Sparse identification of differential equations aims to compute the analytic expressions from the observed data explicitly. However, there exist two primary challenges. Firstly, it exhibits sensitivity to the noise in the observed data, particularly for the derivatives computations. Secondly, existing literature predominantly concentrates on single-fidelity (SF) data, which imposes limitations on its applicability due to the computational cost. In this paper, we present two novel approaches to address these problems from the view of uncertainty quantification. We construct a surrogate model employing the Gaussian process regression (GPR) to mitigate the effect of noise in the observed data, quantify its uncertainty, and ultimately recover the equations accurately. Subsequently, we exploit the multi-fidelity Gaussian processes (MFGP) to address scenarios involving multi-fidelity (MF), sparse, and noisy observed data. We demonstrate the robustness and effectiveness of our methodologies through several numerical experiments.

翻译：微分方程的稀疏识别旨在从观测数据中显式计算解析表达式。然而，这面临两个主要挑战：首先，观测数据中的噪声（特别是导数计算中的噪声）会显著影响识别结果；其次，现有研究主要聚焦于单保真数据，其计算成本限制了方法的适用性。本文从不确定性量化的角度提出两种新方法解决上述问题。我们构建基于高斯过程回归的代理模型，以削弱观测数据中的噪声影响、量化其不确定性，并最终精确恢复方程。进一步，我们利用多保真高斯过程处理涉及多保真、稀疏及含噪观测数据的复杂场景。通过多项数值实验验证了所提方法的鲁棒性和有效性。