数值谱关联：基于带重采样的Koopman-切比雪夫近似识别控制偏微分方程
Numerical Spectrum Linking: Identification of Governing PDE via Koopman-Chebyshev Approximation with Resampling
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中文总结 AI 辅助
该研究提出一种基于带重采样的Koopman-切比雪夫近似的数值框架，可从任意采样观测数据中准确识别控制PDE，为PDE识别提供了实用指导。
中文摘要 AI 辅助
提出一种数值框架，用于从观测数据中识别控制偏微分方程（PDE），方法是在公共切比雪夫谱域中建立观测驱动与方程驱动的Koopman算子之间的关联。与动态模式分解（DMD）等数据驱动方法不同，后者近似Koopman算子时未明确将其与微分算子关联，该框架利用切比雪夫谱表示构造有限维Koopman算子，从而实现数据导出动力学与候选控制PDE的直接比较。引入统一观测模型及最小二乘系数恢复公式，用于从任意采样网格获取的观测中恢复切比雪夫谱系数，为实际观测与基于切比雪夫的Koopman分析提供数值一致的接口。在直接切比雪夫、均匀及不规则采样配置下的数值实验表明，该框架可从观测中准确识别控制PDE；观测密度研究显示，一旦获得足够的独立观测以实现稳定系数恢复，即可持续实现可靠的PDE识别，为实际应用提供了指导。
英文摘要
A numerical framework is proposed for identifying governing partial differential equations (PDEs) from observational data by establishing a link between observation-driven and equation-driven Koopman operators in a common Chebyshev spectral domain. In contrast to data-driven approaches such as dynamic mode decomposition (DMD), which approximate Koopman operators without explicitly relating them to differential operators, the proposed framework constructs finite-dimensional Koopman operators using Chebyshev spectral representations, thereby enabling direct comparison between data-derived dynamics and candidate governing PDEs. A unified observation model together with a least-squares coefficient recovery formulation is introduced to recover Chebyshev spectral coefficients from observations obtained on arbitrary sampling grids. This provides a numerically consistent interface between practical observations and Chebyshev-based Koopman analysis. Numerical experiments under direct Chebyshev, uniform, and irregular sampling configurations demonstrate that the proposed framework accurately identifies the governing PDE from observations. An observation-density study shows that reliable PDE identification is consistently achieved once sufficient independent observations are available for stable coefficient recovery, providing a practical guideline.
