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Friedrichs' systems (FS) are symmetric positive linear systems of first-orderpartial differential equations (PDEs), which provide a unified framework fordescribing various elliptic, parabolic and hyperbolic semi-linear PDEs such asthe linearized Euler equations of gas dynamics, the equations of compressiblelinear elasticity and the Dirac-Klein-Gordon system. FS were studied toapproximate PDEs of mixed elliptic and hyperbolic type in the same domain. Forthis and other reasons, the versatility of the discontinuous Galerkin method(DGM) represents the best approximation space for FS. We implement adistributed memory solver for stationary FS in deal.II. Our focus is modelorder reduction. Since FS model hyperbolic PDEs, they often suffer from a slowKolmogorov n-width decay. We develop two approaches to tackle this problem. Thefirst is domain decomposable reduced-order models (DD-ROMs). We will show thatthe DGM offers a natural formulation of DD-ROMs, in particular regardinginterface penalties, compared to the continuous finite element method. We alsodevelop new repartitioning strategies to obtain more efficient localapproximations of the solution manifold. The second approach involves graphneural networks used to infer the limit of a succession of projection-basedlinear ROMs corresponding to lower viscosity constants: the heuristic behind isto develop a multi-fidelity super-resolution paradigm to mimic the mathematicalconvergence to vanishing viscosity solutions while exploiting to the mostinterpretable and certified projection-based ROMs.

Friedrichs' systems discretized with the Discontinuous Galerkin method:  domain decomposable model order reduction and Graph Neural Networks  approximating vanishing viscosity solutions