  In this paper, we investigate the training process of generative networks
that use a type of probability density distance named particle-based distance
as the objective function, e.g. MMD GAN, Cram\'er GAN, EIEG GAN. However, these
GANs often suffer from the problem of unstable training. In this paper, we
analyze the stability of the training process of these GANs from the
perspective of probability density dynamics. In our framework, we regard the
discriminator $D$ in these GANs as a feature transformation mapping that maps
high dimensional data into a feature space, while the generator $G$ maps random
variables to samples that resemble real data in terms of feature space. This
perspective enables us to perform stability analysis for the training of GANs
using the Wasserstein gradient flow of the probability density function. We
find that the training process of the discriminator is usually unstable due to
the formulation of $\min_G \max_D E(G, D)$ in GANs. To address this issue, we
add a stabilizing term in the discriminator loss function. We conduct
experiments to validate our stability analysis and stabilizing method.
