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We introduce Gaussian Process Regression (GPR) as an enhanced method of thermodynamic extrapolation and interpolation. The heteroscedastic GPR models that we introduce automatically weight provided information by its estimated uncertainty, allowing for the incorporation of highly uncertain, high-order derivative information. By the linearity of the derivative operator, GPR models naturally handle derivative information and, with appropriate likelihood models that incorporate heterogeneous uncertainties, are able to identify estimates of functions for which the provided observations and derivatives are inconsistent due to the sampling bias that is common in molecular simulations. Since we utilize kernels that form complete bases on the function space to be learned, the estimated uncertainty in the model takes into account that of the functional form itself, in contrast to polynomial interpolation, which explicitly assumes the functional form to be fixed. We apply GPR models to a variety of data sources and assess various active learning strategies, identifying when specific options will be most useful. Our active-learning data collection based on GPR models incorporating derivative information is finally applied to tracing vapor-liquid equilibrium for a single-component Lennard-Jones fluid, which we show represents a powerful generalization to previous extrapolation strategies and Gibbs-Duhem integration. A suite of tools implementing these methods is provided at https://github.com/usnistgov/thermo-extrap., (© 2023 Public Domain.)

Variational autoencoders (VAEs) are rapidly gaining popularity within molecular simulation for discovering low-dimensional, or latent, representations, which are critical for both analyzing and accelerating simulations. However, it remains unclear how the information a VAE learns is connected to its probabilistic structure and, in turn, its loss function. Previous studies have focused on feature engineering, ad hoc modifications to loss functions, or adjustment of the prior to enforce desirable latent space properties. By applying effectively arbitrarily flexible priors via normalizing flows, we focus instead on how adjusting the structure of the decoding model impacts the learned latent coordinate. We systematically adjust the power and flexibility of the decoding distribution, observing that this has a significant impact on the structure of the latent space as measured by a suite of metrics developed in this work. By also varying weights on separate terms within each VAE loss function, we show that the level of detail encoded can be further tuned. This provides practical guidance for utilizing VAEs to extract varying resolutions of low-dimensional information from molecular dynamics and Monte Carlo simulations.

Thermodynamic extrapolation has previously been used to predict arbitrary structural observables in molecular simulations at temperatures (or relative chemical potentials in open-system mixtures) different from those at which the simulation was performed. This greatly reduces the computational cost in mapping out phase and structural transitions. In this work, we explore the limitations and accuracy of thermodynamic extrapolation applied to water, where qualitative shifts from anomalous to simple-fluid-like behavior are manifested through shifts in the liquid structure that occur as a function of both temperature and density. We present formulas for extrapolating in volume for canonical ensembles and demonstrate that linear extrapolations of water's structural properties are only accurate over a limited density range. On the other hand, linear extrapolation in temperature can be accurate across the entire liquid state. We contrast these extrapolations with classical perturbation theory techniques, which are more conservative and slowly converging. Indeed, we show that such behavior is expected by demonstrating exact relationships between extrapolation of free energies and well-known techniques to predict free energy differences. An ideal gas in an external field is also studied to more clearly explain these results for a toy system with fully analytical solutions. We also present a recursive interpolation strategy for predicting arbitrary structural properties of molecular fluids over a predefined range of state conditions, demonstrating its success in mapping qualitative shifts in water structure with density.

A tetrahedral structure resulting from hydrogen bonding is a hallmark of liquid water and plays a significant role in determining its unique thermophysical properties. This water feature has helped understand anomalous properties and physically interpret and model hydrophobic solvation thermodynamics. Tetrahedrality is well described by the geometric relationship of any central water molecule with two of its nearest neighbors in the first coordination shell, as defined by the corresponding "three-body" angle. While order parameters and even full water models have been developed using specific or average features of the three-body angle distribution, here we examine the distribution holistically, tracking its response to changes in temperature, density, and the presence of model solutes. Surprisingly, we find that the three-body distribution responds by varying primarily along a single degree of freedom, suggesting a remarkably simplified view of water structure. We characterize three-body angle distributions across temperature and density space and identify principal components of the variations with state conditions. We show that these principal components embed physical significance and trace out transitions between tetrahedral and simple-fluid-like behavior. Moreover, we find that the ways three-body angles vary within the hydration shells of model colloids of different types and sizes are nearly identical to the variations seen in bulk water across density and temperature. Importantly, through the principal directions of these variations, we find that perturbations to the hydration-water distributions well predict the thermodynamics associated with colloid solvation, in particular, the relative entropy of this process that captures indirect, solvent-mediated contributions to the hydration free energy.