  Control barrier functions are widely used to synthesize safety-critical
controls. The existence of Gaussian-type noise may lead to unsafe actions and
result in severe consequences. While studies are widely done in safety-critical
control for stochastic systems, in many real-world applications, we do not have
the knowledge of the stochastic component of the dynamics. In this paper, we
study safety-critical control of stochastic systems with an unknown diffusion
part and propose a data-driven method to handle these scenarios. More
specifically, we propose a data-driven stochastic control barrier function
(DDSCBF) framework and use supervised learning to learn the unknown stochastic
dynamics via the DDSCBF scheme. Under some reasonable assumptions, we provide
guarantees that the DDSCBF scheme can approximate the It\^{o} derivative of the
stochastic control barrier function (SCBF) under partially unknown dynamics
using the universal approximation theorem. We also show that we can achieve the
same safety guarantee using the DDSCBF scheme as with SCBF in previous work
without requiring the knowledge of stochastic dynamics. We use two non-linear
stochastic systems to validate our theory in simulations.
