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.

通过不确定性量化的角度，本文提出了两种新的方法来解决稀疏辨识微分方程的两个主要挑战：噪声对观测数据的影响以及关于计算成本的单一可靠性数据的限制。通过构建高斯过程回归模型来缓解观测数据中噪声的影响，量化不确定性并最终准确地恢复方程；然后，利用多可靠性高斯过程来处理涉及多可靠性、稀疏和噪声观测数据的情况。通过多个数值实验展示了我们方法的稳健性和有效性。