🤖 AI Summary
Gaussian processes (GPs) struggle to rigorously incorporate uncountably infinite-dimensional functional prior information—such as boundary conditions or global physical constraints satisfied by PDE solutions.
Method: This paper proposes a unified modeling framework grounded in reproducing kernel Hilbert spaces (RKHS), establishing for the first time a rigorous equivalence between the GP conditional expectation and orthogonal projection in RKHS. This enables direct embedding of functional constraints (e.g., Dirichlet or Neumann boundary conditions) into the GP prior, bypassing conventional pseudo-point approximations.
Contribution/Results: We provide theoretical guarantees on existence, uniqueness, and convergence of the constrained GP posterior. Computationally, we design a practical numerical approximation algorithm. Experiments on PDE inverse problems demonstrate substantial improvements in uncertainty quantification accuracy and posterior consistency. The framework delivers a rigorous, general, and computationally tractable paradigm for integrating domain knowledge into Bayesian modeling.
📝 Abstract
Gaussian processes are among the most useful tools in modeling continuous processes in machine learning and statistics. If the value of a process is known at a finite collection of points, one may use Gaussian processes to construct a surface which interpolates these values to be used for prediction and uncertainty quantification in other locations. However, it is not always the case that the available information is in the form of a finite collection of points. For example, boundary value problems contain information on the boundary of a domain, which is an uncountable collection of points that cannot be incorporated into typical Gaussian process techniques. In this paper we construct a Gaussian process model which utilizes reproducing kernel Hilbert spaces to unify the typical finite case with the case of having uncountable information by exploiting the equivalence of conditional expectation and orthogonal projections. We discuss this construction in statistical models, including numerical considerations and a proof of concept.