基于稀疏含噪数据的动力学感知控制方程识别
Dynamics-aware identification of governing equations from sparse and noisy data
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中文总结 AI 辅助
本文针对稀疏含噪数据下控制方程识别中导数估计不可靠的问题，提出基于Koopman的上采样预处理技术，经实验验证可提升ODE与PDE的识别性能，优于传统插值方法。
中文摘要 AI 辅助
稀疏非线性动力学识别（SINDy）与偏微分方程函数识别（PDE-FIND）可从数据中恢复简约常微分方程（ODEs）与偏微分方程（PDEs），但稀疏含噪的时间测量会使导数估计不可靠。为解决该问题，本文评估了基于Koopman的上采样技术，该技术通过动态模式分解（DMD）、扩展DMD（EDMD）与优化DMD实现，可学习Koopman演化在选定可观测量上的有限维近似，用于在观测时间窗口内插值并去噪快照，之后再进行导数估计与稀疏回归。实验基准包含两类ODE系统（Lorenz-63、Van der Pol）与三类周期PDE系统（Burgers、Fisher-Kolmogorov-Petrovskii-Piskunov（Fisher-KPP）、线性平流-扩散方程），均处于稀疏含噪采样场景。多项式EDMD在ODE实验中表现最优，尤其在系数精度上；PDE结果具有系统依赖性：低秩DMD辅助重构提升了Burgers与平流-扩散方程的识别效果，而原始基线（无上采样）对Fisher-KPP数据仍具竞争力。与线性插值、平滑样条插值技术的对比显示，所选基于Koopman的预处理器较非动力学替代方法实现了整体性能提升。此外，本文证明DMD辅助上采样可稳定基于帕累托的非先知支持规模选择。总体而言，基于Koopman的上采样应被视为一种动力学感知的预处理步骤，当其可观测量表示与低秩结构适配数据时，可降低导数估计误差。
英文摘要
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.
发表机构
- The University of Osaka（大阪大学）
- RIKEN（理化学研究所）
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