Sparse discovery of differential equations based on multi-fidelity
  Gaussian process
Yuhuang Meng Yue Qiu
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.