多尺度动力学的优化采样
Optimized Sampling for Multiscale Dynamics
AI总结：
本文提出了一种基于动态模态分解的多尺度动力学分析方法，通过稀疏采样实现对复杂动力学的分层多分辨率分解，并用于动态区域分类和全状态重建。
摘要：
利用数据驱动分析对间歇性、多尺度和瞬态动力学进行表征仍是一个开放性挑战。本文展示了将动态模态分解（DMD）与稀疏采样相结合，用于多尺度物理诊断分析的应用。DMD是一种理想的空间时间矩阵分解方法，能够将计算或实验数据的空间特征与周期性时间行为相关联。DMD可以被修改为多分辨率分析，以将复杂动力学分解为多分辨率时间尺度的层次结构，其中每个层次将动力学分为不同的背景（慢）和前景（快）时间尺度。多分辨率DMD能够通过递归分解系统状态为低秩空间模式及其时间傅里叶动力学，以无方程的方式表征非线性动力系统。此外，这些多分辨率DMD模式可用于确定稀疏采样位置，这些位置几乎最优用于动态区域分类和全状态重建。具体而言，通过DMD库的QR列主元，高效地选择优化传感器，从而避免了NP难的选取过程。本文在多个例子上展示了该方法的有效性，包括全球海面温度数据，并表明仅需少量传感器即可实现准确的全球重建和厄尔尼诺事件的分类。
英文摘要：
The characterization of intermittent, multiscale and transient dynamics using data-driven analysis remains an open challenge. We demonstrate an application of the Dynamic Mode Decomposition (DMD) with sparse sampling for the diagnostic analysis of multiscale physics. The DMD method is an ideal spatiotemporal matrix decomposition that correlates spatial features of computational or experimental data to periodic temporal behavior. DMD can be modified into a multiresolution analysis to separate complex dynamics into a hierarchy of multiresolution timescale components, where each level of the hierarchy divides dynamics into distinct background (slow) and foreground (fast) timescales. The multiresolution DMD is capable of characterizing nonlinear dynamical systems in an equation-free manner by recursively decomposing the state of the system into low-rank spatial modes and their temporal Fourier dynamics. Moreover, these multiresolution DMD modes can be used to determined sparse sampling locations which are nearly optimal for dynamic regime classification and full state reconstruction. Specifically, optimized sensors are efficiently chosen using QR column pivots of the DMD library, thus avoiding an NP-hard selection process. We demonstrate the efficacy of the method on several examples, including global sea-surface temperature data, and show that only a small number of sensors are needed for accurate global reconstructions and classification of El Niño events.
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