用多精度数据提升非线性系统建模精度，低质数据也能帮上忙。
Multi-Fidelity SINDy: Sparse Discovery of Nonlinear Dynamical Systems with Fidelity-Weighted Measurements
- 基于加权回归融合高/低精度测量数据，自动识别噪声差异。
- 在双摆系统预测中，多次低成本低精度数据优于单次高精度数据。
- 适合实验数据质量不一的物理建模场景，如传感器误差大或成本受限。
模拟与实验数据通常存在异方差噪声，即不同观测、设备或实验阶段的测量精度不一。本文将稀疏非线性动力系统识别（SINDy）框架扩展至多精度场景，结合集成SINDy与弱SINDy，通过广义最小二乘推导出加权回归方法，并提供统计依据支持权重策略。在多个基准系统（含常微分与偏微分方程）上验证了该方法的有效性。结果表明，该方法可有效缓解异方差噪声影响；在双摆系统预测中，重复的低成本低精度测量能显著提升模型恢复效果，甚至在某些情况下超越仅使用高精度数据的重构表现。
原文摘要 · Abstract (English)
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.
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