  We apply two data assimilation (DA) methods, a smoother and a filter, and a
model-free machine learning (ML) shallow network to forecast two weakly
turbulent systems. We analyse the effect of the spatial sparsity of
observations on accuracy of the predictions obtained from these data-driven
methods. Based on the results, we divide the spatial sparsity levels in three
zones. First is the good-predictions zone in which both DA and ML methods work.
We find that in the good-predictions zone the observations remain dense enough
to accurately capture the fractal manifold of the system's dynamics, which is
measured using the correlation dimension. The accuracy of the DA methods in
this zone remains almost as good as for full-resolution observations. Second is
the reasonable-predictions zone in which the DA methods still work but at
reduced prediction accuracy. Third is the bad-predictions zone in which even
the DA methods fail. We find that the sparsity level up to which the DA methods
work is almost the same up to which chaos synchronisation of these systems can
be achieved. The main implications of these results are that they (i) firmly
establish the spatial resolution up to which the data-driven methods can be
utilised, (ii) provide measures to determine if adding more sensors will
improve the predictions, and (iii) quantify the advantage (in terms of the
required measurement resolution) of using the governing equations within
data-driven methods. We also discuss the applicability of these results to
fully developed turbulence.
