AI 中文总结
研究在 Koopman 算子框架下，基于跟踪理论评估离散动力系统数据驱动计算分析的反向稳定性，通过反向误差分析利用计算特征对进行谱分析，汇总残差为算子扰动，传递影响到系统映射和初始条件，最终实现伪轨迹被精确轨迹跟踪。
AI 中文摘要
本文在 Koopman 合成算子和扩展动态模式分解的框架下，引入了基于跟踪理论的离散动力系统数据驱动计算分析的反向稳定性评估。利用 Koopman 算子的近似（计算得到的）特征对进行动力学的数据驱动谱分析，通过反向误差分析来实现。将计算得到的特征对的各个残差汇总为 Koopman 算子的反向扰动，使得计算得到的特征对对于一个不是合成算子的扰动算子是精确的。然后，算子中的扰动影响被传递到原始系统的映射和初始条件，表明计算得到的近似值恰好对应于系统的一个伪轨迹。最后，伪轨迹被系统的精确轨迹跟踪。这种对误差的解释特别适用于数据被噪声污染的数据驱动场景，为数值跟踪提供了新的见解。
英文摘要
This paper introduces shadowing theory based backward stability assessment of data driven computational analysis of discrete dynamical systems in the framework of the Koopman composition operator and the Extended Dynamic Mode Decomposition. Data driven spectral analysis of the dynamics using the approximate (computed) eigenpairs of the Koopman operator is cast in terms of the backward error analysis. The individual residuals of the computed eigenpairs are aggregated in a backward perturbation of the Koopman operator, so that the computed eigenpairs are exact for a perturbed operator that is not a composition operator. Then, the impact of the perturbation in the operator is carried over to the map and the initial condition of the original system, showing that the computed approximations correspond exactly to a pseudo--trajectory of the system. In the final step, the pseudo--trajectory is shadowed by an exact trajectory of the system. This interpretation of errors is in particular suitable in data driven scenarios where the data is contaminated by noise. New insights into the numerical shadowing are provided.
Comments18 pages, 4 figures
