This work introduces a novel approach for data-driven model reduction of time-dependent parametric partial differential equations. Using a multi-step procedure consisting of proper orthogonal decomposition, dynamic mode decomposition and manifold interpolation, the proposed approach allows to accurately recover field solutions from a few large-scale simulations. Numerical experiments for the Rayleigh-B\'{e}nard cavity problem show the effectiveness of such multi-step procedure in two parametric regimes, i.e.~medium and high Grashof number. The latter regime is particularly challenging as it nears the onset of turbulent and chaotic behaviour. A major advantage of the proposed method in the context of time-periodic solutions is the ability to recover frequencies that are not present in the sampled data.
翻译:这项工作引入了一种新颖的方法,用数据驱动模型来减少基于时间的参数部分差异方程式。采用由适当的正心分解、动态模式分解和多重内插组成的多步程序,拟议方法能够准确地从几个大规模模拟中找到实地解决办法。雷利-B\'{e}nard洞穴问题的数字实验表明,在两个参数系统中,即~中度和高度Grashof数中,这种多步程序是有效的。后一种制度由于接近动荡和混乱行为的开始,因此特别具有挑战性。在时间-周期解决办法中,拟议方法的主要优点是能够恢复抽样数据中不存在的频率。