A data-driven method for parametric PDE Eigenvalue Problems using Gaussian Process with different covariance functions

TL;DR

This work addresses efficient surrogate modeling for parametric PDE eigenvalue problems by coupling a POD-based offline reduced basis with Gaussian Process Regression (GPR) for online predictions of eigenvalues and eigenvectors. By systematically comparing four covariance functions (including Matérn variants and Squared Exponential) and relating kernel regularity to data smoothness and eigenstructure (e.g., eigenvalue crossings and non-affine dependence), the authors quantify when GPR outperforms spline-based interpolation. The results show that absolute exponential kernels perform best for non-smooth or discontinuous behavior, while squared exponential kernels excel for very smooth dependence, with Matérn kernels offering advantages at intermediate regularity; spline methods remain competitive in certain regimes. The proposed data-driven MOR framework enables fast parametric modal analyses and points to adaptive kernel strategies as a promising avenue for handling mixed regularity in pEVPs.

Abstract

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 Matérn 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.

Figures (21)

• Figure 1: Performance comparison of spline and GPR in the uniform case
• Figure 2: Comparison between the function $f(\mu)$ (black) and estimates using three different methods. Blue circles are the training data points.
• Figure 3: First six eigenvalues of the eigenvalue problem \ref{['eq:crs']}.
• Figure 4: GPR corresponding to 2nd eigenvalues of EVP \ref{['eq:crs']} using different kernels.
• Figure 5: GPR corresponding to 1st coefficient of the reduced 2nd eigenvector of EVP \ref{['eq:crs']} using different kernels.

Theorems & Definitions (3)

• Example 1
• Example 2
• Example 3