  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.
