Learning reduced-order dynamics for parametrized shallow water equations from data,International Journal for Numerical Methods in Fluids

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Learning reduced-order dynamics for parametrized shallow water equations from data
International Journal for Numerical Methods in Fluids ( IF 1.7 ) Pub Date : 2021-05-09 , DOI: 10.1002/fld.4998
Süleyman Yıldız ₁ , Pawan Goyal ₂ , Peter Benner ₂ , Bülent Karasözen _{1,

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Affiliation

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.

中文翻译：

从数据中学习参数化浅水方程的降阶动力学

本文讨论了一种非侵入式数据驱动模型降阶方法，该方法可以学习参数化浅水方程的低维动力模型。我们考虑非传统形式的浅水方程 (NTSWE)。我们专注于以非侵入性的方式学习低维模型。这意味着，我们假设无法以任何形式访问离散形式的 NTSWE。相反，我们有可以使用黑盒求解器获得的快照。因此，我们的目标是仅从快照中学习降阶模型。准确地说，降阶模型是通过在低维子空间中解决适当的最小二乘优化问题来学习的。此外，我们讨论了特别是由病态优化问题引起的计算挑战。而且，我们将非侵入式模型降阶框架扩展到参数情况，在那里我们利用偏微分方程级别的参数依赖性。我们说明了所提出的非侵入式方法为 NTSWE 构建降阶模型的效率，并将其与侵入式方法（适当的正交分解）进行了比较。我们进一步讨论了两种模型在训练数据范围之外的预测能力。

更新日期：2021-07-01

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