Abstract: Data-driven Gappy proper orthogonal decomposition (POD), namely GP algorithm, is a method to solve inverse problems in physical phenomena such as flow and heat transfer. However, the actual data is usually polluted by various noises, thus affecting the accuracy of the GP algorithm. The database was built on the Burgers equation because it contained some important forms of the flow and heat transfer equations. The reconstruction accuracy and stability of the GP algorithm for Gaussian noise and random noise based on ordinary least squares (OLS), weighted least squares (WLS), total least squares (TLS), and L1 regularization were studied. The results show that the GP algorithm can reconstruct the one-dimensional Burgers equation with high accuracy with only a small amount of incomplete data. Compared with those of GP-OLS, the root-mean-square error and maximum error of GP-WLS, GP-TLS, and GP-L1 are significantly reduced, and the correlation coefficient is closer to 1. Under the noise condition, the root-mean-square error of GP-WLS is reduced to 1/27 that of GP-OLS, with improved reconstruction accuracy. The root-mean-square error and maximum error of GP-TLS reconstruction are the smallest, which are 0.014 1 and 0.013 0, respectively. The reconstruction performance is the best when the data matrix and observation vector are noisy. The correlation coefficient of GP-L1 reconstruction is close to 1, which improves the trend prediction ability of the algorithm. Before and after adding noise, the reconstruction ability of GP-L1 does not change much, indicating that the GP-L1 algorithm has strong anti-interference ability against outliers and noise and improves the robustness of the model.

Key words: Gappy POD algorithm, least squares method, regularization, limited measurement point, noise processing