Model order reduction (MOR) techniques have always struggled in compressing information for advection dominated problems. Their linear nature does not allow to accelerate the slow decay of the Kolmogorov $N$--width of these problems. In the last years, new nonlinear algorithms obtained smaller reduced spaces. In these works only the offline phase of these algorithms was shown. In this work, we study MOR algorithms for unsteady parametric advection dominated hyperbolic problems, giving a complete offline and online description and showing the time saving in the online phase. We propose an arbitrary Lagrangian--Eulerian approach that modifies both the offline and online phases of the MOR process. This allows to calibrate the advected features on the same position and to strongly compress the reduced spaces. The basic MOR algorithms used are the classical Greedy, EIM and POD, while the calibration map is learned through polynomial regression and artificial neural networks. In the performed simulations we show how the new algorithm defeats the classical method on many equations with nonlinear fluxes and with different boundary conditions. Finally, we compare the results obtained with different calibration maps.

中文翻译：

---

ALE 框架中对流主导双曲线问题的模型简化：离线和在线阶段

模型降阶 (MOR) 技术一直在为平流主导问题压缩信息方面苦苦挣扎。它们的线性性质不允许加速这些问题的 Kolmogorov $N$-宽度的缓慢衰减。在过去几年中，新的非线性算法获得了更小的缩减空间。在这些作品中，只显示了这些算法的离线阶段。在这项工作中，我们研究了非定常参数对流主导的双曲线问题的 MOR 算法，给出了完整的离线和在线描述，并显示了在线阶段的时间节省。我们提出了一种任意的拉格朗日-欧拉方法，它修改了 MOR 过程的离线和在线阶段。这允许在同一位置校准平流特征并强烈压缩减少的空间。使用的基本 MOR 算法是经典的 Greedy、EIM 和 POD，而校准图是通过多项式回归和人工神经网络学习的。在执行的模拟中，我们展示了新算法如何在具有非线性通量和不同边界条件的许多方程上击败经典方法。最后，我们比较了不同校准图获得的结果。