用随机偏微分方程分析扩散模型的得分误差，揭示生成模型稳定性机制。
Modeling Score Approximation Errors in Diffusion Models via Forward SPDEs
- 将得分误差建模为随机源，驱动福克-普朗克方程演化概率密度场。
- 提出基于径向测试函数投影的二次变差评估指标，前10%采样轨迹即有效。
- 从几何稳定性和位移凸性角度解释生成模型鲁棒性，适合研究生成模型理论者。
本研究通过将得分估计误差视为驱动福克-普朗克方程的随机源，探究得分生成模型（SGMs）的动力学特性。不同于以粒子为中心的随机微分方程分析，本文采用随机偏微分方程（SPDE）框架，建模在随机漂移扰动下概率密度场的演化。在简化设定下，借助该框架从几何稳定性与位移凸性视角解析生成模型的鲁棒性。进一步，提出一个候选评估指标，基于SPDE解在径向测试函数上的投影二次变差。初步观察表明，该指标仅使用采样轨迹的初始10%即可保持有效性，暗示潜在计算效率优势。
原文摘要 · Abstract (English)
This study investigates the dynamics of Score-based Generative Models (SGMs) by treating the score estimation error as a stochastic source driving the Fokker-Planck equation. Departing from particle-centric SDE analyses, we employ an SPDE framework to model the evolution of the probability density field under stochastic drift perturbations. Under a simplified setting, we utilize this framework to interpret the robustness of generative models through the lens of geometric stability and displacement convexity. Furthermore, we introduce a candidate evaluation metric derived from the quadratic variation of the SPDE solution projected onto a radial test function. Preliminary observations suggest that this metric remains effective using only the initial 10% of the sampling trajectory, indicating a potential for computational efficiency.
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