more sophisticated model for non-polar energy is used with GBNSR6 and CHA-GB, see details in the Supporting Information. The choice for the only representative of the explicit solvent class – TIP3P – is further justified below. First, based on available, albeit limited published data,⁸⁶ we do not expect the widely used, reasonably fast (fixed-charge) explicit solvation models to provide nearly as wide a range of HFE accuracy as do the implicit solvent models, so selecting just one member of the explicit solvent class should be enough for our purposes. While water models other than TIP3P, including special-purpose models,^135,136 can often predict water properties more accurately than TIP3P, see, e.g., Ref.¹¹ for a review, it is not obvious¹³⁷ whether these necessarily provide substantial accuracy gains with respect to our main accuracy metric, which is the RMSE of HFE predictions of small molecules to experiment. At the same time, more accurate water models, such as polarziable ones, would make computing HFEs for a large enough data set much more computationally expensive than is already the case with TIP3P, Table 1. Since our goal is to evaluate our general strategy, rather than evaluating specific water models, we believe that limiting the representative examples to the above five models is appropriate. Note that we are deliberately not considering the wide class of HFE-predictive models that are QM-based, see, e.g., Ref.¹³⁸ for a comprehensive review of this class of models. The reason for the exclusion of QM-based approaches is two-fold: most importantly, unlike classical models of solvation, which necessarily miss some physics of the process, Quantum Mechanics is capable of predicting molecular properties essentially exactly in principle. Also, from a practical perspective, QM-based models generally represent a different computational complexity class, at least compared to fast classical implicit solvent models.

Table 1:

The accuracy-efficiency range offered by the physics-based water models considered in this work. Shown are the RMSE (kcal/mol) relative to the experiment and approximate simulation time (for TIP3P model see Ref.¹³⁹) to obtain a converged estimate of the HFE of a small molecule using one CPU core for each of the physics-based water models. The numbers are averages over the entire FreeSolv database.

Model	RMSE	Approximate Simulation Time
TIP3P	1.54	2 day/molecule139
AASC	2.51	100 ms/molecule
CHA-GB	1.72	30 ms/molecule
GBNSR6	1.67	30 ms/molecule
IGB5	2.84	5 ms/molecule

2.2. The Neural Network

The GraphConvModel implemented in TensorFlow by DeepChem⁹⁵ and based on the graph convolutions described by Duvenaud et al.¹⁴⁰ is used to predict the difference between experimental values and predictions of the physics models. As input, the model takes graph representations of molecules that have been created from SMILES strings using the ConvMolFeaturizer supplied by DeepChem. This represents each molecule as a graph, with atoms as nodes and bonds as edges. The atoms have 75 features, all of which are one-hot encoding of either atomic number, atom node degree, implicit valence, aromaticity, or number of attached hydrogens.¹⁴⁰

The model itself consists of two convolutional layers and an atom-level dense layer with ReLU activation used for each layer as seen in Fig. 2. The first convolutional layer identifies small-scale patterns and starts with a graph convolution that aggregates information about graph nodes and their neighboring nodes. Next, a fraction of randomly chosen output nodes of the graph convolution are ignored in the dropout stage (a regularization technique used to improve the robustness of the model while training¹⁴¹). Note this step is skipped for all layers when the model is used to make predictions for new molecules. Then, graph pooling is applied, which decreases the size of the graph representation while maintaining the most important information. The second convolutional layer captures more high-level structural patterns in the molecule. It takes the output from the first graph pool layer as input and follows the same procedure as the first convolutional layer. The output from the second graph pool layer is used as input to the atom-level dense layer which transforms and combines the information from the convolutional layers into a better representation for regression. The graph gather takes the output of the dense layer after dropout and combines the information from the graph to a fixed-size representation. This representation is then used as input for the regression step which linearly transforms the data to give its final prediction. A diagram of the architecture of the model can be seen in Fig. 2.

Figure 2:

Schematic of the architecture of the GraphConvModel implemented in this paper. The model takes a SMILES string representing the molecule, which has been converted into a graph representation as its input. The model is composed of two convolutional layers and one atom-level dense layer which each use ReLU activation. Each convolutional layer has three main steps: the graph convolution with optimized convolutional layer size, dropout of an optimized fraction of nodes, and graph pooling which transforms the output for the next layer. The dense layer occurs after both convolutional layers and starts with the atom-level dense layer with optimized dense layer size. Dropout of an optimized fraction of nodes occurs next before the graph gather transforms the data into a fixed size. After this, the output of the graph gather is linearly transformed to the final output in the regression step.

2.3. Data sets

The ML-corrected solvation models are trained and evaluated using version 0.52 of the FreeSolv database¹¹⁸ which is found at the following URL: https://github.com/MobleyLab/FreeSolv. This database, described in Table 2, is a collection of experimental HFEs for 642 small neutral molecules. The experimental uncertainties on the HFE values listed in the database range from 0.03 to 1.93 kcal/mol, with a mean of 0.57 kcal/mol; for the majority (459 out of 642) of the molecules listed uncertainty is 0.6 kcal/mol. Higher uncertainties are generally correlated with higher absolute HFEs, so that an average relative error over the entire data set is ~20%. At the extremes, some molecules have relative uncertainty of the measured HFE as high as ~50%. Molecules with |HFE| < 1 kcal/mol were excluded from the relative error estimates above.

Table 2:

FreeSolv database version 0.52: experimental hydration free energies for small neutral molecules.

Number of molecules in the dataset	642
Average number of heavy (non-H) atoms	8 ± 4.5
Largest molecule number of atoms (non-H)	24
Average hydration free energy, μ ± σ	−3.80 ± 3.85 kcal/mol
Hydration free energy range	(−25.47, 3.43) kcal/mol
Average experimental uncertainty, μ ± σ	0.57 ± 0.31 kcal/mol
Experimental uncertainty range	(0.03, 1.93) kcal/mol
Elements (most to least frequent)	H, C, O, Cl, N, F, S, Br, P, I

Since the size of the FreeSolv set does not approach that of the typical ML tasks, extra care was taken to ensure that the training and test sets were well-balanced,¹⁴² representing the full range of HFEs present in FreeSolv. The dataset was separated into a test set of 80 molecules and the remaining 562 molecules, representing the training and validation sets, respectively. The latter was further divided into 20 subsets of 28 or 29 molecules each used for 20-fold cross-validation. These sets were selected using stratified sampling, with groups each representing a different range of HFEs, to ensure that each partition had a similar distribution of HFE as the whole dataset. For more details see the Supporting Information or the code available on GitHub (see the Supporting Information).

2.4. Hyperparameter optimization

We optimize the hyperparameters for the GraphConvModel to find the best combination of high accuracy and low overfitting in predicting the error relative to the experiment for HFEs. Since our goal is a direct comparison of the performance of our proposed approach across 5 physics-based models of solvation, it makes sense to use one and the same neural network architecture and hyperparameters for all of the ML models. Here we chose to optimize the hyperparameters using TIP3P model as the base to which the DNN correction is added. We chose TIP3P as the target for hyperparameter optimization for the following reasons. First, the TIP3P model is among the most accurate physics-based models of solvation we utilize in this work, Table 1, therefore its HFE predictions are likely to be consistent with the main premise of our approach in that the errors remaining after an application of the physics-based model should be relatively small. Second, the development of TIP3P was completely agnostic to HFEs in FreeSolv – just like many other popular, classical atomistic water models, TIP3P was originally parameterized to describe experimental properties of liquid water at ambient conditions. This is yet another reason for why hyperparameters optimized using TIP3P should generalize well to the other physics models. In Section 3.4 we confirm that our conclusions are robust to the choice of hyperparameters, which suggests that optimizing these for each model individually is not necessary, as it would likely result in insignificant change in performance.

Each set of parameters was evaluated using 20-fold cross-validation where the training set partitions (see previous section) take turns being the validation set while the rest of the molecules make up the train set. For each fold, a model was trained on the training set and then used to make predictions for each molecule in the training set and separately on every molecule in the validation set. These predicted corrections were added to the TIP3P predictions for each molecule, and the root mean square error (RMSE) between the machine learning corrected TIP3P values and experiment was calculated for the validation set and the training set for each fold. The average validation RMSE value for each fold over two rounds of 20-fold cross-validation was used as the primary metric to evaluate each set of parameters. Two rounds of cross-validation were used to reduce random variation due to the stochastic nature of the training. The difference between the average RMSE for the validation set and training set was used as a metric to assess overfitting.

Each generated ML model for correction of TIP3P hydration was optimized over 500 epochs with the ADAM optimizer,¹⁴³ as it was observed in preliminary testing (results not shown) that this number of epochs led to a reasonably well-converged loss function, while also mitigating the possibility of overfitting. This choice is further justified by the convergence of models trained on final optimized hyperparameters seen in Fig. 3. The batch size used for each of these models was 100, which represents about 20% of the training data. Batch normalization was disabled because its improvements to convergence speed were negligible due to the relatively small training set compared to typical ML tasks.

Figure 3:

Convergence of the loss function demonstrated for 100 separate ML models learning the TIP3P residuals errors for 500 epochs. The models were trained with optimized hyperparameters using the full training set of 562 molecules used for cross-validation. The L₂ loss function is evaluated on this full training set for each epoch during training.

Next, the value for dropout used for all three layers, and dense layer size, were optimized in conjunction using two rounds of 20-fold cross-validation. The goal was to reduce overfitting to below 0.3 kcal/mol while maximizing accuracy. The value for dropout is the fraction of nodes in each layer randomly chosen each batch to be ignored during training highlighted as orange in Fig. 2. The dense layer size is the width of channels for the atom-level dense layer of the graph convolutional neural network as seen in green in Fig. 2. While optimizing dense layer size and dropout, two convolutional layers of size 32 were used for the neural network. The size of 32 was chosen to be smaller than the default value of 64 to reduce runtime but still large enough to produce accurate results. As seen in Fig. 4a, dropout was tested from 0 to 0.9 in increments of 0.1 and the dense layer size was tested from 10 to 906 nodes, on a 10-value logarithmic scale. The final parameters chosen were the ones with the lowest average validation RMSE, ensuring that the train-validation RMSE difference was less than 0.3 kcal/mol. The final values for dropout and dense layer size were 0.4 and 27 respectively. The accuracy performance of these final parameters relative to the full hyperparameter space searched is shown by the white squares in Fig. 4a.

Figure 4:

Hyperparameter exploration using cross-validation for (a) dropout and dense layer size, and (b) convolutional layer sizes using optimized values for dropout and dense layer size. In each case, the top and middle graphs show how different combinations of the hyperparameters affect model performance on training and validation data respectively. As seen on vertical color bar to the right, more red colors indicate high RMSE values while more purple colors indicate low RMSE values. The bottom graph shows the difference in RMSE between the training and validation sets meaning that more purple colors indicate a low difference in RMSE and as a result, low overfitting. The white squares indicate the final selected values for those hyperparameters. The thick black lines indicate the overfitting boundary; squares to the left of the black line for (a) and to the lower right for (b) indicate overfitting greater than 0.3 kcal/mol.

After that, the sizes of convolutional layers, shown in purple in Fig. 2, were optimized using two rounds of 20-fold cross-validation. The convolutional layer sizes are the widths of channels for the convolution layers. For this step, the optimized values of dropout and dense layer size were used. As seen in Fig. 4b, the sizes of the two convolutional layers were varied independently, each tested from 2 to 107 nodes, with 10 values growing proportional to x^1.8 with x going from 1 to 10. The final values chosen for the sizes of convolutional layers were 53 and 38 for the first and second convolutional layers respectively. These values maximized the accuracy of the ML predictions while ensuring that overfitting remained below 0.3 kcal/mol. Fig. 4b shows how model accuracy and overfitting change as the sizes of the convolutional layers are changed.

In summary, this hyperparameter optimization resulted in the final hyperparameters of a dropout of 0.4, a dense layer size of 27 nodes, and convolutional layers with sizes of 53 and 38, respectively. Once the hyperparameters had been optimized, we used the same hyperparameter values for training ML models to correct all of the physics-based models. Doing so keeps the dimension of the neural net consistent for all of the comparisons made in this work.

2.5. Final training and testing

After final hyperparameters were determined, ML models were trained to correct for errors of each of the physics-based models. For each physics-based model, the target values used in the training were the difference between the experiment and HFE prediction based on that specific physics model. We also trained ML alone to predict HFE directly (technically, this model is trained to predict the difference between the experiment and the Null model assuming all HFEs to be zero); in what follows this model is referred to as “ML alone”. In total, 100 ML models were individually trained to make corrections for each physics-based model individually. In each case, the training set was the 562 molecules used for cross-validation, and the models were evaluated on the 80 molecule test set.

The ML models were evaluated using the RMSE between the experimental values and the final ML corrected physics-based model predictions for the training and test sets. These results, see Table S2 in the SI, were analyzed to gain a better understanding of the variation in performance caused by the stochastic optimization used by the GraphConvModel. Convergence of the loss function over 500 epochs for the 100 trained models for TIP3P is shown in Fig. 3.

One final ML model for each physics-based model was chosen as the ML model with the RMSE of the training set closest to the mean training set RMSE of the 100 models. Results for the final chosen ML models can be seen in Table 3.

Table 3:

Performance of the physics-based hydration models without ML corrections and with final trained ML corrections on the TEST set of 80 molecules and on the full TRAINING set of 562 small neutral molecules. The method for choosing the final ML model for each physic-based model is described in Section 2.5, and the summary of the average performance of ML models can be seen in Table S2 in the SI. The performance metrics – RMSE relative to the experimental HFEs and mean deviation – are in kcal/mol, and the compute speed is relative to the speed of IGB5 model. The Null “physics model” sets all HFE values to zero; thus, the physics + ML results for Null model effectively represent using ML alone.

		TIP3P	AASC	CHA-GB	GBNSR6	IGB5	Null (ML alone)
Data set	relative compute speed	107	20	6	6	1	N/A
TEST	physics model (RMSE)	1.44	2.65	1.32	1.85	3.11	5.45
	physics + ML (RMSE)	0.88	1.36	0.90	1.50	1.65	1.53
	physics model (mean deviation)	−0.19	0.16	0.03	0.09	1.45	−3.90
	physics + ML (mean deviation)	0.33	−0.15	−0.01	0.40	0.26	−0.52
	physics model (5% outlier RMSE)	3.79	6.79	3.57	4.88	9.04	14.34
	physics + ML (5% outlier RMSE)	2.74	3.71	2.77	5.23	5.60	5.23
TRATNTNG	physics model (RMSE)	1.56	2.49	1.77	1.64	2.80	5.40
	physics + ML (RMSE)	0.63	0.91	0.70	0.67	0.93	0.94
	physics model (mean deviation)	−0.33	0.15	0.18	−0.06	1.38	−3.79
	physics + ML (mean deviation)	0.20	−0.23	−0.01	0.13	0.08	−0.37
	physics model (5% outlier RMSE)	4.83	7.75	5.84	5.07	8.38	14.26
	physics + ML (5% outlier RMSE)	1.98	2.96	2.38	2.16	3.19	2.68

Source code for the hyperparameter optimization, the final training and testing of ML models, in addition to all of the saved ML models and data sets used, is available on GitHub (see the SI).

3. Results and Discussion

3.1. Error reduction achieved by Physics + ML

For every physics-based model tested, ML decreases the RMSE to experiment for both training and testing data as seen in Fig. 5a. It is also clear that more accurate physics-based models consistently provide better overall predictive accuracy after ML corrections. In most cases, final RMSE values for both the training and test sets were better for the more accurate physics-based models and worse for the less accurate ones. This highlights the benefit of using more accurate physics-based models for further improvement with ML. However, the relative improvement for more accurate physics-based models was significantly less than that for less accurate models as seen in Fig. 5b. The two least accurate models, IGB5 and AASC, both had over 45% (almost two-fold) relative improvement for test set data while the other more accurate models each had only around 30% relative improvement.

Figure 5:

For every physics model tested, ML reduces the error relative to the experiment for both training and testing data, but the improvements due to ML correction tend to decrease as it is applied to more accurate physics models. Each set of blue (TRAINING) and green (TEST) points with an arrow between them represents a single physics model. (a) Physics + ML model RMSE values plotted against physics model alone. The y=x line (dash) represents no improvement; points below the line show improvement due to ML corrections. (b) Relative improvement due to ML corrections for each of the physics models.

Fig. 6 shows a more detailed view of how the ML-corrected physics models perform compared to the uncorrected models on the test set. As seen in the left column, the ML corrected predictions match up better to experiment and have fewer outliers than uncorrected predictions for each physics model – in particular, the negative trend line bias of IGB5 and AASC is mostly corrected after the ML correction is applied. The center column of Fig. 6 clearly shows that the distribution of error to experiment for each physics model is significantly narrower after ML corrections, although some outliers do persist. Finally, it can be seen in the right column of Fig. 6 that for most, but not all of the molecules in the test set, ML corrections improved the physics model predictions.

Figure 6:

(Previous page.) Error analysis of ML corrections to the five physics-based models of hydration, on the TEST set. (Left:) Scatter plots of predictions of five different physics models before (red) and after (green) ML corrections against experiment. Least squares regression lines are shown as dotted lines. (Center:) The distribution of errors relative to the experiment for five different physics models (unnormalized probability density) before (red) and after (green) their residual errors were predicted and corrected by ML. (Right:) The distribution of how much ML improved predictions (unnormalized probability density), with points to the left of zero indicating that the physics prediction was made worse.

Table 3 shows detailed results for the performance of each physics-based model without and with ML corrections on both the training and test sets. After ML corrections, the two least accurate physics models, IGB5 and AASC, were both within around 0.2 kcal/mol in accuracy (on the test set) from the uncorrected explicit TIP3P model. In both cases, the absolute improvement to the RMSE was over 1.2 kcal/mol and the final test set RMSE was just over half of that for the uncorrected models. A major reason for this significant improvement is that the ML corrections worked well to correct outlier physics model predictions with particularly large errors, greatly reducing the outlier RMSE for the two models seen on the bottom two rows of each section of Table 3. In contrast, the most accurate physics models saw less relative improvement, but the ML corrections were still able to bring TIP3P and CHA-GB RMSE values below 1 kcal/mol. Interestingly, ML-corrected GBNSR6 saw the least improvement and performed comparably to ML-corrected AASC and IGB5 on the test set after ML corrections, despite being a far more accurate physics model than these two. We see at least two possible explanations for the poorer then expected accuracy of GBNSR after the ML corrections. First, recall that the particular ML model chosen is a result of a stochastic optimization. Averaging over multiple optimization outcomes (see Table S2 in the SI) gives a more consistent relative performance ranking of the ML-corrected models, where, on average, GBNSR6 performs better than AASC and IGB5, as expected. In addition, we note that the ZAP9 radii set, used here with GBNSR6, was optimized exclusively using the FreeSolv database and, as a result, may be overfit to the database. Relative overfitting of ML corrections was highest for GBNSR6 at 55%, lowest for CHA-GB at 23%, and remained between 25% and 45% for the other physics models. IGB5 had the highest mean deviation to experiment, but after ML corrections, this systemic error was significantly reduced in the training and test sets as seen in Table 3.

3.2. Error reduction by ML Alone

Based on the accuracy metrics in Table 3, ML alone predictions of HFEs show performance similar to the ML-corrected AASC, GBNSR6, and IGB5 models on the FreeSolv database. The error distribution seen in Fig. 7b is also comparable to those ML-corrected physics models. However, as seen in Fig. 7a, the trend line for ML alone deviates significantly from experiment for molecules with high absolute HFE values. In contrast, the ML-corrected AASC, GBNSR6, and IGB5 models follow the correct trend across the entire range of HFEs in the FreeSolv database as seen in Fig. 6 (left) We expect that ML alone will likely perform significantly worse for molecules that are very different from those in the FreeSolv database; we will test that expectation in the next section.

Figure 7:

(a) Scatter plot of ML alone model predictions for the TEST set against experimental values. The model deviates from experiment near the extremes, as highlighted by the dotted green trend line – linear regression fit y = 0.72x − 0.58, which is significantly off from experiment. (A perfect match to experiment is indicated the y=x grey line). (b) Distribution of the errors of ML alone predictions on the TEST set.

3.3. Application to blocked amino acids

To test how generalizable the ML models trained on small neutral molecules may be, they were used to predict the HFE values for a set of 44 structures of blocked amino acids:^73,144 two conformations for each of 20 blocked amino acids and for 2 additional neutralized ASP and GLU analogs. Unfortunately, there is no experimental data for this set, which includes charged species, therefore TIP3P calculated polar solvation values^73,144 were used as a reasonable proxy. Note that our goal here is limited to identifying gross errors, if any. As seen in Fig. 8, the ML alone model performs very poorly on this amino acid dataset, predicting HFE values of around −10 kcal/mol for all amino acids, whereas the TIP3P calculated values for this set have a mean of −28.9 and a range of −12.0 to −82.7 kcal/mol. It is expected that the ML model performs poorly on the charged amino acids, with their high absolute HFEs – these were completely absent from the training set. However, it is somewhat surprising that the ML alone model performs as poorly as it does on uncharged amino acids, given that there exist several amino acid analogs in the FreeSolv dataset. A possible reason for this poor performance for the ML alone model is that the HFEs for the uncharged blocked amino acids (−30 to −12 kcal/mol) are not within the typical range of values used in the training SM set (−12 to 3 kcal/mol). Only 15 of the 642 molecules in the FreeSolv database have HFE values outside this typical range, meaning that all of the amino acids, even neutral ones, have HFEs that are outside the normal range of molecules used to train the ML models.

Figure 8:

ML alone fails to preserve the physically correct trend for HFEs of blocked amino acids, while physics with ML corrections follows the correct trend. Shown are (a) ML alone model and (b) CHA-GB before (red x’s) and after (green circles) ML corrections on the blocked amino acid data set. In both cases, the ML models are trained on small molecules in the FreeSolv database and evaluated here on a completely separate set of blocked amino acids, see “Methods”.

As seen in Figure 8a, the ML alone model performs very poorly when in predicting HFEs for the amino acids. This result shows that the ML alone model cannot predict HFEs with any useful accuracy for molecules with HFEs significantly outside the range of HFEs it was trained on.

In contrast, when the ML correction model is applied to predictions of each of the physics models (except TIP3P), the overall correct trend is preserved as seen in Figure 8b and Figure S1 in the SI, which is in stark contrast to the results using ML alone. We refrain from quantifying possible improvement of accuracy that the ML corrections may provide for physics-based HFEs of blocked amino acids. This is mainly because we would have to compare these against available TIP3P calculated values, which have their own error relative to the experiment. A minor technical point is that the reference TIP3P HFE values available to us for the blocked amino-acids are for the polar solvation part of the total hydration energy, rather than the total HFE values that the models were trained for. While the difference between the two is relatively small for these highly polar molecules, which justifies using the polar component to make the qualitative statements above, it is not completely negligible.

3.4. Impact of hyperparameter optimization

So far, the hyperparameters used for all the final ML models were optimized for TIP3P for reasons outlined in “Methods”. Distinct, but computationally inexpensive ML models using these hyperparameters were then trained and evaluated for each of the physics-based models. We find that despite the hyperparameters being optimized for TIP3P, each of the final physics + ML models works to significantly reduce the error to experiment for its respective physics-based model, Table 3. To test how the hyperparameters may impact performance for the different ML-corrected physics models, we re-trained the ML models using two additional hyperparameters sets and compared them to the results based on the previously optimized hyperparameters. For both sets of hyperparameters, the differences in test and training set accuracy compared to the optimized hyperparameters are relatively small as seen in Table S3 of the SI. In general, it appears that a reasonable set of hyperparameter values similar to what is used here works relatively well and does not affect any of the key conclusions made in this work. Based on these results, it does not appear necessary to optimize hyperparameters for each physics model separately. A more detailed analysis of these results can be found in Section 5 of the SI.

4. Conclusions

This work investigates use of ML (DNN) to improve the accuracy of estimating hydration free energies (HFEs) of small molecules by predicting and correcting the errors of physics-based atomistic water models. This strategy is explored for several popular implicit and explicit water models and evaluated mainly on experimental HFEs from the FreeSolv database, arguably the largest publicly available database of small neutral molecules. The graph convolutional deep neural network (DNN) implemented as the GraphConvModel in DeepChem is utilized to try to learn the remaining error that a physics model alone fails to properly account for. A distinct feature of the approach, which sets it apart it from many previous works that combined physics and ML to improve accuracy of HFE (or other molecular property) predictions is a complete separation of the physics and ML (DNN) parts of the overall workflow: the output of the physics-based model becomes the input of the ML part, the latter being a “pure” DNN, not a physics-motivated ML model.¹¹⁰ That is the proposed strategy aims to render unto clear and traceable physics all that is physics’.

We find that for every physics-based water model employed to test our approach, the subsequent ML treatment reduces the RMSE relative to the experiment, improving individual physics-based HFE predictions for most molecules in the test set. The ML treatment also reduces outlier errors for most physics-based models. With the ML correction, even relatively low-accuracy implicit solvation models achieve the final accuracy comparable to the uncorrected explicit solvent TIP3P model, which is orders of magnitude more computationally expensive. A noteworthy trend that emerges from this investigation is that the relative improvement of the physics-based HFE accuracy due to subsequent application of the ML tends to be lower for higher accuracy physics models. Moreover, the amount of relative accuracy improvement does not appears to be sensitive to the type of the physics model; for example, the relative benefit of the ML correction is essentially the same for the explicit solvent model TIP3P and the implicit solvent model CHA-GB, which have very comparable accuracy on the small molecule sets considered here. For all of the physics models, the ML correction carries negligible additional computational expense. A noteworthy feature of our approach is that the remaining error that the DNN has to correct is relatively small. Therefore, the final ML-corrected solution retains most of the “physicality” of the underlying physics-based model, and likely contains only a relatively small amount of “untraceable physics” often attributable to purely DNN solutions.

For practical application, the strategy we propose requires no additional effort to integrate the DNN post-processing correction with the physics-based model. With any of the five physics-based solvent models considered in this work, regardless of their specific implementation, the approach could be called “plug-and-play”, in that the HFE of any new small molecule can be predicted by simply running one of the pre-trained DNN models as a post-processing step, following a completely independent pre-processing by the chosen physics model. For a physics model not yet on the list, hyperparameter re-optimization is not required, which makes the task of using the approach still relatively straightforward. Given predictions for the FreeSolv database by the new physics model, the corresponding DNN correction model can be trained in roughly 30s on a CPU core. We see this computational facility as one of the advantages of the approach, which we hope will encourage further evaluation. The source code for the hyperparameter optimization, the final training and testing of ML models, and all the saved ML models and data sets are available on GitHub, see the Supporting Information.

We have also compared, in some detail, the “physics + ML” with the corresponding “ML alone” approach, the comparison showing several noteworthy trends. First, the computationally inexpensive ML alone predicts HFE with similar RMSE accuracy as several of the physics-based models after the ML correction; these include AASC, IGB5 (GB-OBC), and GBNSR6 (the latter using ZAP9 radii) implicit solvent models. However, the ML alone has significantly worse accuracy than the ML corrected TIP3P and CHA-GB models, the latter two representing the upper edge of the accuracy range of the physics-based solvent models explore here. Second, significant deviations from experimental HFE values become apparent for the “ML alone” solution at the edges of the FreeSolv database: the overall trend line predicted by the ML is clearly off the mark, unlike that of all of the physics-based, or “physics + ML” models tested.

When used on molecules outside of the range of HFEs in the FreeSolv data set, ML alone performs very poorly, while even the least accurate “physics + ML” models reproduce the correct overall trends, even well outside the training range. This comparison emphasizes a key advantage of the approach, in which ML is used only to correct the relatively small error remaining after the physics-based prediction has already been made, independently. The approach guarantees that the correct overall trend (if present in the physics-based model) persists in the final prediction: if the molecule’s HFE is close to the range of HFEs in the training set, the physics-based prediction is likely further finessed by the ML follow-up. Given the above considerations, about the only clear advantage to the ML alone, compared to the physics+ML approach, would be a significantly faster run time in the case of relatively slow physics models such as TIP3P. However, this advantage becomes less obvious in the case of the implicit solvent physics models that are already fast enough.

The proposed approach has several limitations. First, as mentioned above, the benefit of the ML correction for the accuracy of HFE predictions is expected to be minimal or non-existent for molecules that differ significantly from those in the FreeSolv database: for these molecules, the accuracy of the final prediction is close to that of the physics model alone. Without further investigation, this particular limitation is hard to avoid unless the training set is expanded significantly. In the near future, this limitation can be potentially be mitigated somewhat by applying a more sophisticated train/test partitioning of the existing data set of small molecule HFEs. Second, all the HFE values used for the training have experimental uncertaintties, but these uncertainties were not considered when training the ML models. The uncertainty sets a limit for the achievable accuracy, and also means that the ML models may learn systematic experimental error. The concern is mitigated by an observation that the errors of the final predicted HFEs are still appreciably higher than the average experimental uncertainty of the FreSolve set. In the future, one can consider accounting for the experimental uncertainties in the training procedure. Third, some inevitable over-fitting is still present in the final models. The degree of over-fitting varies between the models, for more accurate physics models it is close to, or even smaller than, the target of 30%, but becomes larger for less accurate models, including ML alone. In the future, one can explore setting a tighter over-fitting threshold at the hyper-parameter optimization stage. Forth, a limitation of the specific featurizer (part of DeepChem) employed here, is that it is agnostic to conformation of the molecule, sacrificing exact positional data for generalizability. As a consequence, changes in the HFE due to conformational transitions in the same molecule are only accounted for by the physics part of the procedure. We believe that this specific limitation may also be overcome in the future. Finally, given the trend of lower relative improvement for more accurate physics models, it remains to be seen whether the proposed approach would have a noticeable impact for physics models significantly more accurate than those (TIP3P or CHA-GB) at the upper end of the accuracy range of HFE predictive power explored in this work.

In summary, we have proposed a strategy for improving predictions made by physics-based water models by using deep neural networks to make corrections as an independent post-processing step. The strategy, tested on improving accuracy of hydration free energy predictions, shows promise, making it worthwhile to pursue further exploration of this approach, while also encouraging its adoption for immediate practical applications.