用随机插值与正向-反向随机微分方程实现高维分布的高效采样。
Sampling in High-Dimensions using Stochastic Interpolants and Forward-Backward Stochastic Differential Equations
- 构建时间依赖的概率密度路径，连接高斯分布与目标分布。
- 通过求解汉密尔顿-雅可比-贝尔曼偏微分方程生成扩散过程。
- 结合机器学习方法解决复杂高维分布采样难题，适用于传统方法失效场景。
我们提出一类基于扩散的算法，用于从给定非归一化密度的高维概率分布中采样。理想情况下，该方法可在有限时间内将样本从高斯分布传输至指定目标分布。方法基于随机插值框架，定义一个时间索引的概率密度序列，逐步连接高斯分布与目标分布。随后，推导出在每个时刻均服从该概率密度的扩散过程。获得该扩散过程需求解特定的汉密尔顿-雅可比-贝尔曼（HJB）偏微分方程，我们利用正向-反向随机微分方程（FBSDE）理论与基于机器学习的方法求解这些方程。数值实验表明，该算法能有效从传统方法难以处理的分布中采样。
原文摘要 · Abstract (English)
We present a class of diffusion-based algorithms to draw samples from high-dimensional probability distributions given their unnormalized densities. Ideally, our methods can transport samples from a Gaussian distribution to a specified target distribution in finite time. Our approach relies on the stochastic interpolants framework to define a time-indexed collection of probability densities that bridge a Gaussian distribution to the target distribution. Subsequently, we derive a diffusion process that obeys the aforementioned probability density at each time instant. Obtaining such a diffusion process involves solving certain Hamilton-Jacobi-Bellman PDEs. We solve these PDEs using the theory of forward-backward stochastic differential equations (FBSDE) together with machine learning-based methods. Through numerical experiments, we demonstrate that our algorithm can effectively draw samples from distributions that conventional methods struggle to handle.
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