用动态模型预处理稀疏噪声数据，提升微分方程发现的准确性。
Dynamics-aware identification of governing equations from sparse and noisy data
- 基于柯尔曼算子的上采样技术，先重构数据动态再估计导数。
- 在洛伦兹与范德波尔系统中，多项式EDMD显著提高系数精度。
- 适合有低秩结构的系统，尤其在噪声和稀疏数据下表现优异。
稀疏非线性动力学识别（SINDy）与偏微分方程函数识别（PDE-FIND）可从数据中恢复简洁的常微分方程（ODE）和偏微分方程（PDE）。然而，稀疏且含噪的时间测量会导致导数估计不可靠。为此，本文评估了基于柯尔曼算子的上采样方法，包括动态模态分解（DMD）、扩展DMD（EDMD）及优化DMD。这些方法学习选定观测量上的柯尔曼演化有限维近似，用于在观测时间窗内插值和去噪，以改善导数估计与稀疏回归。实验涵盖两个ODE系统（洛伦兹-63、范德波尔）和三个周期性PDE系统（伯格斯、费希尔-科尔莫戈罗夫-佩特罗夫斯基-皮斯基诺夫、线性对流扩散），覆盖稀疏与噪声采样场景。多项式EDMD在ODE任务中表现最佳，尤其在系数精度方面；PDE结果则依赖系统：低秩DMD辅助重建提升了伯格斯与对流扩散方程的发现效果，而费希尔-KPP数据中原始基线仍具竞争力。与线性及平滑样条插值相比，所选柯尔曼预处理方法整体性能更优。此外，DMD辅助上采样能稳定帕累托最优支持集选择。总体而言，柯尔曼上采样应视为一种动态感知的预处理步骤，在可观测表示与低秩结构适配数据时，可有效降低导数估计误差。
原文摘要 · Abstract (English)
Sparse identification of nonlinear dynamics (SINDy) and PDE functional identification (PDE-FIND) recover parsimonious ordinary and partial differential equations (ODEs and PDEs) from data. However, sparse and noisy temporal measurements can make derivative estimates unreliable. To address this problem, we evaluate Koopman-based upsampling techniques implemented with dynamic mode decomposition (DMD), extended DMD (EDMD), and optimized DMD. These methods learn finite-dimensional approximations of Koopman evolution on selected observables and are used to interpolate and denoise snapshots inside the observed time window before derivative estimation and sparse regression. The empirical benchmark comprises two ODE systems, Lorenz-63 and Van der Pol, and three periodic PDE systems, Burgers, Fisher-Kolmogorov-Petrovskii-Piskunov (Fisher-KPP), and linear advection-diffusion, over sparse and noisy sampling regimes. Polynomial EDMD gives the strongest ODE results, especially in coefficient accuracy. The PDE results are system-dependent: low-rank DMD-assisted reconstructions improve Burgers and advection-diffusion discovery, while the raw baseline (without upsampling) remains competitive for the Fisher-KPP data. A comparison against linear and smoothing-spline interpolation techniques shows that the selected Koopman-based preprocessors provide overall performance gains over these non-dynamical alternatives. We also demonstrate that DMD-assisted upsampling can stabilize Pareto-based non-oracle support-size selection. Overall, Koopman-based upsampling is best viewed as a dynamics-aware preprocessing step that can reduce derivative-estimation error when its observable representation and low-rank structure are appropriate for the data.
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