Data-Driven Learning of Reduced-order Dynamics for a Parametrized Shallow Water Equation

TL;DR

This paper introduces a non-intrusive, data-driven approach to learn reduced-order models for a parametrized shallow water equation using snapshot data, addressing computational challenges and extending to parametric cases.

Contribution

It presents a novel non-intrusive method for reduced-order modeling of parametrized PDEs from snapshot data, including handling ill-conditioned optimization and extending to parametric dependencies.

Findings

The non-intrusive method effectively constructs reduced models for NTSWE.

The approach compares favorably with intrusive POD in accuracy.

The models demonstrate good predictive capabilities outside training data.

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 are 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.