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.
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.
Dynamic optimal transport unifies optimal transport, fluid mechanics, and gradient-flow theory within a continuous dynamical framework, offering a geometry-aware language for applications across physics, biology, and machine learning. However, conventional formulations cast it as a constrained optimization problem that must explicitly satisfy the continuity equation, hindering the reconstruction of the underlying dynamics directly from data. We propose the energetic variational method for dynamic optimal transport (EVMDOT), which reformulates it within an energetic variational framework by combining the flow map, the least action principle, and the maximum dissipation principle. The flow map recasts the constrained problem as an unconstrained one by automatically enforcing the continuity equation, while the balance between the conservative and dissipative forces determines the velocity field. Applied to the Fokker--Planck equation, the EVMDOT reconstructs both the energy landscape and the Waddington landscape directly from time-series density data. Through numerical experiments, we reveal that the EVMDOT achieves an intrinsic balance between data quantity and data quality: a sufficient data quantity compensates for limited data quality, making the reconstruction robust to the choice of the observation window. We further apply the EVMDOT to the Alzheimer's Disease Neuroimaging Initiative (ADNI) dataset to infer the potential landscape of amyloid-$\beta$ and tau, revealing two wells corresponding to the cognitively normal and Alzheimer's disease stages and the transition pathway between them.
We study high-dimensional numerical integration with respect to complex target measures using diffusion-based transport maps and randomized quasi-Monte Carlo (RQMC). Score-based diffusion models induce a deterministic probability flow ODE that transports a simple prior to the target, suggesting a principled way to transform low-discrepancy points on the unit cube into informative samples. We construct a cube-to-target map by composing a Gaussian base transformation (the component-wise inverse Gaussian CDF) with an Euler-discretized probability flow ODE. To retain unbiasedness under transport approximation, we formulate integration as importance sampling (IS) on the cube. Our main result provides verifiable conditions under which the resulting IS integrand satisfies the boundary growth condition, implying an $O(N^{-1+\epsilon})$ RMSE for scrambled nets. We then establish these conditions for diffusion probability-flow transport under mild bounded-derivative assumptions on the learned vector field, explicitly controlling the boundary singularities introduced by the inverse Gaussian CDF. Experiments range from a 2D mixture to 784D images and a 40,960D conditional vorticity-assimilation task; in the latter, blocked scrambled Sobol'sampling reduces the randomization standard deviation of nonlinear accuracy metrics at essentially unchanged online denoising cost. Together, these results give a theoretical and empirical foundation for combining diffusion generative modeling with high-precision RQMC integration.
The accuracy of the computational estimation of relative free energies (e.g., for solvation or protein–ligand binding) depends on the smoothness of the phase-space transformation between the two alchemical end-states. A smooth transformation ensures sufficient phase-space overlap between the neighboring intermediate states connecting the two end-states in equilibrium (EQ) simulations and generates less dissipative work in nonequilibrium (NEQ) simulations. The conventional energy interpolation (EI) coupling scheme constructs the intermediate states by linearly combining the end-state potentials. We show that the enveloping distribution sampling (EDS) coupling scheme, a generalization of EI where the corresponding Boltzmann factors are linearly combined, represents a much more flexible alternative. Through the use of a negative smoothing parameter, the EDS scheme increases the local curvature of the sampling phase space along the transformation axis, thereby avoiding phase transitions and creating a smoother transformation. We validate this behavior in increasingly complex settings, from harmonic oscillators and Ising model systems to absolute hydration free-energy (AHFE) calculations on the FreeSolv data set. EDS consistently yields more accurate and statistically robust free-energy estimates compared to the conventional EI scheme for the model system calculations, while a clear advantage is observed for AHFE in the NEQ regime, where less dissipative transitions lead to more reliable free-energy estimates.
Shu-Yu Chen, Enrico Ruijsenaars, P. Hünenberger et al.· Journal of Chemical Theory a...· 0 citations
Rare-event molecular dynamics simulations are limited by the separation between atomic and activated timescales. Hyperdynamics accelerates these processes via bias potentials while preserving correct transition kinetics. The minimum-mode following (MMF) formulation provides a ridge-based bias construction but typically employs an identity approximation for the Jacobian together with local force evaluation, which introduces inconsistency in high-dimensional systems. Here, we develop a Jacobian-corrected MMF (J-MMF) method that improves both accuracy and consistency. The method introduces a path-informed semi-analytical Jacobian based on the sequential evolution of minimum-mode vectors and an orthogonal projection that removes spurious force components along the reaction coordinate, without additional force evaluations. Benchmark simulations of adatom diffusion on the Cu(100) surface (1-201 mobile atoms) demonstrate improved accuracy and robust, largely size-independent acceleration compared with standard MMF and bond-boost methods. J-MMF provides a scalable and consistent framework for rare-event simulations in complex atomistic systems.
Lixiang Qian, Liang Zhang· Journal of Chemical Physics· 0 citations
Kirigami metamaterials, formed by introducing cuts into planar sheets, can exhibit bistability through geometry-dependent energy landscapes. Designing structures with prescribed energy barriers and target deformation states is therefore essential for programmable mechanical functionality. Here, we present a physics-informed neural network (PINN) framework that does not require pre-collected labeled training datasets and unifies forward prediction and inverse programming of bistable kirigami energy landscapes by embedding equilibrium conditions, energy formulations, and geometric compatibility directly into the learning objective. In the forward setting, the framework predicts continuous energy landscapes with coefficients of determination above 0.99 and barrier errors below 0.1%. In the inverse setting, it identifies kirigami geometries from fully prescribed energy curves with barrier errors below 5%, and further extends to a minimally specified setting in which only the target energy barrier and zero-energy stable states are given. In this underdetermined case, the framework autonomously infers the full energy landscape while achieving a mean barrier error of 0.4%. Finite element analysis and experiments on 3D-printed prototypes confirm that the programmed ordering of energy barriers is preserved. By assembling unit cells with different programmed barriers, we further demonstrate sequential actuation, in which units with lower barriers transform first during tensile loading. This work establishes an efficient physics-informed approach for programming bistable energy landscapes in planar kirigami systems, with potential applications in deployable structures, soft robotic actuators, and impact mitigation devices.
Sukheon Kang, Sukkyung Kang, Sanha Kim et al.· Materials Horizons· 0 citations
