We propose a fast probabilistic framework for identifying differential equations governing the dynamics of observed data. We recast the SINDy method within a Bayesian framework and use Gaussian approximations for the prior and likelihood to speed up computation. The resulting method, Bayesian-SINDy, not only quantifies uncertainty in the parameters estimated but also is more robust when learning the correct model from limited and noisy data. Using both synthetic and real-life examples such as Lynx-Hare population dynamics, we demonstrate the effectiveness of the new framework in learning correct model equations and compare its computational and data efficiency with existing methods. Because Bayesian-SINDy can quickly assimilate data and is robust against noise, it is particularly suitable for biological data and real-time system identification in control. Its probabilistic framework also enables the calculation of information entropy, laying the foundation for an active learning strategy.

翻译：我们提出了一种快速概率框架，用于识别控制观测数据动力学的微分方程。将该方法重新纳入贝叶斯框架，并利用先验和似然的高斯近似加速计算。由此产生的贝叶斯稀疏动力学识别方法（Bayesian-SINDy）不仅能量化参数估计的不确定性，还能在从有限且含噪声的数据中学习正确模型时展现出更强的鲁棒性。通过山猫-野兔种群动力学等合成与真实案例，我们验证了新框架在正确学习模型方程方面的有效性，并比较了其与现有方法的计算效率和数据效率。由于Bayesian-SINDy能够快速吸收数据且对噪声具有鲁棒性，特别适用于生物数据及控制中的实时系统辨识。其概率框架还能计算信息熵，为主动学习策略奠定了基础。