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Scalable sampling of molecular states in thermodynamic equilibrium is a long-standing challenge in statistical physics. Boltzmann generators tackle this problem by pairing normalizing flows with importance sampling to obtain uncorrelated…

Machine Learning · Computer Science 2026-01-21 Charlie B. Tan , Avishek Joey Bose , Chen Lin , Leon Klein , Michael M. Bronstein , Alexander Tong

Boltzmann Generators have emerged as a promising machine learning tool for generating samples from equilibrium distributions of molecular systems using Normalizing Flows and importance weighting. Recently, Flow Matching has helped speed up…

Machine Learning · Statistics 2025-10-21 Lorenz Vaitl , Leon Klein

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…

Computational Physics · Physics 2026-03-31 Emil Hoffmann , Maximilian Schebek , Leon Klein , Frank Noé , Jutta Rogal

Normalizing flows are a class of deep generative models that are especially interesting for modeling probability distributions in physics, where the exact likelihood of flows allows reweighting to known target energy functions and computing…

Machine Learning · Statistics 2023-11-27 Leon Klein , Andreas Krämer , Frank Noé

Efficient sampling from the Boltzmann distribution given its energy function is a key challenge for modeling complex physical systems such as molecules. Boltzmann Generators address this problem by leveraging continuous normalizing flows to…

Machine Learning · Computer Science 2025-10-17 Rishal Aggarwal , Jacky Chen , Nicholas M. Boffi , David Ryan Koes

Flows are exact-likelihood generative neural networks that transform samples from a simple prior distribution to the samples of the probability distribution of interest. Boltzmann Generators (BG) combine flows and statistical mechanics to…

Machine Learning · Statistics 2019-10-03 Jonas Köhler , Leon Klein , Frank Noé

Sampling from unnormalized target distributions, e.g.\ Boltzmann distributions $\mu_{\text{target}}(x) \propto \exp(-E(x)/T)$, is fundamental to many scientific applications yet computationally challenging due to complex, high-dimensional…

Machine Learning · Statistics 2026-04-22 Niclas Dern , Lennart Redl , Sebastian Pfister , Marcel Kollovieh , David Lüdke , Stephan Günnemann

Simulation-free training frameworks have been at the forefront of the generative modelling revolution in continuous spaces, leading to large-scale diffusion and flow matching models. However, such modern generative models suffer from…

Machine Learning · Computer Science 2025-10-31 Danyal Rehman , Oscar Davis , Jiarui Lu , Jian Tang , Michael Bronstein , Yoshua Bengio , Alexander Tong , Avishek Joey Bose

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…

Statistical Mechanics · Physics 2025-10-23 Maximilian Schebek , Frank Noé , Jutta Rogal

Sampling equilibrium molecular configurations from the Boltzmann distribution is a longstanding challenge. Boltzmann Generators (BGs) address this by combining exact-likelihood generative models with importance sampling, but practical…

Machine Learning · Computer Science 2026-05-29 Weilong Chen , Bojun Zhao , Jan Eckwert , Julija Zavadlav

Efficient sampling of the Boltzmann distribution of molecular systems is a long-standing challenge. Recently, instead of generating long molecular dynamics simulations, generative machine learning methods such as normalizing flows have been…

Machine Learning · Computer Science 2024-08-06 Henrik Schopmans , Pascal Friederich

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…

Statistical Mechanics · Physics 2024-11-26 Maximilian Schebek , Michele Invernizzi , Frank Noé , Jutta Rogal

Sampling physical systems with rough energy landscapes is hindered by rare events and metastable trapping. While Boltzmann generators already offer a solution, their reliance on the reverse Kullback--Leibler divergence frequently induces…

Machine Learning · Computer Science 2026-04-08 Guang Lin , Christian Moya , Di Qi , Xuda Ye

Diffusion models have shown promising potential for advancing Boltzmann Generators. However, two critical challenges persist: (1) inherent errors in samples due to model imperfections, and (2) the requirement of hundreds of functional…

Machine Learning · Computer Science 2024-09-12 Fengzhe Zhang , Jiajun He , Laurence I. Midgley , Javier Antorán , José Miguel Hernández-Lobato

Boltzmann generators approach the sampling problem in many-body physics by combining a normalizing flow and a statistical reweighting method to generate samples in thermodynamic equilibrium. The equilibrium distribution is usually defined…

Statistical Mechanics · Physics 2022-09-07 Manuel Dibak , Leon Klein , Andreas Krämer , Frank Noé

In this work, we consider the problem of training a generator from evaluations of energy functions or unnormalized densities. This is a fundamental problem in probabilistic inference, which is crucial for scientific applications such as…

Machine Learning · Computer Science 2024-08-30 Dongyeop Woo , Sungsoo Ahn

The generation of equilibrium samples of molecular systems has been a long-standing problem in statistical physics. Boltzmann Generators are a generative machine learning method that addresses this issue by learning a transformation via a…

Machine Learning · Statistics 2025-02-04 Leon Klein , Frank Noé

Generative models based on invertible transformations provide a physics-aware route to sample equilibrium configurations directly from the Boltzmann distribution, enabling efficient exploration of complex thermodynamic landscapes. Here, we…

Statistical Mechanics · Physics 2026-03-06 Luigi de Santis , John Russo , Andrea Ninarello

Normalizing flows can transform a simple prior probability distribution into a more complex target distribution. Here, we evaluate the ability and efficiency of generative machine learning methods to sample the Boltzmann distribution of an…

Soft Condensed Matter · Physics 2024-09-16 Gerhard Jung , Giulio Biroli , Ludovic Berthier

Sampling molecular conformations from the Boltzmann distribution is essential for computational chemistry, but iterative diffusion methods are prohibitively slow. Drifting Models offer one-step generation, yet their equilibrium matches the…

Chemical Physics · Physics 2026-03-09 Pipi Hu
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