Model Reduction for Advection Dominated Hyperbolic Problems in an ALE Framework: Offline and Online Phases

TL;DR

This paper introduces a novel model order reduction approach for advection dominated hyperbolic problems using an ALE framework, enhancing online efficiency and space compression through learned calibration maps.

Contribution

It develops a complete offline-online MOR framework with an ALE approach and learned calibration, improving efficiency for advection dominated problems.

Findings

The new method outperforms classical MOR techniques on various equations.

Calibration maps via neural networks improve space compression.

Significant online time savings demonstrated in simulations.

Abstract

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.