用物理约束高斯过程求解线性算子特征值问题，突破传统方法的退化困境。
Physics-informed Gaussian Process Regression in Solving Eigenvalue Problem of Linear Operators
- 基于系统辨识思想构造转移函数指标，解决无源项导致的预测均值退化问题。
- 后验协方差仅在λ为真实特征值时非退化，样本自动落入特征空间。
- 适用于线性和非线性特征值问题，数值实验验证了方法有效性。
将物理信息高斯过程回归应用于线性算子的特征值问题 $(\mathcal{L}-λ)u = 0$ 时，因源项为零导致预测均值平凡且边际似然退化。受系统辨识启发，我们利用物理信息高斯过程后验构建了未知特征值/特征函数的转移函数型指标。结果表明，仅当 $λ$ 为微分算子 $\mathcal{L}$ 的特征值时，后验协方差才非退化，反映非平凡特征空间的存在；后验任意样本均属于该线性算子的特征空间。通过多个线性与非线性特征值问题的数值实验，验证了所提方法的有效性。
原文摘要 · Abstract (English)
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.
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