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.

翻译：本文提出了一种"一体化"建模框架，用于从稀缺、不完整且含噪声的状态观测数据中学习常微分方程系统。该方法将针对函数库的常微分方程稀疏恢复策略，与再生核希尔伯特空间理论中的状态估计及方程离散化技术相结合。数值实验表明，所提策略在方程学习和未知状态估计两方面，均能显著提升精度、样本效率及噪声鲁棒性。本工作不仅展现了远超现有主流算法的性能，同时拓展了近期方程发现研究中其他方法的建模灵活性。