Liquids exhibit collective behavior that depends sensitively on thermodynamic conditions, interfaces and confinement, yet predicting each new state commonly requires a separate atomistic simulation. Classical density functional theory offers a reusable variational description, but its central excess free-energy functional is generally unknown, and learned approximations have largely remained restricted to planar or lower-dimensional settings. Here we show that this functional can be learned directly from fully three-dimensional equilibrium density fields while preserving spatial symmetry and variational consistency, without free-energy or chemical-potential labels. A single learned functional transfers across temperatures, system sizes and statistical ensembles, and recovers structure factors, the equation of state, liquid--vapor coexistence and interfacial broadening, none of which are used as training targets. Applied to complex three-dimensional geometries, it predicts the non-monotonic force associated with formation and rupture of a solvent-depleted bridge between colloids and adsorption in an interconnected gyroid pore. These results demonstrate that equilibrium density data can be converted into a transferable thermodynamic generator connecting microscopic liquid structure to response, phase behavior and collective phenomena.
We construct a neural classical density functional that acts directly on unrestricted three-dimensional density profiles. As a computationally tractable test bed, we consider parallel hard cubes of side length three on a simple cubic lattice. A fully convolutional network learns the one-body direct-correlation functional $c^{(1)}[\rho]$ from data obtained with grand-canonical Monte Carlo simulations in randomized external potentials. Complete profiles are used during both training and inference; a stochastic Bernoulli mask on the output sites makes full-profile training effective without explicitly extracting and storing overlapping local density windows. Averaging the first-layer kernels over all 48 rotations and reflections of the cubic point group additionally imposes exact cubic equivariance without data augmentation. We compare the learned functional with independent simulation data and with the lattice fundamental-measure functional of Lafuente and Cuesta. The neural functional markedly improves the homogeneous equation of state and the density profile at a planar hard wall. For the anisotropic pair distribution around a fixed particle, both functionals reproduce the principal packing shells, with their relative accuracy depending on crystallographic direction. These results demonstrate neural density-functional calculations on complete three-dimensional profiles while also identifying accurate full-dimensional training data, thermodynamic consistency, and structural correlations as the central challenges for extensions to continuum fluids.
Jens Weimar, Martin Oettel, Alessandro Simon· 1 citation
We evaluate machine learning force fields derived from different density functional theory exchange correlation functionals using the full six-dimensional pair correlation function of liquid water, three-body structural descriptors, excess entropy, and transport properties. The predicted microscopic structure and dynamics depend strongly on the underlying functional: neglecting dispersion produces pronounced overstructuring, overly negative excess entropy, and suppressed diffusion. Translational and orientational entropy contributions are tightly coupled and together exhibit a clear relationship with the reduced selfdiffusion coefficient. Among the tested models, RPBE-D3 provides the most consistent agreement with experiment across structural, thermodynamic, and transport properties. The classical SPC/E model serves as an additional reference and displays notable similarities to RPBE-D3, consistent with the comparable Born effective and partial charges of the two models.
A. Kretschmer, Florian Altmann, Nader Nour et al.· 0 citations
Free-energy surfaces govern the populations of metastable states and the barriers that control transitions between them, making their direct optimization a central challenge in molecular and materials design. In this work, we introduce Gradient-Based Free Energy Surface Optimization (GB-FESO), an inverse design framework that uses a trained conditional diffusion model as a differentiable surrogate for the ensemble distribution. After training, the diffusion model is frozen, and the conditioning variables defining the system are optimized so that the generated ensemble reproduces a prescribed target free-energy surface. The optimization is carried out by backpropagating a distribution-level loss, based on kernel density estimates of the Kullback-Leibler divergence, through a deterministic diffusion sampling trajectory. We first validate GB-FESO on one-dimensional Gaussian ensembles, demonstrating that both continuous and relaxed discrete conditioning variables can be optimized to recover target distributions, including those outside the training domain. We then apply the method to a four-particle Lennard-Jones toy peptide exhibiting multiple metastable conformational states. In this more physically motivated setting, GB-FESO successfully optimizes the interaction parameters to reproduce target free-energy landscapes in the majority of test cases, with optimization performed either in the full internal-coordinate space or in a reduced collective-variable representation. These results establish GB-FESO as a promising first step toward an ensemble-level inverse design framework for molecular systems with prescribed thermodynamic and kinetic behavior.
A novel anisotropic machine learning CG potential is introduced that extends the point particle representation of atomic nuclei to massive ellipsoidal beads with orientation-dependent features, enabling the learning of energies, forces, and torques directly from atomistic data.
High-entropy alloys (HEAs) exhibit exceptional structural and functional properties arising from their complex local chemical environments, and their vast compositional space offers considerable flexibility to further tune and optimize these properties. Atomistic simulations based on density functional theory (DFT) have played a central role in elucidating the thermodynamic, mechanical, magnetic, and defect-related properties of HEAs. However, DFT simulations are severely limited by the intrinsic chemical and configurational complexity of these alloys, particularly because reliable predictions require extensive statistical sampling over chemically diverse configurations and access to extended spatial and temporal scales. In this review, we summarize recent advances in atomistic simulations of HEAs, with particular emphasis on machine learning interatomic potentials (MLIPs), which extend beyond conventional DFT approaches. We discuss how MLIPs enable statistically robust simulations with near-DFT accuracy while dramatically reducing computational cost, thereby allowing explicit treatment of chemical short-range order, vibrational contributions to Gibbs energies, point defects, diffusion, dislocation behavior, grain boundaries, and hydrogen absorption in chemically complex alloys. Particular attention is devoted to the role of local chemical environments, many-body interactions, and configurational sampling in determining HEA properties. We further review recent developments in universal/foundation MLIPs trained on chemically diverse datasets and discuss their potential for rapid and transferable atomistic simulations of HEAs across broad compositional and configurational spaces. We discuss current limitations and open challenges, including transferability to highly distorted defect configurations, treatment of magnetic and charge degrees of freedom, incorporation of finite-temperature excitations, and construction of representative training datasets for chemically and structurally complex systems. This review aims to provide a comprehensive perspective on the ongoing transition from conventional DFT-based simulations toward scalable, statistically rigorous, and predictive atomistic modeling frameworks for HEAs and related compositionally complex materials.
Yuji Ikeda, Xiang Xu, Pranav Kumar et al.· Journal of Materials Science· 0 citations
The prediction of stable alloys forming solid-state solutions across large portions of the composition space is a serious theoretical challenge, since one has to evaluate the Gibbs free energy, including both configurational and vibrational contributions. This requires an energy theory capable of extremely high throughput. By taking the Ni-Pd system as prototype, we construct an efficient Jacobi-Legendre machine-learning potential based on density-functional-theory data, which provides accurate energies and forces across the entire composition space. Based on a cluster expansion up to three-body terms and only 873 trainable parameters, this allows us to compute the partition function by directly integrating all accessible microstates, differing for composition, atomic configuration and thermal agitation. We confirm that Ni and Pd are fully miscible, forming an $fcc$ solid-state solution. This is only metastable at room temperature, while becomes thermodynamically stable at around 600~K, with the stability achieved first at the Pd-rich end of the composition range. Interestingly, entropy and heat capacity analysis reveal a competition between the solid-state solution and two intermetallic phases with long-period L1$_0$ structure for NiPd and NiPd$_3$. All in all, our approach offers a powerful and high-throughput workflow for the study of disordered alloys, an approach that can be extended to multi-component systems such as high-entropy alloys.
Rutchapon Hunkao, U. Patil, S. Sanvito· 0 citations
