Sampling Boltzmann distributions via normalizing flow approximation of transport maps

2026-03-15

AI Summary

This work addresses the challenge of sampling high-dimensional, low-regularity Boltzmann distributions arising from Coulomb and Lennard-Jones interactions in molecular dynamics. The authors propose a method based on normalizing flows to approximate the Moser transport map, thereby constructing an invertible transformation between a reference measure and the target distribution. They establish, for the first time, a rigorous mathematical foundation for approximating such low-regularity Boltzmann distributions with normalizing flows, proving the existence of transport maps achieving arbitrary accuracy in Wasserstein distance. Furthermore, they demonstrate that RealNVP architectures can effectively capture metastable dynamics. Leveraging neural network approximation theory in Sobolev spaces, the approach is validated on model systems and alanine dipeptide, showing excellent agreement between the generated samples and the true Boltzmann distribution, as well as accurate reproduction of metastable behavior.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Sampling/Simulation-based Search
• Natural Language Processing: Safety and Robustness

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Security and Privacy: Large-scale security measurements

Abstract

In a celebrated paper \cite{noe2019boltzmann}, Noé, Olsson, Köhler and Wu introduced an efficient method for sampling high-dimensional Boltzmann distributions arising in molecular dynamics via normalizing flow approximation of transport maps. Here, we place this approach on a firm mathematical foundation. We prove the existence of a normalizing flow between the reference measure and the true Boltzmann distribution up to an arbitrarily small error in the Wasserstein distance. This result covers general Boltzmann distributions from molecular dynamics, which have low regularity due to the presence of interatomic Coulomb and Lennard-Jones interactions. The proof is based on a rigorous construction of the Moser transport map for low-regularity endpoint densities and approximation theorems for neural networks in Sobolev spaces. Numerical simulations for a simple model system and for the alanine dipeptide molecule confirm that the true and generated distributions are close in the Wasserstein distance. Moreover we observe that the RealNVP architecture does not just successfully capture the equilibrium Boltzmann distribution but also the metastable dynamics.

Problem

• Boltzmann distribution
• normalizing flow
• transport map
• molecular dynamics
• Wasserstein distance

Innovation

• normalizing flow
• Boltzmann distribution
• Moser transport map
• Wasserstein distance
• low regularity