基于少测点噪声数据重构问题的改进Gappy POD算法

doi:10.3969/j.issn.0255-8297.2025.05.003

摘要/Abstract

摘要： 基于数据驱动的Gappy POD（GP）算法是一种解决流动、传热等物理现象中反问题的方法，然而实际数据通常因受到多种噪声污染而影响了Gappy POD算法的精度。伯格斯方程含有流动和传热方程的部分重要形态，以此构建数据库，并对高斯噪声和无规则噪声下，基于普通最小二乘法（ordinary least squares,OLS）、加权最小二乘法（weighted leastsquares,WLS）、总体最小二乘法（total least squares,TLS）及L1正则化的Gappy POD算法的重构精度与稳定性展开研究。研究结果表明，Gappy POD算法可以在仅利用少量残缺数据条件下对一维伯格斯方程实现较高精度的重构；与GP-OLS相比，GP-WLS、GP-TLS和GP-L1的均方根误差显著降低，最大误差明显减小，相关系数更接近于1；在噪声条件下，GP-WLS重构均方根误差比GP-OLS减小到原来的1/27，提高了重构精度； GP-TLS重构均方根误差和最大误差均为最小，分别是0.014 1和0.013 0，数据矩阵和观测向量均存在噪声时的重构性能最好； GP-L1重构的相关系数接近1，提高了算法趋势预测能力，且从添加噪声前后来看，其重构能力变化程度不大，说明GP-L1算法对于异常值和噪声具有较强的抗干扰能力，提高了模型的鲁棒性能。

关键词: Gappy POD算法, 最小二乘法, 正则化, 有限测点, 噪声处理

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

中图分类号:

TB115
TP391

韩佳洁, 苑清扬, 张博, 赵鑫, 兰天, 李郁. 基于少测点噪声数据重构问题的改进Gappy POD算法[J]. 应用科学学报, 2025, 43(5): 740-756.

HAN Jiajie, YUAN Qingyang, ZHANG Bo, ZHAO Xin, LAN Tian, LI Yu. Improved Gappy POD Algorithm for Noisy Data Reconstruction Problems Based on Few Measurement Points[J]. Journal of Applied Sciences, 2025, 43(5): 740-756.

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