用扩散模型改进玻尔兹曼生成器，提升高维多峰分布采样精度。
Diffusion-based Annealed Boltzmann Generators : benefits, pitfalls and hopes
- 基于退火蒙特卡洛与扩散模型结合，构建新型生成器采样框架。
- 二阶去噪核可显著提升性能，但一阶方法在多峰场景下系统失败。
- 提出确定性积分路径，适合对精度要求高、计算资源充足的场景。
在统计物理中，从热力学平衡态采样是核心挑战。玻尔兹曼生成器（BGs）通过生成模型结合蒙特卡洛校正步骤，实现对未归一化目标分布的渐近无偏采样。现有大多数BGs依赖经典蒙特卡洛机制（如重要性采样），需生成模型具备可计算似然，且在高维多峰分布中表现不佳。本文研究基于退火蒙特卡洛（aMC）的BGs，其通过一系列中间密度连接简单参考分布与目标分布，克服上述限制。扩散模型（DMs）作为强大生成模型，可通过扩散诱导密度路径融入aMC校准框架，成为理想骨架。我们在可控多峰高斯混合分布上进行实证元分析（调节模式分离度、模式数和维度），明确区分推理与学习效应：即使使用完美训练的DM，仅采用一阶随机去噪核的标准积分仍系统性失败；而利用协方差信息的二阶去噪核可显著改善。此外，我们提出基于一阶确定性传输映射的aMC积分方法，在更高计算成本下优于随机一阶变体。在真实数据训练的DM设置下，所有DM-aMC变体均难以生成准确的BGs，主要瓶颈在于DM对对数似然估计不准。代码已开源：https://github.com/h2o64/dabg。
原文摘要 · Abstract (English)
Sampling configurations at thermodynamic equilibrium is a central challenge in statistical physics. Boltzmann Generators (BGs) tackle it by combining a generative model with a Monte Carlo (MC) correction step to obtain asymptotically unbiased samples from an unnormalized target. Most current BGs use classic MC mechanisms such as importance sampling, which both require tractable likelihoods from the backbone model and scale poorly in high-dimensional, multi-modal targets. We study BGs built on annealed Monte Carlo (aMC), which is designed to overcome these limitations by bridging a simple reference to the target through a sequence of intermediate densities. Diffusion models (DMs) are powerful generative models and have already been incorporated into aMC-based recalibration schemes via the diffusion-induced density path, making them appealing backbones for aMC-BGs. We provide an empirical meta-analysis of DM-based aMC-BGs on controlled multi-modal Gaussian mixtures (varying mode separation, number of modes, and dimension), explicitly disentangling inference effects from learning effects by comparing (i) a perfectly learned DM and (ii) a DM trained from data. Even with a perfect DM, standard integrations using only first-order stochastic denoising kernels fail systematically, whereas second-order denoising kernels can substantially improve performance when covariance information is available. We further propose a deterministic aMC integration based on first-order transport maps derived from DMs, which outperforms the stochastic first-order variant at higher computational cost. Finally, in the learned-DM setting, all DM-aMC variants struggle to produce accurate BGs; we trace the main bottleneck to inaccurate DM log-density estimation. Code available at https://github.com/h2o64/dabg.
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