  Friedrichs' systems (FS) are symmetric positive linear systems of first-order
partial differential equations (PDEs), which provide a unified framework for
describing various elliptic, parabolic and hyperbolic semi-linear PDEs such as
the linearized Euler equations of gas dynamics, the equations of compressible
linear elasticity and the Dirac-Klein-Gordon system. FS were studied to
approximate PDEs of mixed elliptic and hyperbolic type in the same domain. For
this and other reasons, the versatility of the discontinuous Galerkin method
(DGM) represents the best approximation space for FS. We implement a
distributed memory solver for stationary FS in deal.II. Our focus is model
order reduction. Since FS model hyperbolic PDEs, they often suffer from a slow
Kolmogorov n-width decay. We develop two approaches to tackle this problem. The
first is domain decomposable reduced-order models (DD-ROMs). We will show that
the DGM offers a natural formulation of DD-ROMs, in particular regarding
interface penalties, compared to the continuous finite element method. We also
develop new repartitioning strategies to obtain more efficient local
approximations of the solution manifold. The second approach involves graph
neural networks used to infer the limit of a succession of projection-based
linear ROMs corresponding to lower viscosity constants: the heuristic behind is
to develop a multi-fidelity super-resolution paradigm to mimic the mathematical
convergence to vanishing viscosity solutions while exploiting to the most
interpretable and certified projection-based ROMs.
