  This contribution extends the localized training approach, traditionally
employed for multiscale problems and parameterized partial differential
equations (PDEs) featuring locally heterogeneous coefficients, to the class of
linear, positive symmetric operators, known as Friedrichs' operators.
Considering a local subdomain with corresponding oversampling domain we prove
the compactness of the transfer operator which maps boundary data to solutions
on the interior domain. While a Caccioppoli-inequality quantifying the energy
decay to the interior holds true for all Friedrichs' systems, showing a
compactness result for the graph-spaces hosting the solution is additionally
necessary. We discuss the mixed formulation of a convection-diffusion-reaction
problem where the necessary compactness result is obtained by the
Picard-Weck-Weber theorem. Our numerical results, focusing on a scenario
involving heterogeneous diffusion fields with multiple high-conductivity
channels, demonstrate the effectiveness of the proposed method.
