通过传输映射的归一化流近似采样玻尔兹曼分布
Sampling Boltzmann distributions via normalizing flow approximation of transport maps
AI总结：
本文通过传输映射的归一化流近似方法，建立了参考测度与真实玻尔兹曼分布之间的归一化流，证明其在Wasserstein距离下的存在性，并通过数值模拟验证了该方法的有效性。
摘要：
在一篇著名的论文中，Noé等人提出了一种高效的采样高维玻尔兹曼分布的方法，该方法基于传输映射的归一化流近似。本文在此基础上建立了坚实的数学基础。我们证明了在Wasserstein距离下，参考测度与真实玻尔兹曼分布之间存在归一化流，误差可以任意小。该结果涵盖了分子动力学中的一般玻尔兹曼分布，这些分布由于存在原子间库仑和伦敦-琼斯相互作用而具有低正则性。证明基于对低正则性端点密度的Moser传输映射的严格构造以及Sobolev空间中神经网络的近似定理。对于简单的模型系统和丙氨酸二肽分子的数值模拟表明，真实分布和生成分布之间的Wasserstein距离接近。此外，我们还观察到RealNVP架构不仅成功捕捉了平衡的玻尔兹曼分布，还捕捉了亚稳态动力学。
英文摘要：
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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