Uncertainty quantification for kinetic equations with stochastic parameters presents major computational challenges due to the high dimensionality and nonlinearity of the models. Traditional Monte Carlo methods suffer from slow convergence and high variance, particularly when dealing with complex nonlinear models such as the Boltzmann equation. In this talk, we present a novel approach that integrates structure-preserving neural networks within a multi-fidelity asymptotic-preserving framework to efficiently address these issues. We introduce deep neural networks, trained in a physics-informed fashion, as low-fidelity surrogate models to reduce variance while preserving key physical properties of the solution, such as non-negativity and entropy dissipation. In particular, we focus on a structure-preserving PINN, which enhances standard PINN approaches by embedding conservation properties directly into the learning process. The resulting hybrid algorithm, combining data-driven surrogates and high-fidelity solvers, allows for significant computational savings while maintaining accuracy in the estimation of statistical observables. Numerical experiments, including homogeneous and non-homogeneous problems with uncertain initial conditions, demonstrate the effectiveness of our methodology. We show that the use of multiple control variates and structure-preserving neural networks can substantially outperform classical MC methods in terms of error reduction and computational cost.