We use a Gaussian Process Regression (GPR) strategy that was recently developed [3,16,17] to analyze different types of curves that are commonly encountered in parametric eigenvalue problems. We employ an offline-online decomposition method. In the offline phase, we generate the basis of the reduced space by applying the proper orthogonal decomposition (POD) method on a collection of pre-computed, full-order snapshots at a chosen set of parameters. Then, we generate our GPR model using four different Mat\'{e}rn covariance functions. In the online phase, we use this model to predict both eigenvalues and eigenvectors at new parameters. We then illustrate how the choice of each covariance function influences the performance of GPR. Furthermore, we discuss the connection between Gaussian Process Regression and spline methods and compare the performance of the GPR method against linear and cubic spline methods. We show that GPR outperforms other methods for functions with a certain regularity.

翻译：我们采用最近发展的一种高斯过程回归策略[3,16,17]来分析参数特征值问题中常见的各类曲线。我们采用离线-在线分解方法。在离线阶段，通过在选定参数集上对预计算的全阶快照集合应用本征正交分解方法，生成降维空间的基。随后，我们使用四种不同的Matérn协方差函数构建高斯过程回归模型。在线阶段，我们利用该模型预测新参数下的特征值与特征向量。我们进一步阐释了不同协方差函数的选择如何影响高斯过程回归的性能表现。此外，我们探讨了高斯过程回归与样条方法之间的关联，并将高斯过程回归方法与线性及三次样条方法进行性能比较。研究表明，对于具有特定正则性的函数，高斯过程回归方法优于其他方法。