We propose a non-intrusive, reduced-basis, and data-driven method for approximating both eigenvalues and eigenvectors in parametric eigenvalue problems. We generate the basis of the reduced space by applying the proper orthogonal decomposition (POD) approach on a collection of pre-computed, full-order snapshots at a chosen set of parameters. Then, we use Bayesian linear regression (a.k.a. Gaussian Process Regression) in the online phase to predict both eigenvalues and eigenvectors at new parameters. A split of the data generated in the offline phase into training and test data sets is utilized in the numerical experiments following standard practices in the field of supervised machine learning. Furthermore, we discuss the connection between Gaussian Process Regression and spline methods, and compare the performance of GPR method against linear and cubic spline methods. We show that GPR outperforms other methods for functions with a certain regularity. To this end, we discuss various different covariance functions which influence the performance of GPR. The proposed method is shown to be accurate and efficient for the approximation of multiple 1D and 2D affine and non-affine parameter-dependent eigenvalue problems that exhibit crossing of eigenvalues.

翻译：我们提出一种非侵入式、降阶基、数据驱动的方法，用于近似参数化特征值问题中的特征值和特征向量。通过对选定参数集合上预计算的全阶快照进行本征正交分解，生成降阶空间的基。在在线阶段，我们采用贝叶斯线性回归（即高斯过程回归）预测新参数下的特征值和特征向量。在数值实验中，遵循监督机器学习领域的标准实践，将离线阶段生成的数据划分为训练集和测试集。此外，我们探讨了高斯过程回归与样条方法之间的联系，并将高斯过程回归方法与线性样条及三次样条方法进行了性能比较。结果表明，高斯过程回归在处理具有特定正则性的函数时优于其他方法。为此，我们讨论了影响高斯过程回归性能的不同协方差函数。对于存在特征值交叉的一维和二维仿射及非仿射参数依赖特征值问题的近似求解，所提出方法展现出高精度和高效率。