Advancements in Gaussian Process Learning for Uncertainty Quantification

Gaussian processes are among the most useful tools in modeling continuous processes in machine learning and statistics. The research presented provides advancements in uncertainty quantification using Gaussian processes from two distinct perspectives. The first provides a more fundamental means of constructing Gaussian processes which take on arbitrary linear operator constraints in much more general framework than its predecessors, and the other from the perspective of calibration of state-aware parameters in computer models. 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. State-aware calibration is a novel approach in describing the relationship between properties of a system and experimental data by allowing calibration parameters to vary across the input domain as functions rather than remaining constant. Typical formulations in literature on the subject assume that it is already known whether calibration parameters are state-aware, but this is likely not the case in practice. Making incorrect assumptions on whether parameters are state-aware can produce confounding which misrepresents properties of a system and increases prediction error. We propose a means of determining the state of parameters which leverages the effect of the covariance function of Gaussian processes on their variation throughout the parameter space. We then apply the methodology to the analysis of interphase properties of composite materials and compare our results with previous studies