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

arXiv: https://arxiv.org/abs/2601.06462 · HTML: https://mlsanity.com/paper/2601.06462/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 24 references that are in the map.
- First posted on arXiv 10 Jan 2026.

## Builds on (most cited first, 6 of 6)

- [Machine Learning of Linear Differential Equations using Gaussian Processes](https://mlsanity.com/paper/1701.02440/) (2017)
- [Hilbert Space Methods for Reduced-Rank Gaussian Process Regression](https://mlsanity.com/paper/1401.5508/) (2014)
- [Physics-Informed Gaussian Process Regression Generalizes Linear PDE Solvers](https://mlsanity.com/paper/2212.12474/) (2022)
- [Gaussian Process Priors for Systems of Linear Partial Differential Equations with Constant Coefficients](https://mlsanity.com/paper/2212.14319/) (2022)
- [Gaussian processes for Bayesian inverse problems associated with linear partial differential equations](https://mlsanity.com/paper/2307.08343/) (2023)
- [Probabilistic Numerical Methods for Partial Differential Equations and Bayesian Inverse Problems](https://mlsanity.com/paper/1605.07811/) (2016)

## Abstract (arXiv)

Applying Physics-Informed Gaussian Process Regression to the eigenvalue problem (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 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.