Do Machine-Learning Atomic Descriptors and Order Parameters Tell the Same Story? The Case of Liquid Water

Do Machine-Learning Atomic Descriptors and Order Parameters Tell the Same Story? The Case of Liquid Water. Edward Danquah Donkor,†,‡Alessandro Laio,∗,‡and Ali Hassanali∗,† †The Abdus Salam International Center for Theoretical Physics, Strada Costiera 11, 34151 Trieste, Italy. ‡SISSA – via Bonomea 265, 34136 Trieste, Italy. E-mail: [email protected]; [email protected] Abstract Machine-learning (ML) has become a key workhorse in molecular simulations. Building an ML model in this context, involves encoding the information of chemical environments using local atomic descriptors. In this work, we focus on the Smooth Overlap of Atomic Positions (SOAP) and their application in studying the properties of liquid water both in the bulk and at the hydrophobic air-water interface. By using a statistical test aimed at assessing the relative information content of different distance measures defined on the same data space, we investigate if these descriptors provide the same information as some of the common order parameters that are used to characterize local water structure such as hydrogen bonding, density or tetrahedrality to name a few. Our analysis suggests that the ML description and the standard order parameters of local water structure are not equivalent. In particular, a combination of these order parameters probing local water environments can predict SOAP similarity only approximately, and viceversa, the environments that are similar according 1 arXiv:2211.16196v1 [cond-mat.soft] 29 Nov 2022 to SOAP are not necessarily similar according to the standard order parameters. We also elucidate the role of some of the metaparameters entering in the SOAP definition in encoding chemical information. Introduction The last decade has seen a tremendous spurt in both the development and application of machine-learning (ML) approaches to study molecular systems.1–3 ML has become a mainstream component of atomistic modeling. In particular, for the construction of interaction potentials,1,4,5 and for the analysis of molecular dynamics simulations.6–9 Establishing an understanding of the underlying physical and chemical principles that make ML useful, accurate and in short, avoiding its black box usage, remains an open challenge. A critical first step in designing a ML-based model for molecular systems, is identifying atomic descriptors that encode information of the local environments of a chemical moiety.10,11 Over the recent years, various flavours of atomic descriptors have been developed including atomic-density ones which focus on characterizing local environments,12,13 topology based descriptors relying on graph-theoretical approaches14 to extract the connectivity patterns in data and many-body tensor network representations which serve to characterize the global structure of materials,15 to name a few. Most of these descriptors require human intervention in the selection of various parameters. For example, the size of the local environment or the number of basis functions one needs to describe the local configurations. Since ML-based descriptors are designed to capture the physical and chemical nature of molecular systems, interpreting and understanding their meaning is important, and can lead to a more rational and also physical choice of the hyper-parameters entering their definition. In this work, we address the issue of interpretability, focusing on a specific class of localatomic descriptors namely; the Smooth Overlap of Atomic Positions (SOAP).13,16 SOAP has become widely popular for identifying structural finger prints in systems such as liquid water17–19 and inorganic crystals20,21 and more recently, to develop coarse-grained intermolec2 ular potentials.22 Here we examine how SOAP fingerprints are related to standard order parameters (Figure 1, right panel) that are typically used to describe the local structure of liquid water.23 Moreover, we study how this relationship changes from bulk liquid water to the air-water interface; a prototypical system used to study the environmental effects of hydrophobic interactions.24–26 To measure the relationship between SOAP-based descriptors and standard order parameters, we deploy a recently developed technique, dubbed as the Information Imbalance27 (IB). IB is a statistical test that determines the relative information content between different distance measures defined on the same data space. If applied to two different distances, it allows determining if these distances are equivalent, unrelated, or if one of the two is more predictive than the other. Specifically, we explore the IB between SOAP descriptors and a wide variety of order parameters28–33 that have been built on physical and chemical intuition to characterize aqueous environments. Using the IB, we first investigate if a suitable combination of the order parameters is able to predict the similarity of local environments measured using SOAP features. We find that even the best combination of order parameters is able to predict the SOAP similarity only approximately. The quality of the prediction is better for configurations close to the surface of water, and is significantly improved by choosing a SOAP length scale parameter σ= 0.25 ˚ A, which is much smaller than the value typically used (σ= 1 ˚ A). This result may not be too surprising as the structural information embedded in SOAP descriptors likely contains a richer characterisation of the local environments in water than the standard order parameters which are often fine-tuned to capture specific chemical interactions. We also investigate the reverse problem, namely if SOAP descriptors are able to predict the similarity as measured using chemical-intuition based order parameters. Rather surprisingly, we find that most of these standard order parameters can be predicted rather approximately. For example, the IB between the SOAP similarity and the similarity measured by the number of hydrogen bonds or the Local Structural Index (LSI), appears to be 3 close to the value observed for distances which are unrelated to each other. Interestingly, SOAP predicts the standard coordination number the most accurately out of all the order parameters we examine. The IB can be improved rather marginally by reducing the value of σ. We provide some perspectives on the possible origins of these discrepancies. The paper is organized as follows. We begin in Section 1 with the Methods employed in this work, including a summary of both SOAP and order parameters we study, as well as the Information Imbalance technique. We then move on in Section 2 to the Results where we illustrate the relationships we unravel between the SOAP and order parameters that are obtained using the IB method for both bulk and interfacial water. We then end in Section 3 with some conclusions of our work. Methods Molecular Dynamics Simulations Molecular Dynamics (MD) simulations are carried out using the LAMMPS package.34 We use the TIP4P/200535 rigid water model for most of our work. We also repeat some of our analysis using the TIP3P36 water model which is commonly used in bio-molecular simulations. Our initial simulation setup consists of a bulk water system with 729 water molecules equilibrated at ambient temperature and pressure in a box with sides 27.9×27.9×27.9 ˚ A. To construct an interface, we add a vaccuum region of 139.5 ˚ A in the z-direction and then equilibrate within the NVT ensemble for 10 ns at 300 K with a timestep of 2 fs. This is followed by a production run of 20 ns. The velocity-rescaling37 thermostat is used with a time constant of 100 fs. In our simulations, the real space cut-off for the Coloumb and Lennard Jones (LJ) interactions is 15 ˚ A. Long range corrections are treated using the ParticleParticle Particle-Mesh (PPPM)38 solver for both the Coulomb and LJ interactions. In order to validate the use of our model for the air-water interface, we computed the surface tension 4 for our simulation, obtaining a value of (67.98 ±0.74)mN/m, consistent with previous reports.39 For the TIP3P model, we obtain a surface tension of (47.29 ±0.40) mN/m, which is also consistent with previous studies.39 Identifying Bulk vs Surface Water Environments Using our MD simulations, a total of approximately 5000 local environments are sampled from the water-surface system, on which the SOAP power spectrum is computed using the DScribe software package.40 These environments are chosen by randomly selecting a water molecule every 4 ps. Since we were interested in understanding how the the relationship between SOAP fingerprints and order parameters evolves from bulk water to the hydrophobic surface of water, we characterised the interface using the Willard-Chandler Interface (WCI).41 In brief, a coarse-grained density field is defined as a sum of Gaussian functions, with a specified smoothening parameter (ξ) centered on all the atoms. The interface is then chosen as the set of points for which the coarse-grained density is half of the bulk density. The ξvalue used for the WCI construction was 2.4 ˚ A consistent with previous studies. The water density as a function of distance from the WCI is then built, yielding the distribution shown in the Supporting Information (SI Figure 1). For our analysis, we define various layers from the density profile as performed in several prior works,42,43 which allows for identifying surface and bulk water molecules. In most of our analysis, we focus on comparing the bulk and surface as defined by those waters in layer 4 (Bulk) and layer 1 (L1) respectively (see SI Figure 1 for a visual depiction of these layers relative to the WCI). Descriptors for the Local Structure of Water In this work, we examine the relationship between chemically inspired order parameters and SOAP-based descriptors using the Information Imbalance method. In the following, we begin by summarizing the theory underlying the construction of SOAP descriptors, (see 5 Figure 1: SOAP descriptors and order parameters (left and right respectively) used in this work. The left panel (top) shows a cartoon of the orientational and radial extent of the SOAP descriptors and the bottom panel summarizes the definition of the SOAP power spectrum - see main text for more details. The right panel shows snapshots of local water environments and some of the species, distance and angular based criteria that goes into defining the various order parameters. The symbols next to each variable is what is used throughout the manuscript to refer to each variable. Section 1.3.1) and subsequently, the collection of different order parameters that we examine (see Section 1.3.2). Finally, we also discuss the principles behind the Information Imbalance method (see Section 1.4). Figure 1 illustrates the SOAP and order parameters along with the respective symbols that are used throughout the manuscript. Smooth Overlap of Atomic Positions (SOAP) SOAP has emerged in the last few years as a powerful method to describe the local environments of atoms and molecules, allowing for a wide range of applications from the study of structural properties of organic molecules13,44,45 and very recently, the properties of liquid water.17–19 In the context of SOAP, the local density of an atomic environment χis written as a sum of Gaussian functions with variance σ2, centered on all species that are neighbours of 6 the central atom: ρχ(~r) = X i∈χ exp −|~ri−~r|2 2σ2(1) In this work, we will show that the choice of the value of σparameter plays a very important role in the ability of SOAP to predict chemical properties. Specifically, this parameter determines the resolution of chemical details of the water hydrogen bond network. The default value of σin the DScribe package is 1.0 ˚ A which to the best of our knowledge, is the value used in previous studies using SOAP to study the structure of liquid-water.17,18 The atomic neighbour density in equation 1 can be expanded on a basis of radial basis functions and real spherical harmonics such that: ρχ(~r) = nmax X n=0 lmax X l=0 l X m=−l cnlmgn(r)Ylm(θ, φ) (2) For practical purposes, one defines the environment χby a cut-off radius (rcut) and also limits the number of radial and angular basis functions (nmax,lmax) used. In the following work, we have examined the sensitivity of our results to changing rcut from 3.7 to 5.5 ˚ A as well as varying nmax and lmax between 10-12 and 6-8 respectively (see SI Figures 2-5). We find a marginal drop of chemical information contained in the SOAP descriptors (as reflected in the increase in the IB between SOAP and the order parameters) as we increase rcut, owing to the fact that the order parameters we have examined, consider fluctuations only within the first solvation shell of a given water environment. We also find that there is no significant dependence of our results to the choice of nmax and lmax. Unless otherwise stated, the rcut, nmax and lmax are 3.7 ˚ A, 10 and 6 respectively. By accumulating the expansion coefficients, a rotationally invariant power spectrum can be defined such that, pnn0l(χ) = πr8 2l+ 1 X m (cnlm)†cn0lm (3) 7 Equation 3 defines the components of the SOAP features we will use in our analysis. Order parameters Orientational Tetrahedral Order (qtet): The qtet order parameter is one of the most used local structural quantities to describe liquid water.28,29,46–49 It measures how much a reference water environment deviates from an ordered tetrahedron whose vertices are defined by the bond vectors between the four nearest neighbouring oxygen atoms of a central one. It takes as input the angles between the O-O bond vectors and gives a value of 1 for a perfectly tetrahedral environment and closer to zero for environments which are not tetrahedral. Specifically, the tetrahedrality of a reference water molecule is defined as: qtet = 1 −3 8 3 X i=1 4 X j=i+1 cos φij +1 32 (4) Where φij is the angle between the oxygen molecule of the reference water and its nearest four neighbours (indexed with iand j) Translational Tetrahedral Order (Sk): The translational tetrahedral order parameter is another measure of how much a reference water environment deviates from a regular tetrahedron.29 Whereas qtet focuses on the angles between the O-O bond vectors, Skis computed as the variance between the O-O distances of the four nearest water molecules to a central water: Sk= 1 −1 3 4 X k=1 (rk−¯r)2 4¯r2(5) rkis the distance between a reference water molecule and its kth neighbour and ¯ris the mean of these distances. It has been shown in previous works that Skis more sensitive to the local density variations compared to qtet.50 Local Structural Index (LSI): 8 The LSI is another important variable which has been used to study the structure of water in the bulk under different thermodynamic conditions.30,51,52 It is obtained by ranking the O-O distances from an ith central water molecule such that r1< r2< ... < ri< ri+1 < ...rn<3.7˚ A< rn+1, and then subsequently, the LSI is computed as: LSI = 1 n n X i=1 ∆(i)−¯ ∆2(6) where ∆(i) = ri+1 −riand ¯ ∆ is the arithmetic mean of ∆(i). A high value of LSI implies a larger separation between the first and second solvation shell and points to a more ordered water environment, while a lower value is interpreted as a more disordered environment. Distance to the fifth Oxygen (d5): The d5is defined as the distance between a central oxygen atom and its fifth closest oxygen atom.31,53,54 A large value of d5points to a high separation between the first and second solvation shell of a water environment and is interpreted as a locally ordered structure. A smaller value indicates a smaller separation between the first and second solvation shells and with a similar logic, a disordered local environment. Coordination Number (C.N.): The coordination number quantifies the average number of atoms that surround a chosen central site within some radial cutoff. It can be computed from the radial integral of the radial distribution function (RDF). In order to have a smooth and continuous definition of the coordination number, we use a switching function, commonly done in the construction of different types of collective variables:55,56 C.N. = N X j=1 1−rj rcut 12 1−rj rcut 28 (7) where rjis the distance between a central oxygen atom and atom j. Number of Hydrogen Bonds (NH.B.): We adopt the definition of hydrogen bonding by Luzar and Chandler.32,33 This definition 9 ing. We begin with the top panel comparing SOAP and qtet. Within an rcut of 3.7 ˚ A, there are two interstitial oxygen atoms (purple arrow). Since the definition of qtet only looks at the first 4 nearest oxygens, the two environments are flagged as non tetrahedral (left) and highly tetrahedral (right). Due to the restriction of focusing on the nearest 4 oxygens, the qtet parameter picks up an angle that clearly deviates from tetrahedrality. In fact, when one restricts the SOAP computation to only the first four neighbours, the SOAP space is completely predictive of qtet (see SI Figure 10). Figure 5: Snapshot of two sets of environments χ,χ0and χ00,χ000 which are nearest neighbours in SOAP space but are distant in qtet (top panel) and NH.B. (bottom panel) space. Since the definition of qtet focuses on the first four neighbours of a central atom within the first solvation shell, the two environments are flagged as not tetrahedral (top left) and tetrahedral (top right), while the SOAP space predicts the two environments to be similar. For NH.B., its strict angular cut-off picks up slight angle changes (∼2◦) which labels the two environments as having different number of hydrogen bonds. However, these are clearly two environments that are more similar than different. In similar spirit to the analysis presented for qtet, the bottom panel shows two environments that are again close in SOAP space, but far in terms of hydrogen bonding. Specifically, 16 the bottom left panel shows an environment that accepts and donates two hydrogen bonds (NH.B. = 4) while the bottom right shows a small change in the local geometry where the angle used for the hydrogen bonding criterion changes from 29◦to 31◦resulting in a defect that now accepts two but donates only one hydrogen bond (NH.B. = 3). In hydrogen-bonding space, these configurations are topologically different but in SOAP space, these are, unsurprisingly very similar. Figure 6: Convergence of the IB between SOAP and selected order parameters, as a function of the number of SOAP vector components. The compression is done for all the layers of the density profile depicted in the supplementary material. These results confirm previous reports showing that most of the significant structural changes in the water environments occurs in the topmost layer. Predictions Within Order Parameters Our analysis shows that there is a complex interplay of different order parameters and that they need to be used in combination in order to predict SOAP similarity both in the bulk and at the air-water interface. In the field of aqueous science where the structure and dynamics of liquid water is studied in terms of different local structures, it is quite common to synonymously associate the different order parameters such as d5, LSI and qtet . However, these order parameters are not equivalent. 17 Figure 7: Information Imbalance plane27 for the space of order parameters, showing the IB obtained between different pairs of order parameters. In the Bulk (left) the lowest IB obtained is ∼0.7 which is between C.N. and d5. The rest of the variables are all not informative of each other. In the Surface (right), the lowest IB obtained ∼0.4 is also between C.N. and d5. In general, most of the variables are uninformative of each other both in the Bulk and on the Surface. Figure 7 shows the IB between all pairs of order parameters for both bulk and surface environments. For bulk environments, we observe that the lowest imbalance obtained is approximately 0.7; between the coordination number and d5. Essentially all the other order parameters are not informative of each other consistent with previous work from our group.19 Moving to the surface environments marginally improves the predictability for the coordination number and d5pair. All in all, it is clear that the different order parameters used to characterized local water structure contain very little information about each other. For example, considering the IB between tetrahedrality (qtet) and local density (ρvoro) which is closer to 1, shows that they provide different information about the local environment. Thus, an environment that is more tetrahedral may not have a lower local density and vice-versa. 18 Diagnostics for Water Structure We have used the IB approach to examine the coupling between ML-based local atomic descriptors and chemistry-inspired order parameters. We then explored whether this could be used as a diagnostic tool to study the length-scale of the perturbations induced by the prescence of the surface. Earlier, we compared the IB for bulk and surface water environments, where the latter focused on interfacial waters that reside only within ∼3.0 ˚ A of the WCI. Examining the relative density of water with respect to the WCI (see SI Figure 1), one observes that the correlations albeit weaker, extend up to 1 nanometer from the surface. The extent of the thickness of this interface as probed by both surface sensitive vibrational spectroscopy experiments and simulations, and how it is manifested in terms of local structure, continues to be a topic of active research.42,62,63 Since the IB appears to provide a sensitive measure of the surface water environments, we performed the compression of the full SOAP space for selected order parameters separately for different layers of water with respect to the WCI (see SI Figure 1 for an illustration of how the layers are defined). The left and right panels of Figure 6 shows the IB between SOAP and the coordination number and ρvoro respectively, as a function of the number of SOAP components. Our results show that the IB in the second and third layer are essentially indistinguishable from the bulk. This shows that there is essentially only a very thin layer of water (∼3.0 ˚ A near the surface) whose local structural properties are significantly different, consistent with previous studies.42 Conclusions In this work, we have used a recently developed statistical test (IB) to examine the relationships between the SOAP descriptors and chemically inspired order parameters that are used to characterize local water structure. We focus our analysis on how these relationships 19 evolve as one moves from bulk water to the hydrophobic surface of water. Examining the IB between the full SOAP space and order parameters shows that these two classes of variables do not contain the same information. While the chemical information encoded in the SOAP descriptors can be improved by reducing the Gaussian smearing, the IB between the order parameters and the SOAP descriptors obtained is far from optimal. The predictability improves marginally for water environments sampled at the interface. Given that the SOAP descriptors are designed to be generic and do not probe specific chemical details, this observation may not be too surprising. The consistency between SOAP and the order parameters could be improved by including features that probe directly the fluctuations of the hydrogen bond network. Order parameters like the ones we have investigated here to probe the local structure of water, are used ubiquitously for analyzing molecular dynamics simulations of liquid water. In particular, these are often used to assign whether the water structure arises from a low or high density local environment.19,51,64 However, our analysis suggests that most of these order parameters are somewhat independent of each other and cannot be used synonymously. Furthermore, the SOAP descriptors seem to be probing the local environment of water molecules in a much less specific manner. 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The solid circles represent the optimized IB obtained for the TIP4P/2005 model for the specified color coded chemical variable. These results show very small differences in the IB across the two models and hence are consistent. 33 Figure S10: Information Imbalance between SOAP and qtet for SOAP descriptors computed with 6,5 and 4 nearest neighbours. As the number of neighbours used to compute the SOAP descriptors decreases, the ability of SOAP to predict qtet increases. With 4 nearest neighbours, the SOAP space is completely predictive of qtet 34