In Score based Generative Modeling (SGMs), the state-of-the-art in generative modeling, stochastic reverse processes are known to perform better than their deterministic counterparts. This paper delves into the heart of this phenomenon, comparing neural ordinary differential equations (ODEs) and neural stochastic differential equations (SDEs) as reverse processes. We use a control theoretic perspective by posing the approximation of the reverse process as a trajectory tracking problem. We analyze the ability of neural SDEs to approximate trajectories of the Fokker-Planck equation, revealing the advantages of stochasticity. First, neural SDEs exhibit a powerful regularizing effect, enabling $L^2$ norm trajectory approximation surpassing the Wasserstein metric approximation achieved by neural ODEs under similar conditions, even when the reference vector field or score function is not Lipschitz. Applying this result, we establish the class of distributions that can be sampled using score matching in SGMs, relaxing the Lipschitz requirement on the gradient of the data distribution in existing literature. Second, we show that this approximation property is preserved when network width is limited to the input dimension of the network. In this limited width case, the weights act as control inputs, framing our analysis as a controllability problem for neural SDEs in probability density space. This sheds light on how noise helps to steer the system towards the desired solution and illuminates the empirical success of stochasticity in generative modeling.

该研究论文研究了在生成建模中，基于分数的生成建模方法中，与确定性过程相比，随机反向过程表现更好。研究比较了神经常微分方程（ODEs）和神经随机微分方程（SDEs）作为反向过程的性能，从控制理论的角度分析了神经SDEs近似轨迹的能力，发现了随机性的优势。

逆过程中的噪音增强扩散模型的逼近能力