Data from simulations and experiments are rarely noise-free and often exhibit heterogeneous levels of fidelity. Measurement uncertainty may vary across repeated observations, sensing devices, or even within a single experiment. This work addresses the problem of discovering nonlinear dynamical systems from such inhomogeneous data. We extend the Sparse Identification of Nonlinear Dynamical Systems (SINDy) framework to account for variable noise levels by combining Ensemble SINDy and Weak SINDy within a weighted regression formulation derived from generalized least squares. A statistical justification for the weighting strategy is also provided. The methodology is validated on several benchmark systems, including ordinary and partial differential equations. In addition, we show the benefit of multi-fidelity integration for forecasting the dynamics of a double pendulum system. The results confirm that the proposed approach mitigates the adverse effects of heteroscedastic noise and that repeated, low-cost, low-quality measurements can improve model recovery, in some cases matching or outperforming reconstructions obtained using only high-fidelity data.

翻译：仿真和实验数据很少是无噪声的，且常呈现异质性保真度水平。测量不确定性可能因重复观测、传感设备甚至单个实验内部而有所不同。本文针对从这类异质数据中辨识非线性动力系统的问题，通过结合集成SINDy与弱SINDy，在广义最小二乘推导的加权回归框架下，扩展了稀疏辨识非线性动力系统（SINDy）方法以处理可变噪声水平。同时提供了加权策略的统计学依据。该方法在多个基准系统（包括常微分方程和偏微分方程）上得到验证。此外，我们展示了多保真度集成对双摆系统动力学预测的益处。结果证实，所提方法可缓解异方差噪声的不利影响，且重复的低成本、低质量测量能改善模型恢复效果，在某些情况下其表现可与仅使用高保真度数据获得的重构结果相媲美甚至更优。