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.

翻译：将基于物理信息的Gaussian Process回归应用于特征值问题 $(\mathcal{L}-λ)u = 0$ 时面临一个根本性挑战：零源项会导致平凡的预测均值与退化的边缘似然函数。受系统辨识理论启发，我们利用基于物理信息的Gaussian Process后验分布构建了针对未知特征值/特征函数的传递函数型指示器。我们证明：仅当 $λ$ 对应于偏微分算子 $\mathcal{L}$ 的特征值时，后验协方差才具有非平凡性，这反映了非平凡特征子空间的存在性，且从后验分布中抽取的任何样本均位于该线性算子的特征子空间中。通过若干线性与非线性特征值问题的数值算例，我们验证了所提方法的有效性。