  We report on an experimental observation of classical diffusion
distinguishing between structural universality classes of disordered systems in
one dimension. Samples of hyperuniform and short-range disorder were designed,
characterized by the statistics of the placement of $\mu$m-thin parallel
permeable barriers, and the time-dependent diffusion coefficient was measured
by NMR methods over three orders of magnitude in time. The relation between the
structural exponent, characterizing disorder universality class, and the
dynamical exponent of the diffusion coefficient is experimentally verified. The
experimentally established relation between structure and transport exemplifies
the hierarchical nature of structural complexity --- dynamics are mainly
determined by the universality class, whereas microscopic parameters affect the
non-universal coefficients. These results open the way for non-invasive
characterization of structural correlations in porous media, complex materials,
and biological tissues via a bulk diffusion measurement.
