Matching covariance kernels to numerical models in Gaussian process emulation

Science relies increasingly on complex numerical models normally solving partial differential equations but also 'black box' models from machine learning. If such models are to be used to
make real world decisions the model predictions need to include estimates of uncertainty. One way to calculate such uncertainties is the use of Gaussian process emulators. Such emulators are statistical models fitted to designed experiments of the numerical model. In essence we model the model. The statistical model used is a Gaussian process. Such models require a covariance function (or kernel) to describe how the covariance between points varies with distance. Because we do not use any information about the form of the numerical model these techniques are often known as 'black box' methods. In this PhD scholarship you will be investigating different covariance kernels and in particular investigating whether the covariance kernel can be matched to some properties of underlying partial differential equation. These methods are sometimes referred to as 'grey box' methods as we use partial information about the underlying system. As an example we know that the solution to the heat equation is a Gaussian process with a Matern covariance kernel. However we do not know what covariance kernel corresponds to other partial differential equations, for example to the Navier-Stokes equations for fluid flow (or a linearised version of them). In the PhD project you will investigate how we can match the covariance kernel to the underlying partial differential equations. You will also explore whether we can use the kernels to include constraints present in the underlying model. For example we may know that an output is always positive of that output 1 is always larger than output 2. In most of the examples we will look at, from climate, healthcare or engineering, the equations are much more complex than a simple set of partial differential equations but using the appropriate covariance kernel
should improve the efficiency of the emulator.