Learning reduced order models from data for hyperbolic PDEs

Abstract

Given a set of solution snapshots of a hyperbolic PDE, we are interested in learning a reduced order model (ROM). To this end, we propose a novel decompose then learn approach. We decompose the solution by expressing it as a composition of a transformed solution and a de-transformer. Our idea is to learn a ROM for both these objects, which, unlike the solution, are well approximable in a linear reduced space. A ROM for the (untransformed) solution is then recovered via a recomposition. The transformed solution results from composing the solution with a spatial transform that aligns the spatial discontinuities. Furthermore, the de-transformer is the inverse of the spatial transform and lets us recover a ROM for the solution. We consider an image registration technique to compute the spatial transform, and to learn a ROM, we resort to the dynamic mode decomposition (DMD) methodology. Several benchmark problems demonstrate the effectiveness our method in representing the data and as a predictive tool.

Adjacency-based, non-intrusive model reduction for Vortex-Induced Vibrations

“…From the model reduction point of view, it is of particular interest to construct the fluid dynamics model on domain Ω, following a different approach from [23]. In this way, we would isolate the flowfield snapshots from the solid motion, thus avoiding typical issues of transport-dominated equations projection [36] close to the FSI interface. The ALE formulation (7) of the Navier-Stokes is suitable for this task.…”