Dynamic Mode Decomposition in Adaptive Mesh Refinement and Coarsening Simulations
This work addresses a technical limitation in data-driven analysis for computational fluid dynamics and epidemiological modeling, enabling feature extraction from adaptive mesh simulations, but it is incremental as it adapts an existing method to handle specific data variations.
The paper tackled the problem of applying Dynamic Mode Decomposition (DMD) to simulations with adaptive mesh refinement/coarsening (AMR/C), where snapshots have varying dimensions, by projecting them onto a common reference space. The method was tested on three AMR/C simulations, including a COVID-19 model and a bubble rising problem, showing DMD's ability to reconstruct dynamics and extrapolate in time.
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).