Sparse discovery of differential equations based on multi-fidelity Gaussian process
This work addresses challenges in computational modeling for scientific domains where data is noisy and multi-fidelity, representing an incremental improvement over existing methods.
The paper tackles the problem of sparse identification of differential equations from noisy and multi-fidelity data by using Gaussian process regression to mitigate noise and multi-fidelity Gaussian processes to handle sparse data, demonstrating robustness and effectiveness in numerical experiments.
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.