Towards a machine learned thermodynamics: exploration of free energy landscapes in molecular fluids, biological systems and for gas storage and separation in metal–organic frameworks

Abstract

In this review, we examine how machine learning (ML) can build on molecular simulation (MS) algorithms to advance tremendously our ability to predict the thermodynamic properties of a wide range of systems. The key thermodynamic properties that govern the evolution of a system and the outcome of a process include the entropy, the Helmholtz and the Gibbs free energy. However, their determination through advanced molecular simulation algorithms has remained challenging, since such methods are extremely computationally intensive. Combining MS with ML provides a solution that overcomes such challenges and, in turn, accelerates discovery through the rapid prediction of free energies. After presenting a brief overview of combined MS–ML protocols, we review how these approaches allow for the accurate prediction of these thermodynamic functions and, more broadly, of free energy landscapes for molecular and biological systems. We then discuss extensions of this approach to systems relevant to energy and environmental applications, i.e. gas storage and separation in nanoporous materials, such as metal–organic frameworks and covalent organic frameworks. We finally show in the last part of the review how ML models can suggest new ways to explore free energy landscapes, identify novel pathways and provide new insight into assembly processes.

---

Design, System, Application

Machine learning (ML) has emerged as a remarkably efficient predictive tool for thermodynamic properties, and has the potential to accelerate considerably the screening of molecules, biological systems or of nanoporous materials for specific applications. To achieve this, ML models are optimized using training datasets generated through high-resolution molecular simulations (MS). Here we focus on free energy and examine how combined MS–ML approaches shed light on the properties of molecular systems, leading to the rapid determination, e.g. of their phase behavior. We also discuss how such methods can be extended to biological systems, with the identification, for instance, of the folded states of proteins, and to fluids confined in nanoporous materials, with the fast evaluation of the free energy costs involved in the gas storage and separation of energy-related molecules, such as hydrogen or methane, and of environmental contaminants. These approaches can also be leveraged to explore free energy landscapes, leading to the elucidation of novel pathways for a wide range of activated processes.

1 Introduction

In recent years, machine learning (ML) has emerged as an extremely useful tool to explore and predict complex phenomena.^1–16 Data-driven methods yield excellent results when applied to the parametrization of new force fields and coarse-grained models^17–27 or to facilitate the exploration of the chemical space in inverse design methods.^28–30 ML methods have also been applied in recent years to the reconstruction of complex high-dimensional potential energy surfaces^31–34 and to the prediction of thermodynamic and kinetic properties.^35–37 This considerably accelerates the determination of the key properties for these systems, since their computation via conventional molecular simulation methods often requires an extensive sampling of the phase space, i.e. performing simulations over very large time-scales and length-scales that quickly become extremely computationally intensive. ML can also provide new insights into assembly processes^38–40 and yield predictive models for heterogeneous catalysis.^29,41 Artificial neural networks have also been shown to provide access to free energy landscapes that are difficult to compute. Examples include processes that involve transitions from one state to another, a task for which rare event sampling and enhanced sampling simulations are required.^42–46 Similarly, ML models can be leveraged to predict adsorption isotherms, adsorption free energies, catalytic activities on nanoclusters surfaces⁴⁷ and as a way to accelerate considerably materials discovery. ML algorithms and ensemble learning models yield new routes to quickly screen potential candidates for application in gas storage and separation.^48–50 ML predictions on gas adsorption capabilities can also be carried out on the basis of crystal designs of materials such as metal–organic frameworks (MOFs) and covalent organic frameworks (COFs) at operating conditions.^51–54 Such predictions, in turn, also suggest new ways of tailoring novel materials with enhanced adsorption properties.^55–58

In this review, we focus on how ML can help us predict the key thermodynamic properties that govern the evolution and outcome of a system. This includes the Helmholtz free energy for systems with a constant volume and temperature, the Gibbs free energy when the temperature and pressure are held constant or the entropy in the case of isolated systems. Many thermodynamic properties can be readily measured in experiments, or calculated during the course of molecular simulation (MS) runs. This is the case, e.g., of the volume, temperature, pressure, internal energy or enthalpy of a system. On the other hand, thermodynamic functions like the entropy and the Helmholtz or Gibbs free energy cannot be evaluated directly, as they require information on all the microstates available to the system. This has led to the invention of novel MS algorithms that achieve an extensive sampling of all possible configurations of the system. The next section of this review discusses several of these strategies, as these MS algorithms provide access to free energies and free energy landscapes that serve as the starting point for the construction of ML models. We briefly discuss how MS data can be used to train ML models on the example of a well-established and widely used deep learning algorithm, know as artificial neural networks (ANNs).^1,59–63 We then show how MS–ML approaches lead to accelerated predictions of free energies and extend the discussion to other ML methods for free energy predictions. Then, we examine several applications of ML models to molecular and biological systems, before turning to applications in the field of gas storage and separation. We finally discuss how MS–ML approaches can be leveraged to unravel new pathways for assembly processes and provide new insights into such processes.

2 Building ML models for thermodynamics

2.1 Datasets generation

The first step consists in the generation of data on the free energy for the system under study. This data will then be used to train and teach the ML models and allow them to learn the thermodynamics, and free energy landscape, of the system. To achieve the evaluation of free energy, MS algorithms rely on determining free energy differences along a thermodynamic path connecting the system actually studied and a reference state for the system. Free energy differences can be indeed obtained by thermodynamic integration^64–68 along paths that connect, for instance, a liquid to an ideal gas or a “real” crystal to an Einstein crystal.^69,70 For instance, considering a crystal for a given number of atoms N, volume V and temperature T, the Helmholtz free energy difference ΔF is given byin which U denotes the potential energy of the system and λ is a reaction coordinate along a path that connects the crystal phase of unknown free energy (for λ = 0) to an Einstein crystal of know free energy (for λ = 1). In practice, λ is a multiplying factor which controls switching of harmonic springs that tether the atoms to the lattice sites (λ = 1) and switching off these springs (λ = 0).

Another class of methods, known as enhanced sampling simulations, is required when the system has to overcome large free energy barriers along this thermodynamic path. This is, e.g., the case when a molecular system undergoes a phase transition through the nucleation of a new phase, or when a protein undergoes a conformational change or folding event. Examples of enhanced sampling methods include the umbrella sampling method^71–73 and metadynamics.^74–77 In such cases, an external potential energy ϒ may be added to the system to bolster the sampling of configurations of high free energy, which would not be observed in the absence of the external potential. This potential is often taken to be a harmonic function of a reaction coordinate ϕ that spans the thermodynamic path. Statistics for P(ϕ), the probability of observing configurations with a given value of ϕ, are collected over the course of MS runs. The free energy profile is then obtained by subtracting the external potential energy and by recognizing that the free energy along the profile can be calculated from –k_BT ln P(ϕ). For more details on the derivation of the free energy using the umbrella sampling method, we refer the readers to the excellent references by Torrie and Valleau,⁷¹ as well as recent work by Kästner.⁷⁸ More specifically, examining a process for a given number of atoms N, pressure P and temperature T, the Gibbs free energy difference can be calculated along a path spanned by the reaction coordinate ϕ. In this case, ΔG is given by

ΔG(ϕ) = −k_BT ln P(ϕ) − ϒ(ϕ) (2)

In this equation, P(ϕ) is collected during the umbrella sampling simulation and the extra potential energy ϒ(ϕ) is removed to yield the free energy for the system actually studied, i.e. without the extra potential energy ϒ(ϕ). This approach has been applied extensively to calculate free energy profiles associated with rare events, including the formation of crystal nuclei,^45,79–88 critical droplets and bubbles in supersaturated phases^89–92 and under nanoconfinement.^93–95 Such approaches can also be combined with other methods to determine the rate of, among others, nucleation events.^96,97 Rare events can also be studied by focusing on the transition paths that the system takes, as it travels across the free energy landscape. Methods like the transition path sampling^98,99 or the transition interface sampling^100,101 approaches, generate an ensemble of the transition paths connecting the two states, or free energy minima, and are especially useful for the determination of rate constants. Alternatively, adiabatic free energy dynamics^102,103 can also be employed to generate free energy profiles for molecular and biological systems. Finally, free energy theorems^104–106 based on nonequilibrium statistical mechanics can also be leveraged to determine free energy differences throughin which β = 1/(k_BT), W is the amount of work done on the system to, e.g., pull a protein from one conformation to another and <…> indicates an average carried out over many nonequilibrium trajectories connecting the two conformations. In such a case, ΔF gives access to the free energy differences between two conformations, or equilibrium states.

Free energy can also be determined through another broad class of simulation approaches known as flat histogram methods.^107–113 Such approaches rely on an extensive sampling of all possible configurations of the system, with the aim of determining the density of states g(E) of a given system. One such approach, the Wang–Landau sampling method,^{107,114–116} was initially implemented in the microcanonical (N, V, E) ensemble and consists in performing a random walk in energy space with a probability proportional to the reciprocal of the density of states 1/g(E). At the start of the simulation, g(E) is unknown. Its value is initially set to g(E) = 1 for all E, with g(E) dynamically updated during the simulations until a flat histogram for the number of visits of each E interval is obtained. This method thus yields g(E) up to a multiplicative constant, It can also be extended to various ensembles including the canonical (N, V, T) ensemble¹⁰⁹ or the isothermal–isobaric (N, P, T) ensemble^{110,111,117–120} to determine the corresponding partition functions up to a multiplicative factor. In some cases, the multiplicative factor can be determined, and exact values for the partition functions, and thus for the free energy, can be obtained. This has been achieved with the expanded Wang–Landau (EWL) simulation method^{112,121–124} that performs an extensive sampling of the grand-canonical (μ, V, T) ensemble.^125–128 In this case, the algorithm yields Θ(μ, V, T), the partition function for the grand-canonical ensemble, as well as the underlying canonical partition function Q(N, V, T), as given by

in which μ is the chemical potential. Here the multiplicative constant for the partition function is readily obtained by recognizing that Q(N = 0, V, T). Once the partition function is known, the free energy can be obtained through statistical mechanics relations such as

F(N, V, T) = −k_BT ln Q(N, V, T) (5)

EWL methods apply to all processes where the number of molecules, or composition of the system, is allowed to fluctuate, for instance in mixtures¹²¹ and in the study of adsorption phenomena¹²¹ for gas storage^129–131 and separation applications.¹³² Other flat histogram techniques include the transition matrix Monte Carlo (TMMC) algorithm, which has been recently applied to the determination of adsorption isotherms in nanoporous materials.^133,134 Furthermore, recent work has shown how the output of TMMC simulations could be used to extrapolate the thermodynamic properties through Taylor expansions.¹³⁵

2.2 Model training and validation

The second step consists in using the simulation data to train a ML model. ANNs have emerged as highly versatile deep learning algorithms and are popular choices for ML models.^1,59–63 Thus, to illustrate how ML models can be built to interpolate between, and extrapolate beyond, the conditions covered by the simulation data, we consider an ANN as outlined in the schematic blueprint of Fig. 1 that summarizes how ML models can be built from the simulation data. Popular approaches rely on the training of neural networks with a feed-forward structure and use an optimization algorithm back-propagation error calculation¹³⁶ to optimize the weights, or contributions made by each neuron, to the overall result. As an example, we consider an ANN with 4 layers. The first layer is an input layer, with a series of input neurons that correspond to the key parameters or descriptors for the system. There can be as many input neurons G_i (i = 1, 2,…) as desired. These can be thermodynamic variables, such as T or P, or geometric parameters, such as, for instance, dihedral angles in the case of free energy landscapes for protein folding. The next two layers are known as hidden layers, and contain variable numbers of neurons h₁ and h₂. The last layer is termed the output layer, with output neurons corresponding to the properties the ML model aims to predict including, e.g., the free energy, partition function^137,138 or any other property of interest. If one of the output neurons is for the ML-predicted free energy F^ML, the ML model provides the following analytic equationHere W denotes the weight matrix, f₂, f₃ and f₄ are activation functions and b_i are bias nodes that act as adjustable offsets. The weights are initially chosen as random numbers and popular choices for the activation functions include the sigmoid, tanh or linear functions.

Fig. 1 Schematic blueprint for a combined MS–ML protocol. Molecular simulations (first panel on the left) can be carried out to generate free energy profiles (top of second panel) or partition functions (bottom of second panel). The simulation data is then used to train a ML model (here the ANN of the third panel) and lead to the prediction of free energies for conditions outside of those covered by the simulation data (here a 3D plot of F(ϕ₁, ϕ₂) in the (ϕ₁, ϕ₂) plane) and of free energy minima as shown by the bright regions on the plot (fourth panel on the right).

The ANN is trained by minimizing an error function that quantifies the difference between the free energy from the simulation data F_i and the free energy predicted by the ML model F^ML_i

where N_d is the number of data points included in the training dataset.

Weights can be adjusted after each forward pass, using an optimization algorithm during the backward pass. The process is then repeated iteratively until additional iterations do not lead to a change in the accuracy of the predictions. The back-propagation algorithm is often used to optimize all weights once per iteration, leading to the following equation

in which Wⁿ and Wⁿ⁺¹ are the weights after n and n + 1 iterations, respectively. The partial derivatives of the error function Δ are calculated analytically, and the calculations are repeated until a convergence criterion is satisfied. Possible convergence criteria are based either on the rate of convergence, i.e. by how much Δ improves after each iteration, or on set limits for the value of Δ. The last adjustable parameter λ controls how fast the ANN actually learns, and is thus referred to as a learning rate.

The simulation data is generally split between a training set for the optimization of the ANN weights and a hold-out set for validation purposes. It is often useful to carry out calculations of root-mean square errors for both the training and hold-out sets. Both RMSEs are helpful in determining the accuracy of the ML model, but also to assess if any underfitting or overfitting of the simulation data takes place. For instance, obtaining a small RMSE for the ML model with respect to the training dataset and a large RMSE with respect to the hold-out/validation dataset can be indicative of an overfit. On the other hand, having RMSEs that are small and of the same order for both datasets shows that the ML model performs very well.

The size of the datasets used to train and validate the ML model depends on several factors, such as, e.g., the complexity of the data to be modeled and the type, and architecture, of the ML model. Preliminary tests are often performed to assess how the performance of a ML model scales with the dataset size. In practice, this can be achieved through the use of a learning curve, or training curve, that captures how the performance of the ML model, measured through a training and a validation score, improves as the size of the training dataset is increased. This tool is especially useful in determining if the ML model gives rise to any underfitting or overfitting of the data, or if the ML model performs very well for the task at hand. Typical sizes for datasets used in the training of ML models for the prediction of free energy surfaces include of the order of 10⁴–10⁵ data points.¹³⁹ A set of 10⁴ data points is typical for datasets for partition functions used in the training of ML models.¹³⁷

The preparation of the datasets is also key to the optimization of the ML model. In order to reduce bias, it may be advantageous to adopt an ensemble learning approach, in which the ML predictions are averaged over several ANNs with different W matrices.^50,140 The k members of an ensemble of ANNs can be obtained through several strategies. One such strategy is known as k-fold cross-validation. In this case, a collection of ANNs with exactly the same architecture is trained on randomized subsets of the training dataset, and the k models are then used as the members of an ensemble.¹⁴¹ Another strategy is known as the bootstrap aggregation (bagging)¹⁴² approach with replacement. The idea there is to generate k training subsets with different sample densities, with the aim of emphasizing the weight of different parts of the dataset in the optimization process. Alternatively, a diversity approach can be adopted by using different ANN architectures and varying the number of neurons in the hidden layers.¹⁴³

Several other ML methods can also be applied to build and predict free energy surfaces from the simulation data. Such methods include Bayesian inference,^144–150 Gaussian progress regression,¹⁵¹ kernel ridge regression, support vector machines, weighted neighbor schemes,¹³⁹ dimensionality reduction as well as transfer learning and reinforcement learning.¹⁵² We finally add that ML models perform extremely well when they are used to make predictions for conditions that lie between those covered in the training datasets. Extrapolations beyond the range of conditions included in the datasets can often be carried out reliably, provided that the systems exhibit similar properties, e.g. similar phases for the predictions of fluid properties.^50,137 However, as a general rule, extrapolations using a ML model should be carried out with caution, especially if the system undergoes dramatic changes (e.g. conformational changes for a biomolecule) that are outside the range of possibilities considered in the training set. In that case, direct extrapolations may yield to inaccurate predictions.

3 ML-predictions of free energy landscapes for molecular and biological systems

The introduction of computer simulations in science has advanced tremendously our understanding of the mechanisms that take place at the microscopic level in molecular and biological systems. The ability of simulations to model increasingly complex systems and to bridge between different length scales and time scales via multiscale methods has been instrumental to the unraveling of many chemical processes.^153–155 As simulations have taken into account finer and finer details of the intermolecular interactions, leading, in turn, to increasingly accurate results, CPU and GPU times have become more and more significant. In order to overcome such challenges, an increasing number of researchers implement ML approaches to fast-track the simulations¹⁵⁶ and accelerate chemical discovery.¹⁵⁷ ML methods have been used to screen materials, providing excellent starting points for the computational optimization of catalysts and for the discovery of new trends and behavior.^158,159 They have also been employed to predict the properties of molecules and crystals, with the development, e.g., of novel ML models based on graph networks to obtain the formation energies, band gaps and elastic moduli of crystals with an accuracy comparable to density functional theory when very large dataset are used for training purposes.¹⁶⁰

ML methods can also be leveraged to predict the free energy of molecular and biological systems. This can be achieved, for instance, by providing a way to extrapolate the simulation data beyond the range of conditions covered by the simulations.^{44,137,138,148} Such protocols have been applied to molecular, polymeric and biological systems, which present stringent tests for the methods as the energy landscapes are often rugged. A variety of methods have been developed to tackle such systems. For instance, nonlinear machine techniques have been used to recover single molecule free energy landscapes from molecular simulations.⁴² In this case, the diffusion map nonlinear machine learning technique is used to understand the relation between changes in external conditions, or in molecular chemistry, and the free energy landscape, with applications to the n-eicosane chain and to a family of polyglutamate-derivative homopeptides, In the latter case, the helical stability-side chain length interdependence and the critical side chain length for the helix–coil transition were identified.

Artificial neural networks can also be employed to learn free energy landscapes through the use of adaptive biasing potentials and of Bayesian regularization to increasing the robustness of the approach to hyperparameters and overfitting.¹⁴⁸ In this case, Bayesian regularization penalizes network weights and auto-regulates the number of effective parameters in the network. Alternatively, ANNs can be used to generate the free energy landscapes for the conformational equilibria in complex molecular systems.⁴⁴ Starting from free energy data obtained from enhanced-sampling molecular simulations, ANNs are trained to represent the free energy surfaces of the alanine di- and tripeptides in the gas phase. Another approach consists in using ANNs to predict the partition function of molecular fluids¹³⁷ and, thus, to gain access to all properties of the system, including the free energy as outlined in the previous section. Using simulation data for the partition functions obtained from expanded Wang–Landau simulations, ANNs are trained to predict the free energy and the phase behavior of molecules over a wide range of conditions. In the case of higher-dimensional surfaces, such as in the case of multicomponent mixtures,¹³⁸ the determination of free energy and of the locus for phase transitions can be greatly accelerated by the combined use of ML and MS methods, with combined approaches only requiring about 20% of the computational cost involved in a conventional flat-histogram MS approach to elucidate fully the free energy landscape and phase behavior. In recent work, alternative strategies, including kernel ridge regression, support vector machines and weighted neighbor schemes have been used to learn free energy landscapes and generate accurate ensemble averages for the observable properties of oligopeptides in the gas phase, as well as in an aqueous solution.¹³⁹

In addition to providing predictions for the free energy, ML models can also be used to gain access to the kinetics of processes.¹⁶¹ As the system under study becomes more complex, its underlying kinetic properties become increasingly challenging to determine and interpret. This is most particularly the case of proteins, and the implementation of dimensionality reduction, transfer learning and reinforcement learning is starting to show promising results.¹⁵² ML models can also be used to compute the solvation free energy. Approaches using a global optimization procedure have been developed to identify low-energy molecular clusters for different numbers of explicit solvent molecules,¹⁶² and sketch maps and nonlinear dimensionality reduction algorithms can be leveraged to quantify similarities between solute environments in microsolvated clusters. Hydration free energies have also been determined by combining alchemical free energy calculations with ML, leading to the computation of highly accurate absolute hydration free energies.¹⁶³ Such approaches that combine free energy methods with machine learning show great promise, most particularly for the study of systems of increasing complexity, including the determination of the free energy of protein–ligand binding for drug design and drug development,^164–167 and in the membrane transport cycle.¹⁶⁸

4 ML thermodynamics of adsorption

The development of ML models for adsorption processes has drawn considerable interest in recent years. Several types of applications can indeed be explored through such models. This includes catalysis, with the determination of adsorption energies of molecules such as hydrogen⁴⁷ on metal surfaces and metal nanoalloys through, e.g., the combination of ML with density functional theory (DFT) methods.^56,169–178 ML predictions for the free energy of adsorbed phases, and the corresponding surface phase diagram over a wide range of coverages and adsorbates, has been carried out with a Gaussian process regression model on IrO₂ and MoS₂ surfaces for applications in electrocatalysis.¹⁷⁹

Another broad class of applications deals with the study of porous nanomaterials, such as metal–organic frameworks (MOFs) and covalent organic frameworks (COFs).^58,180 These materials are candidates for gas storage and separation in energy-related processes⁴⁸ and in environmental processes. Examples of systems for such applications include, among others, the storage of methane¹⁸¹ or hydrogen for energy applications, as well as the sequestration¹⁸² of CO₂ or of polycyclic aromatic hydrocarbons (PAHs)¹⁸³ for environmental applications. In these systems, a gas phase is in thermodynamic equilibrium with the gas adsorbed inside the COF or MOF, which means that the two phases share the same molar Gibbs free energy or chemical potential (as well as the same temperature). In this case, the relation between free energy and the properties of the adsorbed phase can be obtained through grand canonical Monte Carlo (GCMC) simulations. These simulations provide access to, for instance, the amount of molecules adsorbed in the framework as a function of the chemical potential, or to the selectivity towards a specific molecule in the case of the adsorption of mixtures.

ML can be combined with GCMC simulations to screen nanoporous materials and identify the best candidates to reach volumetric targets for practical applications. For instance, hydrogen storage is a major challenge for hydrogen use in contemporary applications as a fuel for automobiles and vehicular applications. Here, the idea is to have sufficiently densified hydrogen at a moderate pressure to power vehicles. Nanoporous materials such as, e.g., MOFs and COFs are novel materials that effectively alleviate the issue of storage pressure.^184–188 Recent studies have started to leverage ML methods to model and predict adsorption in such systems. The combined use of grand canonical Monte Carlo simulations and of neural networks⁵³ has led to the proposal of new crystal designs, and their performance has been assessed by determining hydrogen storage capabilities for various pressure swing conditions.⁵¹ Gaussian process regression has also been employed to predict hydrogen adsorption in nanoporous materials using the CoRE-MOF database.¹⁸⁹ Furthermore, strategies combining Monte Carlo approaches with ML methods have yielded very promising results, in their efficiency in screening materials as candidates for the storage of several gases, including methane,¹⁹⁰ carbon dioxide, hydrogen and hydrogen sulfide.¹⁹¹

Another application of ML to adsorption phenomena is the ‘in silico’ discovery of novel materials for carbon capture. Genetic algorithms have been employed to compute interactions between adsorbates like CO₂ and the framework, and to derive accurate force field parameters for molecular simulations.¹⁹² Moreover, deep learning has allowed for the improved assessment of the importance of textural properties in porous carbons for CO₂ adsorption. In this case, ANNs are trained as a generative model to shed light on this interdependence and, as a result, to guide the development of the next generation of porous carbon materials.⁵⁷

ML can also be leveraged to tackle the adsorption of multicomponent gas mixtures. In this case, having multiple components in the adsorbate increases the dimension of the parameter space that needs to be studied. This is indeed necessary to quantify the dependence of the selectivity towards the adsorption of a given component, but also to determine how the thermodynamic properties of adsorption vary as a function of the mole fraction of each of the mixture components. Deep neural network have recently been employed to study binary sorption equilibria. In some cases, ML methods are even used as a screening method before extensive molecular simulations are run. In recent work, a multilayer perceptron was trained to model the adsorption of alchemical species in MOFs. Alchemical species are represented by variables derived from the force field parameters. MOFs are also described by simple descriptors, such as their geometric properties and chemical moieties. This protocol allowed for the prediction of adsorption for systems relevant to chemical separations, including Ar, Kr, Xe, CH₄, C₂H₆ and N₂ in MOFs.⁵⁴ Another approach consists in carrying out Gibbs ensemble Monte Carlo simulations to generate simulation data on desorptive drying processes in zeolites.¹⁹³ The results are then used to train a multi-task deep ANN to predict equilibrium loadings as a function of thermodynamic state variables for (1,4-butanediol or 1,5-pentanediol)/water and 1,5-pentanediol/ethanol mixtures in an all-silica MFI zeolite and for the 1,5-pentanediol/water mixture in an all-silica LTA zeolite, leading to the rapid optimization of the desorption conditions.

Ensemble learning has also been shown to provide a rapid assessment of the performance of nanoporous materials for separation purposes. In this case, the output of several ML models, e.g. the numerical values given by the output neurons of ANNs, are averaged to remove bias, either in the choice of the data used for the training dataset or, for instance, in the choice of a specific ANN architecture. As discussed in section 2, techniques that randomize the choice of the simulation data used to train the ML model, such as bagging and k-fold cross-validation, and diversity approaches, that average the results obtained with different ANN architectures, can greatly improve the accuracy and transferability of the results. Recent results have show that ensemble learning the partition functions of fluid confined in MOFs and COFs (see Fig. 2) leads to accurate predictions for the selectivity and for the free energy of immersion of the gas in the MOF/COF for H₂ storage, CO₂ storage. The prediction of the free energy of immersion is especially relevant to practical applications, since it captures the free energy cost of regeneration of the adsorbent for practical application. This approach was also applied to gas separation on the specific example of methane–ethane gas separation in COFs. More specifically, it allowed for the rapid screening of a series of COFs through the ML prediction of their relative performance towards the selective adsorption of one of the mixture components, and the efficient evaluation of the optimal operating conditions.⁵⁰

Fig. 2 Sequence for a combined MS–ML approach for the determination of the properties of adsorption for gas storage and adsorption in MOFs and COFs. (a) Configurations are generated during MS, for instance here a configuration for a system of H₂ adsorbed in COF-108 (adapted with permission from ref. 130, copyright Taylor & Francis), (b) the logarithm of the canonical partition function Q(N, V, T) is computed during MS and trains a ML model. Here representative ln Q(N, V, T) are shown for CO₂ in IRMOF-1 (adapted with permission from ref. 129, copyright American Chemical Society), (c) the selectivity can then be derived from the canonical partition function for gas separation in ethane–methane mixtures with a methane mole fraction y_M = 0.5 (adapted with permission from ref. 132, copyright American Chemical Society), and (d) immersion free energy for an ethane–methane mixture in COF-102 predicted via the MS-ML approach (adapted with permission from ref. 50, copyright American Chemical Society).

ML has emerged in recent years as an extremely powerful tool to screen a wide range of nanoporous materials for applications in gas storage and separation. Inspired by the development of ML-based materials research,²⁰⁶ the ML-assisted high-throughput computational screening of MOFs and COFs is undergoing tremendous development.^202,207 Examples studied through such large-scale screening methods (see Table 1) include the examination of databases of tens of thousands of MOFs for methane and carbon dioxide storage,¹⁹⁸ the evaluation of thousands of MOFs membranes for the separation of binary gas mixtures²⁰⁵ and the ML-based selection of MOFs arrays for methane sensing applications.²⁰⁸

Table 1 Examples of ML-assisted high-throughput computational screening of nanoporous materials for specific applications. hMOFs denote databases of hypothetical MOFs structures,¹⁹⁴ eMOFs refer to databases of experimental MOFs structures¹⁹⁵ and CoRE-MOFMs refers to the CoRE database¹⁹⁶ for MOFs membrane. Structure codes correspond to the CSD database¹⁹⁵

System	Structure database	Notes
CH4 storage^48	130 938 hMOFs	Training: 10 433
		Testing: 119 965
CH4 storage^197	137 953 hMOFs	Training: 10 000
		Testing: 127 953
CH4 storage^198	137 953 hMOFs
CO2 storage^199	324 500 hMOFs	Training: 32 450
		Testing: 292 050
CO2 storage^200	55 163 hMOFS
CO2 storage^201	400 hMOFs
CO2 storage^202	100 eMOFs
H2 storage^202	100 eMOFs
CH4/CO2 separation^203	324 500 hMOFs	Training: 32 450
		Testing: 292 050
CO2/N2 separation^204	137 953 hMOF database
CO2/CH4 separation^205	6013 CoRE-MOFMs	Top two structures XUZDUS & XEJXER
H2/CH4 separation^205	6013 CoRE-MOFMs	Top candidates TESGUU & ZIJOF
O2/N2 separation^205	6013 CoRE-MOFMs	Top candidates GETXAG & GOLQII
CO2/CH4 separation^205	6013 CoRE-MOFMs	Top candidates YEKWOC & BAHGUN04

---

5 ML-guided exploration of free energy landscapes

As discussed in previous sections, ML models can be trained on datasets generated by molecular simulations, as a way to interpolate and/or extrapolate the data for conditions that are not covered by the simulations. This leads to ML predictions that have an accuracy close to the simulations for only a fraction of the computational costs, and thus as a way to accelerate the discovery of new materials for a wide range of applications, including high entropy alloys,²⁰⁹ novel glass-forming metallic systems,²¹⁰ materials with improved catalytic performance,^158,159 and complex tasks, such as the prediction of activation energies^14,211 or the elucidation of the polymorph selection process.^13,212 Operating conditions for a given system can be fine-tuned almost instantly through ML models, which provides another path towards an acceleration of purely MS schemes.

Very interestingly, recent work has shown that ML methods can go beyond such interpolation/extrapolation tasks and enable the exploration of high-dimensional free energy landscapes,²¹³ defined by cost functions associated with machine learning. In another example, free energy landscapes, associated with complex assembly processes, can be explored using enhanced sampling simulations for which the reaction coordinate is estimated from a combined MS–ML approach.⁴⁵ For instance, crystal nucleation has been simulated with umbrella sampling MS along an entropic pathway, i.e. with an entropic reaction coordinate estimated on-the-fly from a machine learned Helmholtz free energy and from the current MS internal energy for each step of the enhanced sampling simulations. This combined approach allows for the exploration of novel pathways spanned by a ML-guided reaction coordinate, and, in turn, can shed light on the pathways underlying a wide range of activated processes and rare events.

In recent years, combined MS–ML approaches¹⁵⁶ have been developed using either ANNs, recurrent neural networks,^214–217 convolutional neural networks^218–222 or autoencoders.^223–227 In particular, combinations of deep learning methods with molecular simulations have started to demonstrate great potential in understanding the role played by the collective variables (CVs) that underlie the evolution of a molecular system.²²⁸ Such combined approaches can also yield accurate low-dimensional system representations, along which enhanced sampling simulations can be carried out. These methods can be extended to biological systems, as shown in recent work on oligopeptides using different ML strategies such as neural networks, kernel ridge regression, support vector machines and weighted neighbor schemes.¹³⁹ In particular, the use of autoencoders to learn nonlinear CVs, that are differentiable functions of atomic coordinates, and their use in enhanced sampling simulations can greatly accelerate the exploration of folding free energy landscapes in macromolecular and biological systems.²²⁵ New advances in the selection of appropriate CVs for enhanced sampling simulations have also been recently implemented with the use of supervised machine learning.²²⁹ Decision functions in supervised machine learning methods can be used as initial CVs for enhanced sampling, and the distance to the support vector machines-decision hyperplane, the output probability estimates from logistic regression or the outputs from neural network classifiers can be leveraged to sample structural changes. Another approach consists in combining both ML and variational inference²³⁰ to predict and discover collective variables using deep Bayesian models, with applications to polypeptides²³¹ and to chemical reactions.²³²

The sampling of rugged free energy landscapes has been the focus of considerable attention in recent years. For instance, adaptive enhanced sampling by force-biasing using neural networks (FUNN) has been shown to perform especially well for systems as diverse as simple particles, proteins and coarse-grained polymer chains.²³³ In studies of phase transformations, path collective variables can be defined in a space spanned by global classifiers derived from local structural units, identified via a neural-network-based classification scheme.²³⁴ Another approach relies on an analogy with reinforcement learning to explore the configurational space, i.e. carrying out a reinforced dynamics for enhanced sampling. This, in turn, allows to capture accurately the structural changes undergone by proteins in explicit solvent models.²³⁵ The combination of statistical mechanics with generative learning can also result in the formulation of a competing game between sampling engine and virtual discriminator. This approach has been applied to many-body Hamiltonian systems, with a targeted adversarial learning optimized sampling (TALOS) driving the system to a user-defined target distribution in order to bolster the sampling of rare events.²³⁶

Another emerging idea consists in using coarse-grained free energy landscapes, with the aim of reducing the time scales necessary to accurately sample the conformational topology for complex chemical and biological systems. In particular, the back-mapping based sampling method²³⁷ back-maps coarse-grained free energy landscapes to create starting points, with a resolution at the atomic level, for molecular simulations. Applications to oligopeptides have demonstrated a gain in efficiency of an order of magnitude for sampling transitions in heptamers, when compared with purely MS approaches. Similarly, coarse-grained methods can also be used for the conformational sampling of proteins and peptide chains, using a neural network to determine free energy surfaces from MS, and then leveraging the machine learned free energy surfaces to carry out simulations with a resolution at the coarse-grained level.²⁶ Nonlinear manifold learning techniques can also be employed to accelerate the exploration of free energy surfaces, by biasing the MD simulator towards unexplored regions using the smoothness of the geometry of the surface.²³⁸

ML methods have also become increasingly instrumental in guiding free energy simulations when studying solvation environments^162,239 or protein folding.^152,156 Such approaches have also been applied to the determination of free energy landscapes for protein–ligand unbinding via metadynamics.²⁴⁰ Metadynamics can also be combined with a Hamiltonian replica-exchange algorithm (sampling water interfaces through scaled Hamiltonians or SWISH) and a machine learned-pathlike variable to compute the binding free energy for a series of chemically diverse ligands with a complex target (human soluble epoxide hydrolase) and to shed light on the role of water in the binding process.²⁴¹ In recent work, combinations of supervised and unsupervised machine learning techniques have been applied to observables extracted from MS, with the aim of better understanding protein–ligand binding in the context of drug resistance in HIV-1 protease.¹⁶⁶ Reinforcement learning adaptive sampling strategies, like the REinforcement learning based Adaptive samPling (REAP) method, provide on-the-fly estimates for the significance of collective variables for the exploration of the folding free energy landscape of proteins, and promotes the exploration of the landscape along key degrees of freedom.²⁴² Another deep learning approach to study ligand–protein systems is the reweighted autoencoded variational Bayes for enhanced sampling (RAVE).²⁴³ Recent extensions have allowed learning of reaction coordinate expressed as a linear piecewise function, leading to an efficient protocol for the simulation of slow unbinding processes in practical ligand–protein complexes in an automated manner.

6 Conclusions

Data-driven methods have led to recent advances in the discovery of novel materials. Furthermore, combinations of machine learning with molecular simulation algorithms has provided a way to accelerate conventional computational methods, and as such, have significantly increased the number, the range and the complexity of systems that can be studied and screened as candidates for specific applications. This review first focuses on the determination of free energy via combined ML–MS approaches and examines examples of systems encompassing molecular fluids, biological systems, as well as fluid confined in nanoporous materials known as MOFs and COFs. Free energy simulations are often very computationally intensive and generally require the implementation of enhanced sampling methods for an accurate sampling of high-dimensional free energy surfaces. We thus discuss recent approaches to compute the free energy and how ML models can be trained using simulation data as training datasets. As illustrated by a series of examples presented here, the idea of such combined MS–ML approaches is to predict as accurately and as efficiently as possible the free energy, with the aim e.g. of shedding light on phase transition processes in molecular systems or on conformational changes and folding in biological systems. For nanoconfined systems, the goal is to screen as rapidly as possible materials as candidates for gas storage and gas separation applications. Given the huge number of MOFs and COFs structures that can potentially be synthesized, a purely MS computational screening of all possible structures remains a daunting task. In that case, ML-based screening methods²⁴⁴ can narrow down the field of nanoporous materials that are candidates for a specific application, before MS methods,^58,245 or, alternatively, combined MS–ML approaches,²⁴⁶ are employed to test and characterize a smaller set of materials. Combined MS–ML approaches can also considerably accelerate, for instance, the computationally intensive search of the parameter space for the adsorption of multicomponent mixtures, or the determination of the free energy of immersion that quantifies the cost of regenerating the adsorbent for practical applications. The last part of the review examines several new exciting trends in combined MS–ML approaches. Such developments include the ML-guided exploration of free energy landscapes and the ML-aided identification of the crucial parameters and collective variables that underlie the occurrence of rare events in molecular and biological systems and the transitions they undergo between states. These new approaches are especially promising, as they allow for the exploration of as yet unexplored pathways for assembly, chemical reactions or folding processes, and will likely provide new insights in these phenomena.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

Partial funding for this research was provided by NSF through award CHE-1955403.

References

1. J. Behler, J. Chem. Phys., 2016, 145, 170901.
2. K. T. Butler, D. W. Davies, H. Cartwright, O. Isayev and A. Walsh, Nature, 2018, 559, 547–555.
3. J. Hachmann, M. A. F. Afzal, M. Haghighatlari and Y. Pal, Mol. Simul., 2018, 44, 921–929.
4. M. Ceriotti, J. Chem. Phys., 2019, 150, 150901.
5. F. Häse, S. Valleau, E. Pyzer-Knapp and A. Aspuru-Guzik, Chem. Sci., 2016, 7, 5139–5147.
6. B. R. Hough, D. A. Beck, D. T. Schwartz and J. Pfaendtner, Comput. Chem. Eng., 2017, 104, 56–63.
7. W. Beckner and J. Pfaendtner, J. Chem. Inf. Model., 2019, 59, 2617–2625.
8. R. Gómez-Bombarelli, J. N. Wei, D. Duvenaud, J. M. Hernández-Lobato, B. Sánchez-Lengeling, D. Sheberla, J. Aguilera-Iparraguirre, T. D. Hirzel, R. P. Adams and A. Aspuru-Guzik, ACS Cent. Sci., 2018, 4, 268–276.
9. D. K. Duvenaud, D. Maclaurin, J. Iparraguirre, R. Bombarell, T. Hirzel, A. Aspuru-Guzik and R. P. Adams, Advances in neural information processing systems, 2015, pp. 2224–2232.
10. B. Sanchez-Lengeling and A. Aspuru-Guzik, Science, 2018, 361, 360–365.
11. T. Barnard, H. Hagan, S. Tseng and G. C. Sosso, Mol. Syst. Des. Eng., 2020, 5, 317–329.
12. J. P. Janet, L. Chan and H. J. Kulik, J. Phys. Chem. Lett., 2018, 9, 1064–1071.
13. F. Musil, S. De, J. Yang, J. E. Campbell, G. M. Day and M. Ceriotti, Chem. Sci., 2018, 9, 1289–1300.
14. C. A. Grambow, L. Pattanaik and W. H. Green, J. Phys. Chem. Lett., 2020, 11, 2992–2997.
15. G. M. Rotskoff and E. Vanden-Eijnden, Stat, 2018, 1050, 22.
16. G. Carleo, I. Cirac, K. Cranmer, L. Daudet, M. Schuld, N. Tishby, L. Vogt-Maranto and L. Zdeborová, Rev. Mod. Phys., 2019, 91, 045002.
17. C. M. Handley and P. L. Popelier, J. Phys. Chem. A, 2010, 114, 3371–3383.
18. J. Behler, Phys. Chem. Chem. Phys., 2011, 13, 17930–17955.
19. J. Behler, J. Phys.: Condens. Matter, 2014, 26, 183001.
20. B. Jiang, J. Li and H. Guo, Int. Rev. Phys. Chem., 2016, 35, 479–506.
21. J. S. Smith, O. Isayev and A. E. Roitberg, Chem. Sci., 2017, 8, 3192–3203.
22. F. Nüske, L. Boninsegna and C. Clementi, J. Chem. Phys., 2019, 151, 044116.
23. G. C. Sosso, V. L. Deringer, S. R. Elliott and G. Csányi, Mol. Simul., 2018, 44, 866–880.
24. R. Barrett, M. Chakraborty, D. B. Amirkulova, H. A. Gandhi, G. P. Wellawatte and A. D. White, J. Open Source Softw., 2020, 5, 2367.
25. Z. Li, G. P. Wellawatte, M. Chakraborty, H. A. Gandhi, C. Xu and A. D. White, Chem. Sci., 2020, 11, 9524–9531.
26. T. Lemke and C. Peter, J. Chem. Theory Comput., 2017, 13, 6213–6221.
27. F. Häse, I. F. Galván, A. Aspuru-Guzik, R. Lindh and M. Vacher, Chem. Sci., 2019, 10, 2298–2307.
28. C. S. Adorf, J. Antonaglia, J. Dshemuchadse and S. C. Glotzer, J. Chem. Phys., 2018, 149, 204102.
29. J. G. Freeze, H. R. Kelly and V. S. Batista, Chem. Rev., 2019, 119, 6595–6612.
30. Z. M. Sherman, M. P. Howard, B. A. Lindquist, R. B. Jadrich and T. M. Truskett, J. Chem. Phys., 2020, 152, 140902.
31. J. Chen, X. Xu, X. Xu and D. H. Zhang, J. Chem. Phys., 2013, 138, 221104.
32. A. J. Ballard, R. Das, S. Martiniani, D. Mehta, L. Sagun, J. D. Stevenson and D. J. Wales, Phys. Chem. Chem. Phys., 2017, 19, 12585–12603.
33. H. E. Sauceda, S. Chmiela, I. Poltavsky, K.-R. Müller and A. Tkatchenko, J. Chem. Phys., 2019, 150, 114102.
34. M. Haghighatlari and J. Hachmann, Curr. Opin. Chem. Eng., 2019, 23, 51–57.
35. F. Brockherde, L. Vogt, L. Li, M. E. Tuckerman, K. Burke and K.-R. Müller, Nat. Commun., 2017, 8, 1–10.
36. J. Wang, S. Olsson, C. Wehmeyer, A. Pérez, N. E. Charron, G. De Fabritiis, F. Noé and C. Clementi, ACS Cent. Sci., 2019, 5, 755–767.
37. C. Duan, J. P. Janet, F. Liu, A. Nandy and H. J. Kulik, J. Chem. Theory Comput., 2019, 15, 2331–2345.
38. A. W. Long, J. Zhang, S. Granick and A. L. Ferguson, Soft Matter, 2015, 11, 8141–8153.
39. C. S. Adorf, T. C. Moore, Y. J. Melle and S. C. Glotzer, J. Phys. Chem. B, 2019, 124, 69–78.
40. N. E. Jackson, M. A. Webb and J. J. de Pablo, Curr. Opin. Chem. Eng., 2019, 23, 106–114.
41. B. Jiang, M. Yang, D. Xie and H. Guo, Chem. Soc. Rev., 2016, 45, 3621–3640.
42. R. A. Mansbach and A. L. Ferguson, J. Chem. Phys., 2015, 142, 03B607_1.
43. L. Mones, N. Bernstein and G. Csányi, J. Chem. Theory Comput., 2016, 12, 5100–5110.
44. E. Schneider, L. Dai, R. Q. Topper, C. Drechsel-Grau and M. E. Tuckerman, Phys. Rev. Lett., 2017, 119, 150601.
45. C. Desgranges and J. Delhommelle, Phys. Rev. E, 2018, 98, 063307.
46. F. Noé, S. Olsson, J. Köhler and H. Wu, Science, 2019, 365, eaaw1147.
47. M. O. Jäger, E. V. Morooka, F. F. Canova, L. Himanen and A. S. Foster, npj Comput. Mater., 2018, 4, 1–8.
48. M. Pardakhti, E. Moharreri, D. Wanik, S. L. Suib and R. Srivastava, ACS Comb. Sci., 2017, 19, 640–645.
49. A. Ahmed, S. Seth, J. Purewal, A. G. Wong-Foy, M. Veenstra, A. J. Matzger and D. J. Siegel, Nat. Commun., 2019, 10, 1–9.
50. C. Desgranges and J. Delhommelle, J. Phys. Chem. C, 2019, 124, 1907–1917.
51. G. Anderson, B. Schweitzer, R. Anderson and D. A. Gomez-Gualdron, J. Phys. Chem. C, 2018, 123, 120–130.
52. X. Wu, S. Xiang, J. Su and W. Cai, J. Phys. Chem. C, 2019, 123, 8550–8559.
53. N. S. Bobbitt and R. Q. Snurr, Mol. Simul., 2019, 45, 1069–1081.
54. R. Anderson, A. Biong and D. A. Gómez-Gualdrón, J. Chem. Theory Comput., 2020, 16, 1271–1283.
55. G. Bussi and A. Laio, Nat. Rev. Phys., 2020, 1–13.
56. T. Toyao, K. Suzuki, S. Kikuchi, S. Takakusagi, K.-i. Shimizu and I. Takigawa, J. Phys. Chem. C, 2018, 122, 8315–8326.
57. Z. Zhang, J. A. Schott, M. Liu, H. Chen, X. Lu, B. G. Sumpter, J. Fu and S. Dai, Angew. Chem., 2019, 131, 265–269.
58. K. M. Jablonka, D. Ongari, S. M. Moosavi and B. Smit, Chem. Rev., 2020, 120, 8066–8129.
59. M. B. Christopher, Pattern Recognition and Machine Learning, Springer-Verlag, New York, 2016.
60. K. Hansen, G. Montavon, F. Biegler, S. Fazli, M. Rupp, M. Scheffler, O. A. Von Lilienfeld, A. Tkatchenko and K.-R. Müller, J. Chem. Theory Comput., 2013, 9, 3404–3419.
61. P. O. Dral, O. A. von Lilienfeld and W. Thiel, J. Chem. Theory Comput., 2015, 11, 2120–2125.
62. M. Rupp, A. Tkatchenko, K.-R. Müller and O. A. Von Lilienfeld, Phys. Rev. Lett., 2012, 108, 058301.
63. J. B. Witkoskie and D. J. Doren, J. Chem. Theory Comput., 2005, 1, 14–23.
64. T. Straatsma and J. McCammon, J. Chem. Phys., 1991, 95, 1175–1188.
65. J. Kästner and W. Thiel, J. Chem. Phys., 2005, 123, 144104.
66. M. R. Shirts and V. S. Pande, J. Chem. Phys., 2005, 122, 144107.
67. M. Müller and K. C. Daoulas, J. Chem. Phys., 2008, 128, 024903.
68. D. A. Kofke, Fluid Phase Equilib., 2005, 228, 41–48.
69. D. Frenkel and A. J. Ladd, J. Chem. Phys., 1984, 81, 3188–3193.
70. E. J. Meijer, D. Frenkel, R. A. LeSar and A. J. Ladd, J. Chem. Phys., 1990, 92, 7570–7575.
71. G. M. Torrie and J. P. Valleau, J. Comput. Phys., 1977, 23, 187–199.
72. C. Bartels and M. Karplus, J. Comput. Chem., 1997, 18, 1450–1462.
73. P. Virnau and M. Müller, J. Chem. Phys., 2004, 120, 10925–10930.
74. G. Bussi, A. Laio and M. Parrinello, Phys. Rev. Lett., 2006, 96, 090601.
75. A. Barducci, G. Bussi and M. Parrinello, Phys. Rev. Lett., 2008, 100, 020603.
76. J. F. Dama, M. Parrinello and G. A. Voth, Phys. Rev. Lett., 2014, 112, 240602.
77. A. Laio and F. L. Gervasio, Rep. Prog. Phys., 2008, 71, 126601.
78. J. Kästner, Wiley Interdiscip. Rev.: Comput. Mol. Sci., 2011, 1, 932–942.
79. P. R. Ten Wolde, M. J. Ruiz-Montero and D. Frenkel, Phys. Rev. Lett., 1995, 75, 2714.
80. P. Rein ten Wolde, M. J. Ruiz-Montero and D. Frenkel, J. Chem. Phys., 1996, 104, 9932–9947.
81. S. Auer and D. Frenkel, Nature, 2001, 409, 1020–1023.
82. A. Cacciuto, S. Auer and D. Frenkel, Nature, 2004, 428, 404–406.
83. C. Desgranges and J. Delhommelle, Phys. Rev. Lett., 2007, 98, 235502.
84. C. Desgranges and J. Delhommelle, J. Am. Chem. Soc., 2011, 133, 2872–2874.
85. R. Ni and M. Dijkstra, J. Chem. Phys., 2011, 134, 034501.
86. M. Gonzalez, E. Sanz, C. McBride, J. Abascal, C. Vega and C. Valeriani, Phys. Chem. Chem. Phys., 2014, 16, 24913–24919.
87. C. Desgranges and J. Delhommelle, J. Am. Chem. Soc., 2014, 136, 8145–8148.
88. C. Desgranges and J. Delhommelle, Phys. Rev. Lett., 2018, 120, 115701.
89. B. Chen, J. I. Siepmann, K. J. Oh and M. L. Klein, J. Chem. Phys., 2001, 115, 10903–10913.
90. H. Wang, H. Gould and W. Klein, Phys. Rev. E, 2007, 76, 031604.
91. C. Desgranges and J. Delhommelle, J. Chem. Phys., 2016, 145, 204112.
92. C. Desgranges and J. Delhommelle, J. Chem. Phys., 2016, 145, 234505.
93. C. Desgranges and J. Delhommelle, J. Chem. Phys., 2017, 146, 184104.
94. C. Desgranges and J. Delhommelle, Langmuir, 2019, 35, 15401–15409.
95. C. Desgranges and J. Delhommelle, J. Phys. Chem. C, 2019, 123, 11707–11713.
96. L. Filion, M. Hermes, R. Ni and M. Dijkstra, J. Chem. Phys., 2010, 133, 244115.
97. C. Desgranges and J. Delhommelle, Phys. Rev. Lett., 2019, 123, 195701.
98. C. Dellago, P. Bolhuis and P. L. Geissler, Adv. Chem. Phys., 2002, 123, 1–78.
99. C. Dellago and P. G. Bolhuis, in Advanced Computer Simulation Approaches for Soft Matter Sciences III, Springer, 2009, pp. 167–233.
100. T. S. Van Erp and P. G. Bolhuis, J. Comput. Phys., 2005, 205, 157–181.
101. E. E. Borrero, M. Weinwurm and C. Dellago, J. Chem. Phys., 2011, 134, 244118.
102. L. Rosso, J. B. Abrams and M. E. Tuckerman, J. Phys. Chem. B, 2005, 109, 4162–4167.
103. M. A. Cuendet and M. E. Tuckerman, J. Chem. Theory Comput., 2014, 10, 2975–2986.
104. C. Jarzynski, Phys. Rev. Lett., 1997, 78, 2690.
105. G. E. Crooks, Phys. Rev. E, 1999, 60, 2721.
106. D. J. Evans, Mol. Phys., 2003, 101, 1551–1554.
107. M. S. Shell, P. G. Debenedetti and A. Z. Panagiotopoulos, Phys. Rev. E, 2002, 66, 056703.
108. J. R. Errington, J. Chem. Phys., 2003, 118, 9915–9925.
109. G. Gazenmüller and P. J. Camp, J. Chem. Phys., 2007, 127, 154504.
110. C. Desgranges and J. Delhommelle, J. Chem. Phys., 2009, 130, 244109.
111. T. Aleksandrov, C. Desgranges and J. Delhommelle, Fluid Phase Equilib., 2010, 287, 79–83.
112. C. Desgranges and J. Delhommelle, J. Chem. Phys., 2012, 136, 184107.
113. K. S. Rane, S. Murali and J. R. Errington, J. Chem. Theory Comput., 2013, 9, 2552–2566.
114. F. Wang and D. P. Landau, Phys. Rev. E, 2001, 64, 056101.
115. F. Wang and D. Landau, Phys. Rev. Lett., 2001, 86, 2050–2053.
116. Q. Yan, R. Faller and J. J. de Pablo, J. Chem. Phys., 2002, 116, 8745–8750.
117. C. Desgranges and J. Delhommelle, Energy Fuels, 2017, 31, 10699–10705.
118. K. N. Ngale, C. Desgranges and J. Delhommelle, Mol. Simul., 2012, 38, 653–658.
119. T. Aleksandrov, C. Desgranges and J. Delhommelle, Mol. Simul., 2012, 38, 1265–1270.
120. C. Desgranges and J. Delhommelle, J. Chem. Phys., 2018, 149, 072307.
121. C. Desgranges and J. Delhommelle, J. Chem. Phys., 2012, 136, 184108.
122. C. Desgranges and J. Delhommelle, J. Chem. Phys., 2014, 140, 104109.
123. C. Desgranges and J. Delhommelle, J. Chem. Phys., 2016, 144, 124510.
124. C. Desgranges and J. Delhommelle, J. Chem. Phys., 2016, 145, 184504.
125. C. Desgranges and J. Delhommelle, J. Chem. Theory Comput., 2015, 11, 5401.
126. C. Desgranges, A. Margo and J. Delhommelle, Chem. Phys. Lett., 2016, 658, 37–42.
127. C. Desgranges and J. Delhommelle, J. Chem. Eng. Data, 2017, 62, 4032–4040.
128. C. Desgranges and J. Delhommelle, J. Phys. Chem. B, 2014, 118, 3175.
129. J. M. Hicks, C. Desgranges and J. Delhommelle, J. Phys. Chem. C, 2012, 116, 22938–22946.
130. A. R. V. Koenig, C. Desgranges and J. Delhommelle, Mol. Simul., 2014, 40, 71–79.
131. E. A. Hicks, C. Desgranges and J. Delhommelle, Mol. Simul., 2014, 40, 656–663.
132. K. Gopalsamy, C. Desgranges and J. Delhommelle, J. Phys. Chem. C, 2017, 121, 24692–24700.
133. V. K. Shen and D. W. Siderius, J. Chem. Phys., 2014, 140, 244106.
134. M. Witman, N. A. Mahynski and B. Smit, J. Chem. Theory Comput., 2018, 14, 6149–6158.
135. N. A. Mahynski, H. W. Hatch, M. Witman, D. A. Sheen, J. R. Errington and V. K. Shen, Mol. Simul., 2020, 1–13.
136. Y. LeCun, L. Bottou, G. B. Orr and K.-R. Müller, Efficient backprop, Springer, 1998, pp. 9–50.
137. C. Desgranges and J. Delhommelle, J. Chem. Phys., 2018, 149, 044118.
138. C. Desgranges and J. Delhommelle, Chem. Phys. Lett., 2019, 715, 1–6.
139. J. R. Cendagorta, J. Tolpin, E. Schneider, R. Q. Topper and M. E. Tuckerman, J. Phys. Chem. B, 2020, 124, 3647–3660.
140. T. G. Dietterich, Multiple Classifier Systems, MCS 2000, 2000, pp. 1–15.
141. A. Krogh and J. Vedelsby, Adv. Neural Inf. Process Syst., 1995, 231–238.
142. L. Breiman, Mach. Learn., 1996, 24, 123–140.
143. I. Goodfellow, Y. Bengio and A. Courville, Deep Learning, MIT press, 2016.
144. A. L. Ferguson, J. Comput. Chem., 2017, 38, 1583–1605.
145. N. Chopin, T. Lelièvre and G. Stoltz, Stat. Comput., 2012, 22, 897–916.
146. M. Habeck, Phys. Rev. Lett., 2012, 109, 100601.
147. G. Hummer, New J. Phys., 2005, 7, 34.
148. H. Sidky and J. K. Whitmer, J. Chem. Phys., 2018, 148, 104111.
149. L. Maragliano and E. Vanden-Eijnden, J. Chem. Phys., 2008, 128, 184110.
150. L. Cao, G. Stoltz, T. Lelièvre, M.-C. Marinica and M. Athènes, J. Chem. Phys., 2014, 140, 03B610_1.
151. T. Stecher, N. Bernstein and G. Csányi, J. Chem. Theory Comput., 2014, 10, 4079–4097.
152. S. Mittal and D. Shukla, Mol. Simul., 2018, 44, 891–904.
153. M. Karplus, Angew. Chem., Int. Ed., 2014, 53, 9992–10005.
154. H. M. Senn and W. Thiel, Angew. Chem., Int. Ed., 2009, 48, 1198–1229.
155. H. Lin and D. G. Truhlar, Theor. Chem. Acc., 2007, 117, 185.
156. F. Noé, G. De Fabritiis and C. Clementi, Curr. Opin. Struct. Biol., 2020, 60, 77–84.
157. A. Tkatchenko, Nat. Commun., 2020, 11, 1–4.
158. W. Yang, T. T. Fidelis and W.-H. Sun, ACS Omega, 2019, 5, 83–88.
159. A. R. Singh, B. A. Rohr, J. A. Gauthier and J. K. Nørskov, Catal. Lett., 2019, 149, 2347–2354.
160. C. Chen, W. Ye, Y. Zuo, C. Zheng and S. P. Ong, Chem. Mater., 2019, 31, 3564–3572.
161. Y. Wang, J. M. L. Ribeiro and P. Tiwary, Curr. Opin. Struct. Biol., 2020, 61, 139–145.
162. Y. Basdogan, M. C. Groenenboom, E. Henderson, S. De, S. B. Rempe and J. A. Keith, J. Chem. Theory Comput., 2019, 16, 633–642.
163. J. Scheen, W. Wu, A. S. J. S. Mey, P. Tosco, M. Mackey and J. Michel, ChemRXiv, 2020,  DOI:10.26434/chemrxiv.12380612.v1.
164. D. Kilburg and E. Gallicchio, J. Chem. Theory Comput., 2018, 14, 6183–6196.
165. G. Bitencourt-Ferreira and W. F. de Azevedo, Biophys. Chem., 2018, 240, 63–69.
166. T. W. Whitfield, D. A. Ragland, K. B. Zeldovich and C. A. Schiffer, J. Chem. Theory Comput., 2019, 16, 1284–1299.
167. D. A. Rufa, H. E. Bruce Macdonald, J. Fass, M. Wieder, P. B. Grinaway, A. E. Roitberg, O. Isayev and J. D. Chodera, bioRxiv, 2020,  DOI:10.1101/2020.07.29.227959.
168. B. Selvam, S. Mittal and D. Shukla, ACS Cent. Sci., 2018, 4, 1146–1154.
169. S. Nayak, S. Bhattacharjee, J.-H. Choi and S. C. Lee, J. Phys. Chem. A, 2019, 124, 247–254.
170. T. Toyao, Z. Maeno, S. Takakusagi, T. Kamachi, I. Takigawa and K.-I. Shimizu, ACS Catal., 2019, 10, 2260–2297.
171. G. Panapitiya, G. Avendaño-Franco, P. Ren, X. Wen, Y. Li and J. P. Lewis, J. Am. Chem. Soc., 2018, 140, 17508–17514.
172. M. Zafari, D. Kumar, M. Umer and K. S. Kim, J. Mater. Chem. A, 2020, 8, 5209–5216.
173. Z. Li, S. Wang, W. S. Chin, L. E. Achenie and H. Xin, J. Mater. Chem. A, 2017, 5, 24131–24138.
174. A. J. Chowdhury, W. Yang, K. E. Abdelfatah, M. Zare, A. Heyden and G. A. Terejanu, J. Chem. Theory Comput., 2020, 16, 1105–1114.
175. Z. W. Ulissi, A. J. Medford, T. Bligaard and J. K. Nørskov, Nat. Commun., 2017, 8, 1–7.
176. R. Gasper, H. Shi and A. Ramasubramaniam, J. Phys. Chem. C, 2017, 121, 5612–5619.
177. R. A. Hoyt, M. M. Montemore, I. Fampiou, W. Chen, G. Tritsaris and E. Kaxiras, J. Chem. Inf. Model., 2019, 59, 1357–1365.
178. K. Takahashi and I. Miyazato, J. Comput. Chem., 2018, 39, 2405–2408.
179. Z. W. Ulissi, A. R. Singh, C. Tsai and J. K. Nørskov, J. Phys. Chem. Lett., 2016, 7, 3931–3935.
180. S. Chong, S. Lee, B. Kim and J. Kim, Coord. Chem. Rev., 2020, 423, 213487.
181. G. S. Fanourgakis, K. Gkagkas, E. Tylianakis, E. Klontzas and G. Froudakis, J. Phys. Chem. A, 2019, 123, 6080–6087.
182. H. Dureckova, M. Krykunov, M. Z. Aghaji and T. K. Woo, J. Phys. Chem. C, 2019, 123, 4133–4139.
183. H. Sui, L. Li, X. Zhu, D. Chen and G. Wu, Chemosphere, 2016, 144, 1950–1959.
184. N. L. Rosi, J. Eckert, M. Eddaoudi, D. T. Vodak, J. Kim, M. O'Keeffe and O. M. Yaghi, Science, 2003, 300, 1127–1129.
185. J. L. Rowsell, A. R. Millward, K. S. Park and O. M. Yaghi, J. Am. Chem. Soc., 2004, 126, 5666–5667.
186. D. Sun, S. Ma, Y. Ke, D. J. Collins and H.-C. Zhou, J. Am. Chem. Soc., 2006, 128, 3896–3897.
187. H. Furukawa, M. A. Miller and O. M. Yaghi, J. Mater. Chem., 2007, 17, 3197–3204.
188. S. S. Han, H. Furukawa, O. M. Yaghi and W. A. Goddard Iii, J. Am. Chem. Soc., 2008, 130, 11580–11581.
189. A. Gopalan, B. J. Bucior, N. S. Bobbitt and R. Q. Snurr, Mol. Phys., 2019, 117, 3683–3694.
190. S.-Y. Kim, S.-I. Kim and Y.-S. Bae, J. Phys. Chem. C, 2020, 124, 19538–19547.
191. G. S. Fanourgakis, K. Gkagkas, E. Tylianakis and G. Froudakis, J. Phys. Chem. C, 2020, 124, 7117–7126.
192. K. S. Deeg, D. Damasceno Borges, D. Ongari, N. Rampal, L. Talirz, A. V. Yakutovich, J. M. Huck and B. Smit, ACS Appl. Mater. Interfaces, 2020, 12, 21559–21568.
193. Y. Sun, R. F. DeJaco and J. I. Siepmann, Chem. Sci., 2019, 10, 4377–4388.
194. C. E. Wilmer, M. Leaf, C. Y. Lee, O. K. Farha, B. G. Hauser, J. T. Hupp and R. Q. Snurr, Nat. Chem., 2012, 4, 83.
195. F. H. Allen, Acta Crystallogr., Sect. B: Struct. Sci., 2002, 58, 380–388.
196. Y. G. Chung, J. Camp, M. Haranczyk, B. J. Sikora, W. Bury, V. Krungleviciute, T. Yildirim, O. K. Farha, D. S. Sholl and R. Q. Snurr, Chem. Mater., 2014, 26, 6185–6192.
197. M. Fernandez, T. K. Woo, C. E. Wilmer and R. Q. Snurr, J. Phys. Chem. C, 2013, 117, 7681–7689.
198. G. S. Fanourgakis, K. Gkagkas, E. Tylianakis and G. E. Froudakis, J. Am. Chem. Soc., 2020, 142, 3814–3822.
199. M. Fernandez, P. G. Boyd, T. D. Daff, M. Z. Aghaji and T. K. Woo, J. Phys. Chem. Lett., 2014, 5, 3056–3060.
200. Y. G. Chung, E. Haldoupis, B. J. Bucior, M. Haranczyk, S. Lee, H. Zhang, K. D. Vogiatzis, M. Milisavljevic, S. Ling and J. S. Camp, et al. , J. Chem. Eng. Data, 2019, 64, 5985–5998.
201. R. Anderson, J. Rodgers, E. Argueta, A. Biong and D. A. Gómez-Gualdrón, Chem. Mater., 2018, 30, 6325–6337.
202. G. Borboudakis, T. Stergiannakos, M. Frysali, E. Klontzas, I. Tsamardinos and G. E. Froudakis, npj Comput. Mater., 2017, 3, 1–7.
203. M. Z. Aghaji, M. Fernandez, P. G. Boyd, T. D. Daff and T. K. Woo, Eur. J. Inorg. Chem., 2016, 2016, 4505–4511.
204. M. Fernandez and A. S. Barnard, ACS Comb. Sci., 2016, 18, 243–252.
205. W. Yang, H. Liang, F. Peng, Z. Liu, J. Liu and Z. Qiao, Nanomaterials, 2019, 9, 467.
206. T. Zhou, Z. Song and K. Sundmacher, Engineering, 2019, 5, 1017–1026.
207. Z. Shi, W. Yang, X. Deng, C. Cai, Y. Yan, H. Liang, Z. Liu and Z. Qiao, Mol. Syst. Des. Eng., 2020, 5, 725–742.
208. J. A. Gustafson and C. E. Wilmer, ACS Sens., 2019, 4, 1586–1593.
209. Z. Zhou, Y. Zhou, Q. He, Z. Ding, F. Li and Y. Yang, npj Comput. Mater., 2019, 5, 1–9.
210. F. Ren, L. Ward, T. Williams, K. J. Laws, C. Wolverton, J. Hattrick-Simpers and A. Mehta, Sci. Adv., 2018, 4, eaaq1566.
211. O. A. von Lilienfeld, K.-R. Müller and A. Tkatchenko, Nat. Rev. Chem., 2020, 1–12.
212. J. Yang, N. Li and S. Li, CrystEngComm, 2019, 21, 6173–6185.
213. D. J. Wales, Annu. Rev. Physiol., 2018, 69, 401–425.
214. J. Kadupitiya, G. C. Fox and V. Jadhao, ArXiv e-prints, 2020, arXiv:2004.06493.
215. L. Simine, T. C. Allen and P. J. Rossky, Proc. Natl. Acad. Sci. U. S. A., 2020, 117, 13945–13948.
216. J. Wang, C. Li, S. Shin and H. Qi, J. Phys. Chem. C, 2020, 124, 14838–14846.
217. E. Pfeiffenberger and P. A. Bates, PLoS One, 2018, 13, e0202652.
218. T. Fukuya and Y. Shibuta, Comput. Mater. Sci., 2020, 184, 109880.
219. J. Li, W. Zhu, J. Wang, W. Li, S. Gong, J. Zhang and W. Wang, PLoS Comput. Biol., 2018, 14, e1006514.
220. J. Jiménez, M. Skalic, G. Martinez-Rosell and G. De Fabritiis, J. Chem. Inf. Model., 2018, 58, 287–296.
221. K. Ryczko, K. Mills, I. Luchak, C. Homenick and I. Tamblyn, Comput. Mater. Sci., 2018, 149, 134–142.
222. K. Schütt, P.-J. Kindermans, H. E. S. Felix, S. Chmiela, A. Tkatchenko and K.-R. Müller, Adv. Neural Inf. Process Syst., 2017, 991–1001.
223. W. Wang and R. Gómez-Bombarelli, npj Comput. Mater., 2019, 5, 1–9.
224. P. Rajak, A. Krishnamoorthy, A. Nakano, P. Vashishta and R. Kalia, Phys. Rev. B, 2019, 100, 014108.
225. W. Chen and A. L. Ferguson, J. Comput. Chem., 2018, 39, 2079–2102.
226. C. Wehmeyer and F. Noé, J. Chem. Phys., 2018, 148, 241703.
227. A. Moradzadeh and N. R. Aluru, J. Phys. Chem. Lett., 2019, 10, 7568–7576.
228. H. Sidky, W. Chen and A. L. Ferguson, Mol. Phys., 2020, 118, e1737742.
229. M. M. Sultan and V. S. Pande, J. Chem. Phys., 2018, 149, 094106.
230. L. Bonati, Y.-Y. Zhang and M. Parrinello, Proc. Natl. Acad. Sci. U. S. A., 2019, 116, 17641–17647.
231. M. Schöberl, N. Zabaras and P.-S. Koutsourelakis, J. Chem. Phys., 2019, 150, 024109.
232. L. Bonati, V. Rizzi and M. Parrinello, J. Phys. Chem. Lett., 2020, 11, 2998–3004.
233. A. Z. Guo, E. Sevgen, H. Sidky, J. K. Whitmer, J. A. Hubbell and J. J. de Pablo, J. Chem. Phys., 2018, 148, 134108.
234. J. Rogal, E. Schneider and M. E. Tuckerman, Phys. Rev. Lett., 2019, 123, 245701.
235. L. Zhang, H. Wang and E. Weinan, J. Chem. Phys., 2018, 148, 124113.
236. J. Zhang, Y. I. Yang and F. Noé, J. Phys. Chem. Lett., 2019, 10, 5791–5797.
237. S. Hunkler, T. Lemke, C. Peter and O. Kukharenko, J. Chem. Phys., 2019, 151, 154102.
238. E. Chiavazzo, R. Covino, R. R. Coifman, C. W. Gear, A. S. Georgiou, G. Hummer and I. G. Kevrekidis, Proc. Natl. Acad. Sci. U. S. A., 2017, 114, E5494–E5503.
239. P. Zhang, L. Shen and W. Yang, J. Phys. Chem. B, 2018, 123, 901–908.
240. R. Capelli, A. Bochicchio, G. Piccini, R. Casasnovas, P. Carloni and M. Parrinello, J. Chem. Theory Comput., 2019, 15, 3354–3361.
241. R. Evans, L. Hovan, G. A. Tribello, B. P. Cossins, C. Estarellas and F. L. Gervasio, J. Chem. Theory Comput., 2020, 16, 4641–4654.
242. Z. Shamsi, K. J. Cheng and D. Shukla, J. Phys. Chem. B, 2018, 122, 8386–8395.
243. J. M. Lamim Ribeiro and P. Tiwary, J. Chem. Theory Comput., 2018, 15, 708–719.
244. I. Tsamardinos, G. S. Fanourgakis, E. Greasidou, E. Klontzas, K. Gkagkas and G. E. Froudakis, Microporous Mesoporous Mater., 2020, 110160.
245. D. Ongari, L. Talirz and B. Smit, ACS Cent. Sci., 2020, 6, 1890–1900.
246. X. Zhang, K. Zhang and Y. Lee, ACS Appl. Mater. Interfaces, 2019, 12, 734–743.

---