利用生成模型辅助蒙特卡洛采样

中文摘要 AI 辅助

对高维概率分布进行采样是科学计算的核心任务，应用范围涵盖贝叶斯推断、统计物理及分子模拟等领域。尽管经过数十年的方法发展，仍存在两大核心挑战：一是扩展至高维空间的能力，二是高效探索由亚稳态表征的多模态分布。马尔可夫链蒙特卡洛、调温方法或基于集体变量的增强采样等经典方法已取得重大成功，但也存在固有局限性。本教程综述探讨了近期在机器学习与计算统计物理交叉领域兴起的新范式：将生成模型作为采样工具。在此背景下，归一化流、扩散模型等并非用于传统数据驱动场景，而是作为灵活的概率模型，辅助对仅知归一化常数的分布进行采样。本文综述了这一快速发展领域的早期进展，并讨论了若干方法学方向，包括基于生成模型的精确采样器，以及在无数据情况下训练此类模型的策略。本文未尝试对文献进行详尽调查，而是呈现了一系列关键思想与方法，并讨论了其优势与局限性。本综述旨在为物理学和机器学习领域的读者提供易懂的教程，为有意探索这一令人兴奋的研究方向的研究者提供入门基础。

英文摘要

Sampling high-dimensional probability distributions is a central task in scientific computing, with applications ranging from Bayesian inference to statistical physics and molecular simulation. Despite decades of methodological developments, two major challenges remain: scaling to high dimensions and efficiently exploring multimodal distributions characterized by metastable states. Classical approaches such as Markov chain Monte Carlo, tempering methods, or enhanced sampling based on collective variables have achieved major successes, but they also face intrinsic limitations. This tutorial review explores a new paradigm that has recently emerged at the interface of machine learning and computational statistical physics: the use of generative models as tools for sampling. In this context, models such as normalizing flows and diffusion models are not used in their traditional data-driven setting, but rather as flexible probabilistic models that can assist the sampling of distributions known only up to a normalization constant. This manuscript reviews the early development of this rapidly evolving field and discusses several methodological directions, including exact samplers based on generative models and strategies to train such models in the absence of data. While an exhaustive survey of the literature is not attempted, we present a selection of key ideas and methods, along with a discussion of their strengths and limitations. The review is intended to be an accessible tutorial for both physics and machine learning audiences, and it aims to provide a starting point for researchers interested in exploring this exciting area of research.

发表机构

• École normale supérieure（巴黎高等师范学院）
• PSL Research University（巴黎文理研究大学）

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