A unified machine learning framework for ab initio multiscale modeling of liquids

Abstract

Understanding and predicting the behavior of liquid matter across length scales, using only the microscopic interactions encoded in the Schrödinger equation, remains a central challenge in the physical sciences. Achieving this goal requires not only an accurate and efficient description of intermolecular forces but also a consistent framework that bridges the micro-, meso-, and macroscales. Here, by combining machine-learned interatomic potentials (MLIPs) with neural classical density functional theory (neural cDFT), we present such a framework. The underlying idea is simple: MLIPs trained on quantum-mechanical energies and forces are used to generate inhomogeneous microscopic density profiles, which in turn serve as the training data for neural cDFT. The resulting ab initio neural cDFT is not only significantly more computationally efficient than molecular simulations, but also provides a conceptually transparent route to the thermodynamics of both homogeneous and inhomogeneous systems. We demonstrate the approach for both water and carbon dioxide using several exchange-correlation functionals. Beyond accurately reproducing bulk equations of state and liquid-vapor phase diagrams, ab initio neural cDFT predicts, from first principles, how confinement modifies liquid-vapor coexistence in water. It also captures complex behavior in supercritical carbon dioxide such as the Fisher-Widom and Widom lines. Ab initio neural cDFT establishes a general first-principles route to multiscale modeling of fluids within a single unified conceptual framework.

Figures (4)

• Figure 1: Overview of ab initio neural cDFT. Energies and forces from small-scale electronic structure calculations are used to train an MLIP that represents the potential energy surface, enabling efficient sampling of atomic configurations on nanometer length scales. Equilibrium density profiles obtained from molecular simulations with the MLIP under inhomogeneous external potentials then form the training set for neural cDFT. The resulting ab initio neural cDFT can be used to obtain bulk thermophysical properties, liquid--vapor phase equilibria, and to investigate inhomogeneous systems both on large length scales, and under nanoconfinement.
• Figure 2: Accurate and efficient description of liquids with ab initio neural cDFT. A: Typical random $V_{\rm ext}(z)$ and corresponding $\rho(z)$ used to train the neural cDFT, as obtained from molecular simulations. Results are shown for both water and carbon dioxide, with several interatomic potentials, as indicated in the legends. B, left: $P$ vs $\rho_{\rm b}$ for RPBE-D3 water along several isotherms. B, right: $S(k)$ for PBE-D3 carbon dioxide for different $\rho_{\rm b}$ at $T=360$ K. C, top: $\rho(z)$ at liquid--vapor coexistence for water at $T= 500\,$K (left) and carbon dioxide at $T=250\,$K (right), for different interatomic potentials. C, bottom: Liquid--vapor binodals for water (left) and carbon dioxide (right) with the different interatomic potentials, as indicated in the legend. D, left: $\rho(z)$ of SCAN water at different $\mu$ confined between graphene sheets. D, right: Effective pressure of TraPPE carbon dioxide at different $T$, confined between graphene sheets. In B--D, symbols show results from MD simulations and solid lines show results from ab initio neural cDFT.
• Figure 3: Predicting vapor--liquid equilibria of SCAN water upon confinement between graphene sheets. A: The effective pressure $\tilde{P}$ (see Eq. \ref{['eqn:Ptilde']}), for a graphene slit pore in equilibrium with a reservoir ($\rho_{\rm b} = 33$ nm$^{-3}$) at different $T$. $\tilde{P}$ exhibits minima at different $H$, each corresponding to a different number of water layers, as seen in the accompanying density profiles (the dark shaded regions indicate the positions of the graphene sheets). B: Liquid--vapor phase diagram in the $P$--$T$ plane, for the different $H$ indicated. Note that $P$ is the bulk pressure of the reservoir.
• Figure 4: Predicting behavior of supercritical PBE-D3 carbon dioxide. A: $P$--$T$ phase diagram obtained with ab initio neural cDFT (left) and experiment (right) Span1996ProctorBooknist_webbook; overall, good agreement between the two is observed. B: $\rho_{\rm b}$--$T$ phase diagram with $\chi_{T}^{-1}$ superimposed as a heat map (see Eq. \ref{['eqn:chiT']}). The Widom line obtained by $\max\chi_T$ is shown by the dotted line. We also show the Widom line obtained from $\max\xi$, where $\xi$ is the true correlation length (dashed line). The dot-dashed line shows the Fisher--Widom line, indicating a crossover from simple exponential to oscillatory asymptotic decay of the total correlation function. The Widom and Fisher--Widom lines are also plotted in panel A.