  The training process of generative adversarial networks (GANs) is unstable
and does not converge globally. In this paper, we examine the stability of GANs
from the perspective of control theory and propose a universal higher-order
noise-based controller called Brownian Motion Controller (BMC). Starting with
the prototypical case of Dirac-GANs, we design a BMC to retrieve precisely the
same but reachable optimal equilibrium. We theoretically prove that the
training process of DiracGANs-BMC is globally exponential stable and derive
bounds on the rate of convergence. Then we extend our BMC to normal GANs and
provide implementation instructions on GANs-BMC. Our experiments show that our
GANs-BMC effectively stabilizes GANs' training under StyleGANv2-ada frameworks
with a faster rate of convergence, a smaller range of oscillation, and better
performance in terms of FID score.
