Physics-informed Gaussian Process Regression in Solving Eigenvalue Problem of Linear Operators

TL;DR

This work reframes PDE eigenvalue problems $(\mathcal{L}-\lambda)u=0$ within a physics-informed Gaussian process regression (PI-GPR) framework, where the usual posterior mean becomes uninformative due to homogeneity. It introduces a covariance-based eigenvalue criterion: the posterior covariance $K_N(\lambda)$ collapses to zero when $\lambda$ is not an eigenvalue and becomes nontrivial with samples lying in the eigenspace when $\lambda$ is an eigenvalue; the eigenvalues are identified by peaks in $J(\lambda)=\mathrm{Tr}(K_N(\lambda))$ across a scan of $\lambda$. The authors demonstrate this approach on three canonical problems—the 1D Laplacian, a cantilever beam, and a nonlinear-loaded string—showing that the method can accurately locate eigenvalues and, near those values, sample eigenfunctions from the corresponding posterior. The framework is meshless and adaptable, capable of handling nonlinear eigenvalue problems, and is extensible with optimizations such as sparse GPs and problem-specific kernels to improve efficiency and accuracy.

Abstract

Applying Physics-Informed Gaussian Process Regression to the eigenvalue problem $(\mathcal{L}-λ)u = 0$ poses a fundamental challenge, where the null source term results in a trivial predictive mean and a degenerate marginal likelihood. Drawing inspiration from system identification, we construct a transfer function-type indicator for the unknown eigenvalue/eigenfunction using the physics-informed Gaussian Process posterior. We demonstrate that the posterior covariance is only non-trivial when $λ$ corresponds to an eigenvalue of the partial differential operator $\mathcal{L}$, reflecting the existence of a non-trivial eigenspace, and any sample from the posterior lies in the eigenspace of the linear operator. We demonstrate the effectiveness of the proposed approach through several numerical examples with both linear and non-linear eigenvalue problems.

Figures (7)

• Figure 1: The mean and samples of the GPs: (a) Prior; (b) Conditioned on two boundary points; (c) Conditioned on boundary points and $N_f = 3$ collocation points from $f$; (d) Conditioned on two boundary points and $N_f = 8$ from $f$. The grey lines are samples from the posterior, the dark blue dashed lines represent the posterior mean, the orange lines indicate the exact solution Eq.\ref{['eq:true_sol']}, and the purple regions represent the $95\%$ confidence interval.
• Figure 2: For the 1D Laplace equation, trace of the posterior covariance matrix as a function of $\lambda$. Vertical dashed lines indicate analytical eigenvalue solutions. The red line is the $\lambda^{-0.5}$ trend line.
• Figure 3: Confidence interval, samples and normalised samples of the posteriors of the 1D Laplace problem, for two different $\lambda$s ($\lambda_3$ and $\lambda_6$) and their perturbations: (a.1) $\lambda = \lambda_3$, (a.2) $\lambda = 1.05\lambda_3$, (a.3) $\lambda = 1.5\lambda_3$, (b.1) $\lambda = \lambda_6$, (b.2) $\lambda = 1.05\lambda_6$, (b.3) $\lambda = 1.5\lambda_6$. The corresponding analytical eigenfunctions are drawn in red lines. The shaded regions are the $95\%$ confidence intervals. Grey lines are samples drawn from the posteriors. Note that the scale of the vertical axis varies across different subplots.
• Figure 4: Posterior trace for the cantilever beam example. Same key as Fig. \ref{['fig:std']}
• Figure 5: Confidence interval, samples and normalised samples of the posteriors of the cantilever beam example, for different $\lambda$s: (a.1) $\lambda = \lambda_2$, (a.2) $\lambda = 1.05\lambda_2$, (a.3) $\lambda = 1.5\lambda_2$, (b.1) $\lambda = \lambda_4$, (b.2) $\lambda = 1.05\lambda_4$, (b.3) $\lambda = 1.5\lambda_4$. The corresponding analytical eigenfunctions are drawn in red lines. The shaded regions are the $95\%$ confidence intervals. Grey lines are samples drawn from the posteriors.

Theorems & Definitions (1)

• Theorem 1