用镜面反射速度的朗之万动力学，实现带约束的生成模型采样。
Score-based constrained generative modeling via Langevin diffusions with boundary conditions
- 引入镜面反射速度的动能朗之万过程，确保生成样本满足边界约束。
- 提出高效数值采样器，收敛速度达到最优离散化阶数。
- 对比了镜面反射与局部时间约束模型，验证方法有效性。
基于随机微分方程（SDE）的得分生成模型在未知分布采样中表现优异，但常无法满足隐含约束。本文提出一种基于动能（欠阻尼）朗之万动力学的约束生成模型，其速度在定义约束的边界上采用镜面反射。该方法实现了分段连续可导的加噪与去噪过程，其中去噪过程由时间反演的动力学描述，受限于带有镜面边界条件的域内。此外，我们还对现有基于局部时间项的反射SDE约束生成模型进行了补充。通过设计收敛速度达最优离散化阶的高效数值采样器，系统比较了受限（镜面反射动能）朗之万扩散与基于局部时间的反射扩散模型的性能。
原文摘要 · Abstract (English)
Score-based generative models based on stochastic differential equations (SDEs) achieve impressive performance in sampling from unknown distributions, but often fail to satisfy underlying constraints. We propose a constrained generative model using kinetic (underdamped) Langevin dynamics with specular reflection of velocity on the boundary defining constraints. This results in piecewise continuously differentiable noising and denoising process where the latter is characterized by a time-reversed dynamics restricted to a domain with boundary due to specular boundary condition. In addition, we also contribute to existing reflected SDEs based constrained generative models, where the stochastic dynamics is restricted through an abstract local time term. By presenting efficient numerical samplers which converge with optimal rate in terms of discretizations step, we provide a comprehensive comparison of models based on confined (specularly reflected kinetic) Langevin diffusion with models based on reflected diffusion with local time.
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