A Kernel-Based Approach for Modelling Gaussian Processes with Functional Information

Gaussian processes (GPs) are ubiquitous tools for modeling and predicting continuous processes in physical and engineering sciences. This is partly due to the fact that one may employ a Gaussian process as an interpolator while facilitating straightforward uncertainty quantification at other locations. In addition to training data, it is sometimes the case that available information is not in the form of a finite collection of points. For example, boundary value problems contain information on the boundary of a domain, or underlying physics lead to known behavior on an entire uncountable subset of the domain of interest. While an approximation to such known information may be obtained via pseudo-training points in the known subset, such a procedure is ad hoc with little guidance on the number of points to use, nor the behavior as the number of pseudo-observations grows large. We propose and construct Gaussian processes that unify, via reproducing kernel Hilbert space, the typical finite training data case with the case of having uncountable information by exploiting the equivalence of conditional expectation and orthogonal projections in Hilbert space. We show existence of the proposed process and establish that it is the limit of a conventional GP conditioned on an increasing number of training points. We illustrate the flexibility and advantages of our proposed approach via numerical experiments.