  Stochastic Klein--Gordon--Schr\"odinger (KGS) equations are important
mathematical models and describe the interaction between scalar nucleons and
neutral scalar mesons in the stochastic environment. In this paper, we propose
novel structure-preserving schemes to numerically solve stochastic KGS
equations with additive noise, which preserve averaged charge evolution law,
averaged energy evolution law, symplecticity, and multi-symplecticity. By
applying central difference, sine pseudo-spectral method, or finite element
method in space and modifying finite difference in time, we present some charge
and energy preserved fully-discrete scheme for the original system. In
addition, combining the symplectic Runge-Kutta method in time and finite
difference in space, we propose a class of multi-symplectic discretizations
preserving the geometric structure of the stochastic KGS equation. Finally,
numerical experiments confirm theoretical findings.
