AI 中文总结
本研究针对SINDy存在的选择偏差、测量误差及共享噪声问题，提出定制化后选择推断方法PSI-SINDy，通过数据细分分解轨迹，经理论验证和数值实验证实其有效性。
AI 中文摘要
非线性动力学稀疏识别（SINDy）是一种数据驱动框架，用于从时间序列数据中发现动力学规律，方法是从预先指定的候选动力学项库中识别出稀疏的候选项子集。本研究开发了一种统计推断框架，通过假设检验和置信区间来量化SINDy所选动力学项的可靠性。一个关键难点在于，使用同一条含噪轨迹同时进行动力学项选择和统计显著性评估会引入选择偏差。后选择推断为解决此类偏差提供了有原则的框架，我们提出了PSI-SINDy，这是一种专为SINDy定制的后选择推断方法。现有后选择推断技术的直接应用颇具挑战性，因为SINDy的候选项存在测量误差，且响应与设计之间存在共享噪声。为应对这些挑战，PSI-SINDy采用数据 thinning（数据细分）将单条观测轨迹分解为四个相互独立的视图，分别在选择和推断中发挥不同作用。该构造可对所选动力学项进行推断，同时不仅考虑选择偏差，还考虑测量误差和共享噪声的影响。我们在规定条件下证明了PSI-SINDy的理论有效性，并通过对模拟和实验动力学系统数据的数值实验评估了其性能。
英文摘要
Sparse identification of nonlinear dynamics (SINDy) is a data-driven framework for discovering governing dynamics from time-series data by identifying a sparse subset of candidate dynamical terms from a prespecified library. In this work, we develop a statistical inference framework for quantifying the reliability of dynamical terms selected by SINDy through hypothesis tests and confidence intervals. A key difficulty is that using the same noisy trajectory for both selecting dynamical terms and assessing their statistical significance can introduce selection bias. Post-selection inference provides a principled framework for addressing such bias, and we propose PSI-SINDy, a post-selection inference method tailored to SINDy. Direct application of existing post-selection inference techniques is challenging because SINDy involves measurement error in the candidate terms and shared noise between the response and design. To address these challenges, PSI-SINDy uses data thinning to decompose a single observed trajectory into four mutually independent views with distinct roles in selection and inference. This construction enables inference for selected dynamical terms while accounting not only for selection bias but also for measurement-error and shared noise effects. We establish the theoretical validity of PSI-SINDy under stated conditions and evaluate its performance through numerical experiments on simulated and experimental dynamical-system data.
Comments47 pages, 3 figures, 16 tables
