标题： 由边界值问题约束的高斯过程回归 显示英文标题

标题： Gaussian Process Regression constrained by Boundary Value Problems

摘要： 我们开发了一个受边界值问题约束的高斯过程回归框架。 该框架可以用于推断一个适定的边界值问题的解，该问题具有已知的二阶微分算子和边界条件，但对于源项只有零散的观测数据。 零散的解的观测数据也可以用于回归。 该框架结合了协同克里金法与高斯过程的线性变换，以及利用由边界值问题本征函数的谱展开给出的核函数。 因此，它受益于协方差矩阵的低秩性质。 我们证明，与没有边界条件约束的物理信息高斯过程回归相比，该框架产生的解推断更加准确和稳定。 显示英文摘要

摘要： We develop a framework for Gaussian processes regression constrained by boundary value problems. The framework may be applied to infer the solution of a well-posed boundary value problem with a known second-order differential operator and boundary conditions, but for which only scattered observations of the source term are available. Scattered observations of the solution may also be used in the regression. The framework combines co-kriging with the linear transformation of a Gaussian process together with the use of kernels given by spectral expansions in eigenfunctions of the boundary value problem. Thus, it benefits from a reduced-rank property of covariance matrices. We demonstrate that the resulting framework yields more accurate and stable solution inference as compared to physics-informed Gaussian process regression without boundary condition constraints.