Maximilian Schebek

Maximilian Schebek, Jiajun He, Emil Hoffmann et al. · cambridge

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.

Emil Hoffmann, Maximilian Schebek, Leon Klein et al.

Sampling equilibrium distributions is fundamental to statistical mechanics. While flow matching has emerged as scalable state-of-the-art paradigm for generative modeling, its potential for equilibrium sampling in condensed-phase systems remains largely unexplored. We address this by incorporating the periodicity inherent to these systems into continuous normalizing flows using Riemannian flow matching. The high computational cost of exact density estimation intrinsic to continuous normalizing flows is mitigated by using Hutchinson's trace estimator, utilizing a crucial bias-correction step based on cumulant expansion to render the stochastic estimates suitable for rigorous thermodynamic reweighting. Our approach is validated on monatomic ice, demonstrating the ability to train on systems of unprecedented size and obtain highly accurate free energy estimates without the need for traditional multistage estimators.

Maximilian Schebek, Frank Noé, Jutta Rogal

The use of generative models to sample equilibrium distributions of many-body systems, as first demonstrated by Boltzmann Generators, has attracted substantial interest due to their ability to produce unbiased and uncorrelated samples in `one shot'. Despite their promise and impressive results across the natural sciences, scaling these models to large systems remains a major challenge. In this work, we introduce a Boltzmann Generator architecture that addresses this scalability bottleneck with a focus on applications in materials science. We leverage augmented coupling flows in combination with graph neural networks to base the generation process on local environmental information, while allowing for energy-based training and fast inference. Compared to previous architectures, our model trains significantly faster, requires far less computational resources, and achieves superior sampling efficiencies. Crucially, the architecture is transferable to larger system sizes, which allows for the efficient sampling of materials with simulation cells of unprecedented size. We demonstrate the potential of our approach by applying it to several materials systems, including Lennard-Jones crystals, ice phases of mW water, and the phase diagram of silicon, for system sizes well above one thousand atoms. The trained Boltzmann Generators produce highly accurate equilibrium ensembles for various crystal structures, as well as Helmholtz and Gibbs free energies across a range of system sizes, able to reach scales where finite-size effects become negligible.

Maximilian Schebek, Nikolas M. Froböse, Bettina G. Keller et al.

Accurate calculations of solvation free energies remain a central challenge in molecular simulations, often requiring extensive sampling and numerous alchemical intermediates to ensure sufficient overlap between phase-space distributions of a solute in the gas phase and in solution. Here, we introduce a computational framework based on normalizing flows that directly maps solvent configurations between solutes of different sizes, and compare the accuracy and efficiency to conventional free energy estimates. For a Lennard-Jones solvent, we demonstrate that this approach yields acceptable accuracy in estimating free energy differences for challenging transformations, such as solute growth or increased solute-solute separation, which typically demand multiple intermediate simulation steps along the transformation. Analysis of radial distribution functions indicates that the flow generates physically meaningful solvent rearrangements, substantially enhancing configurational overlap between states in configuration space. These results suggest flow-based models as a promising alternative to traditional free energy estimation methods.

Maximilian Schebek, Michele Invernizzi, Frank Noé et al.

The accurate prediction of phase diagrams is of central importance for both the fundamental understanding of materials as well as for technological applications in material sciences. However, the computational prediction of the relative stability between phases based on their free energy is a daunting task, as traditional free energy estimators require a large amount of simulation data to obtain uncorrelated equilibrium samples over a grid of thermodynamic states. In this work, we develop deep generative machine learning models based on the Boltzmann Generator approach for entire phase diagrams, employing normalizing flows conditioned on the thermodynamic states, e.g., temperature and pressure, that they map to. By training a single normalizing flow to transform the equilibrium distribution sampled at only one reference thermodynamic state to a wide range of target temperatures and pressures, we can efficiently generate equilibrium samples across the entire phase diagram. Using a permutation-equivariant architecture allows us, thereby, to treat solid and liquid phases on the same footing. We demonstrate our approach by predicting the solid-liquid coexistence line for a Lennard-Jones system in excellent agreement with state-of-the-art free energy methods while significantly reducing the number of energy evaluations needed.