Learning Density Functionals to Bridge Particle and Continuum Scales

TL;DR

This work integrates physics-informed neural corrections within classical density functional theory to learn corrections to the excess Helmholtz free energy $F_{ex}[ ho]$ for Lennard–Jones fluids. By enforcing the Euler–Lagrange equilibrium conditions via an adjoint optimization, the augmented functional preserves thermodynamic structure while capturing missing correlations, achieving quantitative agreement with MD for planar adsorption, bulk coexistence, and surface tension, and extending accurately to 3D droplet shapes and contact angles. The approach yields transferable predictions across geometry, temperature, and confinement with data-efficient training that exploits the underlying physics, offering a scalable bridge from atomistic simulations to continuum interfacial models. This framework provides a general route to learned thermodynamic functionals applicable to multiscale modeling of wetting, capillarity, and interfacial phenomena in complex fluids.

Abstract

Predicting interfacial thermodynamics across molecular and continuum scales remains a central challenge in computational science. Classical density functional theory (cDFT) provides a first-principles route to connect microscopic interactions with macroscopic observables, but its predictive accuracy depends on approximate free-energy functionals that are difficult to generalize. Here we introduce a physics-informed learning framework that augments cDFT with neural corrections trained directly against molecular-dynamics data through adjoint optimization. Rather than replacing the theory with a black-box surrogate, we embed compact neural networks within the Helmholtz free-energy functional, learning local and nonlocal corrections that preserve thermodynamic consistency while capturing missing correlations. Applied to Lennard-Jones fluids, the resulting augmented excess free-energy functional quantitatively reproduces equilibrium density profiles, coexistence curves, and surface tensions across a broad temperature range, and accurately predicts contact angles and droplet shapes far beyond the training regime. This approach combines the interpretability of statistical mechanics with the adaptability of modern machine learning, establishing a general route to learned thermodynamic functionals that bridge molecular simulations and continuum-scale models.

Figures (21)

• Figure 1: Relationship between the ML corrections to the physics-based DFT approximation and the features of the density profile of the adsorbed fluid. The correction terms $\phi_\theta^{(1, 3)}$ allows us to capture the bulk binodal, resulting in the correct liquid and gas plateaus of the density profile. The corrections $\phi_\theta^{(2, 4)}$ allow us to capture the correct wall-liquid layering and the slope of the wall-vapor interface.
• Figure 2: Illustrative example of the MD data showing liquid, adsorbed on the planar wall, in contact with its saturated vapor. Limited number of such simulations at different temperatures were used to train the augmented DFT: (a) perspective view and (b) frontal view. An equilibration run is performed to relax the system to its equilibrium state. After equilibrium is reached, a production run is carried out to collect the statistical data for analysis
• Figure 3: Schematic of the adjoint training loop for Eq. \ref{['eq:Optimization']}. At each optimization step, (i) the EL equation \ref{['eq:EL-NVT']} is solved and (ii) the equilibrium density field $\rho(\bm{\theta})$ is computed. Then (iii) the target MD data is injected, and the objective function $J$ in Eq. \ref{['eq:J']} is evaluated. Then (iv) the gradient $\nabla_{\bm{\theta}}J$ is computed in $O(1)$ using adjoint equations \ref{['eq:AdjointEquation']}. Finally, (v) the ADAM scheme is used to (vi) update the parameters $\theta$.
• Figure 4: Superposition of wall-normal equilibrium density profiles for validation data at $N=5000,\,T=0.60$: MD reference (black, shaded confidence interval), baseline mean-field cDFT (blue), and ML-augmented cDFT (red). The validation MD profile $\rho_\text{ref}(x)$ is shown with shaded confidence bands $\rho_\text{ref}(x)\pm \sigma_{\rho_\text{ref}}(x)$, where $\sigma_{\rho_\text{ref}}(x)$ is the standard deviation at position $x$. The ML cDFT accurately captures the near-wall layering and the bulk densities at the plateaus of $\rho_\text{ref}(x)$. On the other hand, the baseline DFT misses all these features.
• Figure 5: Heat map of the relative error $\epsilon_\rho(N,T)$\ref{['eq:eps_rho']} across the validation set. The color scale reports the deviation between the augmented cDFT predictions and MD reference density profiles as a function of particle number $N$ and temperature $T$. In particular, the trained model is able to reduce the $L_1$ error between the ML prediction and MD of around $90\%$ with respect to the for a wide range of the testing data set.