自适应网格加密和粗化仿真中的动态模式分解
Dynamic Mode Decomposition in Adaptive Mesh Refinement and Coarsening Simulations
摘要：
动态模式分解（DMD）是一种强大的数据驱动的方法，用于提取时空相干结构，决定一个给定的动力系统。该方法包括堆叠收集的时间快照到一个矩阵和映射的非线性动力学使用线性算子。标准过程认为快照对于所有可观察数据具有相同的维数。然而，这往往不会发生在数值模拟与自适应网格细化/粗化计划（AMR/C）。本文提出了一种策略，使DMD从不同的网格拓扑结构和尺寸的观察，如AMR/C模拟中发现的特征提取。为此，自适应快照被投影到相同的参考函数空间上，使得能够使用基于快照的方法，例如DMD。本策略应用于具有挑战性的AMR/C模拟：COVID-19的连续扩散-反应流行病学模型，密度驱动的重力流模拟和气泡上升问题。我们还评估了DMD重建动力学和一些相关的感兴趣的量的效率。特别是，对于SEIRD模型和泡沫上升的问题，我们评估DMD的能力外推的时间（短期未来的估计）。
英文摘要：
Dynamic Mode Decomposition (DMD) is a powerful data-driven method used to extract spatio-temporal coherent structures that dictate a given dynamical system. The method consists of stacking collected temporal snapshots into a matrix and mapping the nonlinear dynamics using a linear operator. The standard procedure considers that snapshots possess the same dimensionality for all the observable data. However, this often does not occur in numerical simulations with adaptive mesh refinement/coarsening schemes (AMR/C). This paper proposes a strategy to enable DMD to extract features from observations with different mesh topologies and dimensions, such as those found in AMR/C simulations. For this purpose, the adaptive snapshots are projected onto the same reference function space, enabling the use of snapshot-based methods such as DMD. The present strategy is applied to challenging AMR/C simulations: a continuous diffusion-reaction epidemiological model for COVID-19, a density-driven gravity current simulation, and a bubble rising problem. We also evaluate the DMD efficiency to reconstruct the dynamics and some relevant quantities of interest. In particular, for the SEIRD model and the bubble rising problem, we evaluate DMD's ability to extrapolate in time (short-time future estimates).
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