Learning reduced‐order dynamics for parametrized shallow water equations from data
Süleyman Yıldız, Pawan Goyal, Peter Benner, Bülent Karasözen
Abstract
Abstract This paper discusses a non‐intrusive data‐driven model order reduction method that learns low‐dimensional dynamical models for a parametrized shallow water equation. We consider the shallow water equation in non‐traditional form (NTSWE). We focus on learning low‐dimensional models in a non‐intrusive way. That means, we assume not to have access to a discretized form of the NTSWE in any form. Instead, we have snapshots that can be obtained using a black‐box solver. Consequently, we aim at learning reduced‐order models only from the snapshots. Precisely, a reduced‐order model is learnt by solving an appropriate least‐squares optimization problem in a low‐dimensional subspace. Furthermore, we discuss computational challenges that particularly arise from the optimization problem being ill‐conditioned. Moreover, we extend the non‐intrusive model order reduction framework to a parametric case, where we make use of the parameter dependency at the level of the partial differential equation. We illustrate the efficiency of the proposed non‐intrusive method to construct reduced‐order models for NTSWE and compare it with an intrusive method (proper orthogonal decomposition). We furthermore discuss the predictive capabilities of both models outside the range of the training data.