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.

翻译：Friedrichs系统(FS)是一类对称正定的一阶偏微分方程(PDE)线性系统，为描述椭圆型、抛物型和双曲型半线性PDE（如气体动力学的线性化Euler方程、可压缩线性弹性力学方程以及Dirac-Klein-Gordon系统）提供了统一框架。FS被用于逼近同一域内混合椭圆-双曲型PDE。基于此及其他原因，间断Galerkin方法(DGM)的通用性成为FS最优逼近空间。我们在deal.II中实现了稳态FS的分布式内存求解器，重点研究模型降阶。由于FS可建模双曲型PDE，其通常面临Kolmogorov n-宽度衰减缓慢的问题。我们提出两种解决方案：第一种是领域可分解降阶模型(DD-ROMs)。我们将证明，与连续有限元方法相比，DGM为DD-ROMs（特别是界面惩罚项）提供了自然表述框架。同时，我们开发了新的重分区策略以获得解流形更高效的局部逼近。第二种方法利用图神经网络推断对应低粘性常数的投影型线性ROM序列的极限：其启发式思路在于构建多保真超分辨率范式，在最大化利用可解释且可认证的投影型ROM的同时，模拟数学上向消失粘性解的收敛过程。