Sub-seasonal wind speed forecasts provide valuable guidance for wind power system planning and operations, yet the forecast skills of surface winds decrease sharply after two weeks. However, large-scale variables exhibit greater predictability on this time scale. This study explores the potential of leveraging non-linear relationships between 500 hPa geopotential height (Z500) and surface wind speed to improve sub-seasonal wind speed forecast skills in Europe. Our proposed framework uses a Multiple Linear Regression (MLR) or a Convolutional Neural Network (CNN) to regress surface wind speed from Z500. Evaluations on ERA5 reanalysis indicate that the CNN performs better due to its non-linearity. Applying these models to sub-seasonal forecasts from the European Centre for Medium-Range Weather Forecasts, various verification metrics demonstrate the advantages of non-linearity. Yet, this is partly explained by the fact that these statistical models are under-dispersive since they explain only a fraction of the target variable variance. Introducing stochastic perturbations to represent the stochasticity of the unexplained part from the signal helps compensate for this issue. Results show that the perturbed CNN performs better than the perturbed MLR only in the first weeks, while the perturbed MLR's performance converges towards that of the perturbed CNN after two weeks. The study finds that introducing stochastic perturbations can address the issue of insufficient spread in these statistical models, with improvements from the non-linearity varying with the lead time of the forecasts.