Estimating Gross Primary Production (GPP), the gross uptake of CO$_2$ by vegetation, is a fundamental prerequisite for understanding and quantifying the terrestrial carbon cycle. Over the last decade, multiple approaches have been developed to derive spatio-temporal dynamics of GPP combining in-situ observations and remote sensing data using machine learning techniques or semi-empirical models. However, no high spatial resolution GPP product exists so far that is derived entirely from satellite-based remote sensing data. Sentinel-2 satellites are expected to open new opportunities to analyze ecosystem processes with spectral bands chosen to study vegetation between 10m to 20m spatial resolution with 5-days revisit frequency. Of particular relevance is the availability of red-edge bands that are suitable for deriving estimates of canopy chlorophyll content that are expected to be much better than any previous global mission. Here we analyzed whether red-edge-based and near-infrared-based vegetation indices (VIs) or machine learning techniques that consider VIs, all spectral bands and their non-linear interactions could predict daily GPP derived from 58 eddy covariance sites. Using linear regressions based on classic VIs, including Near Infrared Reflectance of Vegetation (NIRv), we achieved prediction powers of R$^2$textsubscript10-fold = 0.51 and an RMSEtextsubscript10-fold = 2.95 [] in a 10-fold cross-validation. Chlorophyll Index Red (CIR), and the novel kernel NDVI (kNVDI) achieved significantly higher prediction powers of around R$^2$textsubscript10-fold $≈$0.61, and a RMSEtextsubscript10-fold $≈$2.57 []. Using all spectral bands and VIs jointly in a machine learning prediction framework allowed us to predict GPP with R$^2$textsubscript10-fold = 0.71, and RMSEtextsubscript10-fold = 2.68 []. Despite the high-power prediction when machine learning techniques are used, under water-stress scenarios or heat-waves, optical information alone is not enough to predict GPP properly. In general, our analyses show the potential of non-linear combinations of spectral bands and VIs for monitoring GPP across ecosystems at a level of accuracy comparable to previous works, which, however, required additional meteorological drivers.