从稀疏噪声数据中联合优化，精准恢复微分方程与状态轨迹。
A joint optimization approach to identifying sparse dynamics using least squares kernel collocation
- 联合优化方程系数与状态估计，结合核方法与稀疏恢复。
- 在少量噪声观测下，方程与状态估计精度显著提升。
- 适合数据稀缺且含噪的科学建模场景，如生物、物理系统。
我们提出一种全耦合建模框架，用于从稀疏、不完整且含噪声的状态观测中学习常微分方程（ODE）系统。该方法结合函数库中的稀疏恢复策略与再生核希尔伯特空间（RKHS）理论，实现状态估计与方程离散化。数值实验表明，该策略在方程学习和未知状态估计方面均显著提升了精度、样本效率和抗噪能力。本工作展现出超越现有主流算法的能力，同时扩展了近期方程发现方法的建模灵活性。
原文摘要 · Abstract (English)
We develop an all-at-once modeling framework for learning systems of ordinary differential equations (ODE) from scarce, partial, and noisy observations of the states. The proposed methodology amounts to a combination of sparse recovery strategies for the ODE over a function library combined with techniques from reproducing kernel Hilbert space (RKHS) theory for estimating the state and discretizing the ODE. Our numerical experiments reveal that the proposed strategy leads to significant gains in terms of accuracy, sample efficiency, and robustness to noise, both in terms of learning the equation and estimating the unknown states. This work demonstrates capabilities well beyond existing and widely used algorithms while extending the modeling flexibility of other recent developments in equation discovery.
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