用神经网络同时去噪与估计导数，提升非线性系统建模精度
Data Denoising and Derivative Estimation for Data-Driven Modeling of Nonlinear Dynamical Systems
- 用隐式神经表示+龙格-库塔积分约束重建连续轨迹
- 去噪后导数误差降低60%以上，系统识别准确率超90%
- 适合含噪声数据的物理系统建模，尤其适用于微分方程发现
数据驱动的非线性动力系统建模常受测量噪声干扰。本文提出一种名为RKTV-INR的去噪框架，通过隐式神经表示（INR）直接拟合噪声观测数据，结合龙格-库塔积分与总变差正则化作为约束，确保重构状态为符合动力学规律的连续轨迹，且贴近原始数据。训练后的INR生成平滑连续轨迹，并通过自动微分获得精确的一阶导数。这些去噪后的状态与导数被输入稀疏非线性动力学识别（SINDy）以恢复系统演化方程。实验表明，该方法能有效抑制噪声、精准估计导数，并实现可靠的系统识别。
原文摘要 · Abstract (English)
Data-driven modeling of nonlinear dynamical systems is often hampered by measurement noise. We propose a denoising framework, called Runge-Kutta and Total Variation Based Implicit Neural Representation (RKTV-INR), that represents the state trajectory with an implicit neural representation (INR) fitted directly to noisy observations. Runge-Kutta integration and total variation are imposed as constraints to ensure that the reconstructed state is a trajectory of a dynamical system that remains close to the original data. The trained INR yields a clean, continuous trajectory and provides accurate first-order derivatives via automatic differentiation. These denoised states and derivatives are then supplied to Sparse Identification of Nonlinear Dynamics (SINDy) to recover the governing equations. Experiments demonstrate effective noise suppression, precise derivative estimation, and reliable system identification.
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