  We present a novel generative modeling method called diffusion normalizing
flow based on stochastic differential equations (SDEs). The algorithm consists
of two neural SDEs: a forward SDE that gradually adds noise to the data to
transform the data into Gaussian random noise, and a backward SDE that
gradually removes the noise to sample from the data distribution. By jointly
training the two neural SDEs to minimize a common cost function that quantifies
the difference between the two, the backward SDE converges to a diffusion
process the starts with a Gaussian distribution and ends with the desired data
distribution. Our method is closely related to normalizing flow and diffusion
probabilistic models and can be viewed as a combination of the two. Compared
with normalizing flow, diffusion normalizing flow is able to learn
distributions with sharp boundaries. Compared with diffusion probabilistic
models, diffusion normalizing flow requires fewer discretization steps and thus
has better sampling efficiency. Our algorithm demonstrates competitive
performance in both high-dimension data density estimation and image generation
tasks.
