Runge-Kutta Gill与Euler框架下SINDy动力学建模

SINDy Dynamic Modeling under the Runge-Kutta Gill and Euler Frameworks

摘要: 近年来，龙格–库塔类方法与深度学习已被验证可从观测数据中高效学习动力系统控制方程，为数据驱动动力学建模提供了有效途径。在此背景下，本文将基于字典的稀疏学习策略与两种经典的数值离散方法相结合，针对稀疏采样、含噪数据开展微分方程的高精度还原研究。工作在Brunton等提出的非线性动力学稀疏辨识(Sparse Identification of Nonlinear Dynamics, SINDy)框架上进行改进与扩展，分别构造了Runge-Kutta Gill SINDy (以下简称RK-Gill SINDy)与Euler SINDy两类辨识算法。通过采用稳定的数值积分格式替代传统有限差分近似，所提方法能够显著削弱损毁数据和噪声对辨识精度的影响，表现出优于经典SINDy的模型重构能力，具备良好的稳定性与噪声鲁棒性。

Abstract: In recent years, Runge-Kutta methods and deep learning have been proven to efficiently learn the governing equations of dynamic systems from observational data, providing an effective approach for data-driven dynamic modeling. Against this background, this paper combines dictionary-based sparse learning strategies with two classic numerical discretization methods to conduct high-precision recovery research on differential equations under sparse sampling and noisy data. Based on the Sparse Identification of Nonlinear Dynamics (SINDy) framework proposed by Brunton et al., this work makes improvements and extensions, and constructs two identification algorithms: Runge-Kutta Gill SINDy (hereinafter referred to as RK-Gill SINDy) and Euler SINDy. By adopting stable numerical integration schemes to replace the traditional finite difference approximation, the proposed methods can significantly reduce the impact of corrupted data and noise on identification accuracy, exhibit better model reconstruction capability than the classic SINDy, and possess satisfactory stability and noise robustness.