Abstract: Gyrokinetic simulations on parallel supercomputers provide the gold standard for theoretically determining turbulent transport in magnetized fusion plasmas. Applications to large and costly future machines, in particular burning plasma devices, call for a proper Uncertainty Quantification (UQ) in order to assess the reliability of certain predictions. However, since UQ requires an ensemble of simulations, the high computational cost of gyrokinetic simulations prevents straightforward applications of conventional UQ approaches. To overcome this, we propose a structure-exploiting, data-driven method based on sparse grid approximations to enable UQ in computationally expensive simulations. By leveraging the fact that the quantities of interest (e.g., heat or particle fluxes) often exhibit strong dependence on only a subset of the uncertain parameters characterized by anisotropic couplings, our method significantly reduces the number of expensive simulations required. We demonstrate this in the context of turbulent transport at the edge of tokamaks driven by electron temperature gradient (ETG) modes. In a nonlinear scenario with eight uncertain inputs, our sparse grid approach requires a mere total of 57 high-fidelity simulations. This efficiency extends to the construction of surrogate transport models, which are crucial for tasks like the design of optimized fusion devices. We will show that our structure-exploiting sparse grid approach can be effectively used to construct a surrogate model for the ETG-driven electron heat flux that delivers predictions with an acceptable level of precision across a wide range of parameter values. Finally, time permitting, we will discuss how our data-driven approach can be extended to multi-fidelity methods. By incorporating hierarchies of high- and low-fidelity models, these methods can significantly accelerate computations while maintaining accuracy, making them particularly promising for complex applications like fusion plasma simulations.