Learned Free-Energy Functionals from Pair-Correlation Matching for Dynamical Density Functional Theory

TL;DR

The paper tackles the challenge of obtaining accurate excess free-energy functionals needed for cDFT and DDFT by training a neural functional from bulk pair-correlation data via pair-correlation matching, and then applying it directly within DDFT to simulate non-equilibrium dynamics of inhomogeneous densities. It extends DDFT to open systems using a gradient-flow framework based on Wasserstein-Fisher-Rao metrics, enabling particle exchange with reservoirs. Across a 3D Lennard-Jones system with planar geometry and MOF-like confinement, the neural DDFT reproduces Brownian dynamics trajectories with high fidelity while offering orders-of-magnitude speedups, and the open-system extension captures breakthrough behavior consistent with BD. These results establish a practical route for accurate, efficient modeling of many-body non-equilibrium systems by reusing bulk-trained neural free-energy functionals.

Abstract

Classical density functional theory (cDFT) and dynamical density functional theory (DDFT) are modern statistical mechanical theories for modeling many-body colloidal systems at the one-body density level. The theories hinge on knowing the excess free-energy accurately, which is however not feasible for most practical applications. Dijkman et al. [Phys. Rev. Lett. 134, 056103 (2025)] recently showed how a neural excess free-energy functional for cDFT can be learned from bulk simulations via pair-correlation matching. In this article, we demonstrate how this same functional can be applied to DDFT, without any retraining, to simulate non-equilibrium overdamped dynamics of inhomogeneous densities. We evaluate this on a 3D Lennard-Jones system with planar geometry under various complex external potentials and observe good agreement of the dynamical densities with those from expensive Brownian dynamic simulations, up to the limit of the adiabatic approximation. We further develop and apply an extension of DDFT based on gradient flows, to a grand-canonical system modeled after breakthrough gas adsorption studies, finding similarly good agreement. Our results demonstrate a practical route for leveraging learned free-energy functionals in DDFT, paving the way for accurate and efficient modeling of many-body non-equilibrium systems.

Figures (6)

• Figure 1: An overview of the proposed neural DDFT framework, and the pair-correlation matching dijdft approach. (Bottom) A neural free-energy functional is trained via pair-correlation matching where its second functional derivative is optimized against the direct pair correlation function computed from simulated uniform bulk densities. (Top) The neural excess free-energy functional $\mathcal{F}_{\theta, \text{exc}}$ is used for DDFT to time evolve the inhomogeneous particle density $\rho(\mathbf{r})$ of a 3D Lennard-Jones system with planar geometry via automatic differentiation and numerical integration, towards equilibrium.
• Figure 2: Time evolution of the one-body density profile $\rho(z, \tau)$ for a system of interacting $3$D LJ particles with planar geometry subjected to a repulsive Gaussian potential $V_\text{ext}(z)$ (grey curve, right axis). The plots compare two neural excess free-energy functionals and an analytical FMT functional, with a reference Brownian dynamics simulation. Initially ($t = 0\tau$), the system is at equilibrium with a flat density profile. Upon application of the external potential, the density profile evolves non-trivially, exhibiting a depletion near the potential maximum and eventually approaching a new equilibrium ($t = 3\tau$). The neural DDFT approaches (red and blue) exhibit good agreement with Brownian dynamics throughout the temporal evolution, notably with the pair-correlation matching approach (red) not having seen any inhomogeneous densities during training.
• Figure 3: The final equilibrium density profiles and their RMSE over time evolution under complex external potentials (see SM supplement for the movie version). (a)-(d) A comparison of the final 1D equilibrium density profiles for a three-dimensional LJ system with planar geometry under different external potentials constructed from sums of repulsive Gaussian and steep well potentials. (e)-(h) show the corresponding RMSE deviation from the Brownian dynamics results over time evolution. The densities obtained with the neural functionals are significantly more accurate than those with the analytical functional, notably with the pair-correlation matching approach (red) not having seen any inhomogeneous densities during training.
• Figure 4: Breakthrough simulations using neural DDFT and Brownian dynamics (see SM supplement for the movie version). (a) LJ particles from the source move towards the sink under two different MOF-like repulsive external potentials. The dynamics of the density evolution and the final evolved density (visualized) obtained from DDFT are in close agreement with BD. (b) Time evolution of the output to input flux ratio, showing the approach towards steady-state or breakthrough condition. The particles reach breakthrough earlier under the MOF-2 external potential. The curves obtained from DDFT are in close agreement with BD, while being smoother and taking only $2\%$ of the compute time of BD.
• Figure 5: An overview of single-body direct correlation matching.