Assessing generative modeling approaches for free energy estimates in condensed matter

TL;DR

This work assesses generative-model-based approaches for estimating free energy differences in condensed matter, focusing on discrete and continuous normalizing flows (TFEP) and FEAT with escorted Jarzynski. By benchmarking on monoatomic water and Lennard-Jones solids with periodic boundaries, the authors quantify accuracy, data efficiency, and scalability relative to traditional estimators. They find that CNFs and FEAT deliver high accuracy with modest data, while coupling flows require more training and can underperform at low budgets; FEAT can excel in data-scarce regimes but has different computational characteristics than CNFs. The study highlights the promise of size- and system-transferable, graph-neural-network–based generative models as scalable alternatives to MBAR/TI in condensed-phase free energy measurements, while also identifying current limitations and directions for transferability and efficiency improvements.

Abstract

The accurate estimation of free energy differences between two states is a long-standing challenge in molecular simulations. Traditional approaches generally rely on sampling multiple intermediate states to ensure sufficient overlap in phase space and are, consequently, computationally expensive. Several generative-model-based methods have recently addressed this challenge by learning a direct bridge between distributions, bypassing the need for intermediate states. However, it remains unclear which approaches provide the best trade-off between efficiency, accuracy, and scalability. In this work, we systematically review these methods and benchmark selected approaches with a focus on condensed-matter systems. In particular, we investigate the performance of discrete and continuous normalizing flows in the context of targeted free energy perturbation as well as FEAT (Free energy Estimators with Adaptive Transport) together with the escorted Jarzynski equality, using coarse-grained monatomic ice and Lennard-Jones solids as benchmark systems. We evaluate accuracy, data efficiency, computational cost, and scalability with system size. Our results provide a quantitative framework for selecting effective free energy estimation strategies in condensed-phase systems.

Figures (4)

• Figure 1: Overview of free energy estimation methods. $p_A$ and $p_B$ denote the equilibrium distributions of two systems $A$ and $B$. (a) Free energy perturbation (FEP) fails when two systems have no overlap. (b) Multi-staged approaches interpolate between $p_A$ and $p_B$ through a sequence of mutually overlapping intermediate distributions $p_i$ (MBAR, TI). (c) and (d) Learned FEP constructs an invertible map $f$ to bridge the two non-overlapping distributions. Discrete flows learn a conditioner $C^\theta$ parameterizing a bijecting function, whereas continuous flows learn a time-dependent transport vector field $v^\theta_t$. (e) Jarzynski equality utilizes the work along stochastic nonequilibrium trajectories with a time-dependent potential energy $U_t$. (f) Escorted Jarzynski equality introduces an additional control term $b_t$ to minimize the dissipation along the nonequilibrium trajectories and, in FEAT, both the score $s^\phi_t = \nabla u^\phi_t$ and control term $b^\theta_t$ can be learned.
• Figure 2: Overview of the featurization used for the different models. In all cases, a GNN encodes environment-dependent information by aggregating data from each particle’s neighborhood. For coupling flows, this information defines the bijector function through the conditioner. For CNFs, it determines the vector field, and for FEAT the control term and score.
• Figure 3: Error in the absolute free energies for (a) cubic and hexagonal mW ice and (b) FCC and HCP LJ as obtained from TFEP using coupling flows and continuous flows, and from escorted Jarzynski equality using FEAT. Uncertainties were obtained by evaluating three independently trained models once each.
• Figure 4: Free energy differences between (a) cubic--hexagonal mW ice and (b) FCC--HCP LJ phases as a function of the number of inference samples for low, medium, and large training budgets. The reference values are $\beta \Delta F_{\text{mW cub-hex}}/N = 0.0074$ and $\beta \Delta F_{\text{LJ FCC-HCP}}/N = 0.0053$, respectively.