用流模型逼近传输映射，实现高维玻尔兹曼分布的精确采样。
Sampling Boltzmann distributions via normalizing flow approximation of transport maps
- 通过构造Moser传输映射，证明流模型可逼近玻尔兹曼分布。
- 在水溶性模型与丙氨酸二肽系统中，生成分布与真实分布的Wasserstein距离极小。
- 不仅捕捉平衡态，还能再现分子的亚稳态动力学，适合复杂系统模拟。
在一篇开创性论文中，Noé、Olsson、Köhler 和 Wu 提出了一种高效方法，利用流模型近似传输映射来采样分子动力学中的高维玻尔兹曼分布。本文为此方法建立了坚实的数学基础：证明了参考测度与真实玻尔兹曼分布之间存在一个流模型，其误差可任意小（以Wasserstein距离衡量）。该结果适用于具有低正则性的通用玻尔兹曼分布，这些分布因原子间库仑和Lennard-Jones相互作用而呈现不规则性。证明基于低正则性端点密度的Moser传输映射严格构造，以及神经网络在Sobolev空间中的逼近定理。数值实验在简单模型系统和丙氨酸二肽分子上均显示，生成分布与真实分布的Wasserstein距离接近。此外，发现RealNVP架构不仅能有效捕获平衡态玻尔兹曼分布，还可重现系统的亚稳态动力学。
原文摘要 · Abstract (English)
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.
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