Calibration of numerical models with deep Gaussian processes. PhD Mathematics

The University of Exeter EPSRC DTP (Engineering and Physical Sciences Research Council Doctoral Training Partnership) is offering up to 4 fully funded doctoral studentships for 2019/20 entry. Students will be given sector-leading training and development with outstanding facilities and resources. Studentships will be awarded to outstanding applicants, the distribution will be overseen by the University’s EPSRC Strategy Group in partnership with the Doctoral College.

Supervisors:
Dr Danny Williamson, Department of Mathematics, College of Engineering, Mathematics and Physical Sciences
Prof Peter Challenor, Department of Mathematics, College of Engineering, Mathematics and Physical Sciences

Project description:
Complex numerical models are used throughout science to encode known physical dynamics in predictive models. Uncertainty Quantification (UQ) is an area of mathematics designed to address the many uncertainties induced when using these models to learn about the world and support decision-making. Perhaps the major problem in UQ is the calibration of these models using data from the real world and simulations – essentially finding values of the model input parameters that lead to simulations that agree well with physical observations. In the calibration problem, we typically use Gaussian processes (GPs) in place of the expensive computer model and these work well in many situations, in particular when the underlying model is relatively smooth. However, more complex non-stationary GPs are required very often in practice. GPs are determined by their covariance functions (kernels) and capturing non-stationarity often requires unusual or even tailored kernels. How these are to be selected and fitted is currently problem dependent and requires the translation of physical knowledge into the kernel. In machine learning, such structure might be uncovered automatically using “deep learning” techniques.

In this PhD scholarship you will be investigating applying deep Gaussian processes and other machine learning techniques to
“automatically” emulate complex non-stationary numerical models, and then to aid in calibrating them. You will discover how the hyper-parameters of the deep Gaussian process can be used to induce different types of non-stationarity, and develop algorithms for efficiently embedding deep GPs with desirable structures into calibration routines based on MCMC. This project compliments 2 projects with the Alan Turing Institute for Data Science and Artificial Intelligence (ATI) that the primary supervisor is involved with, giving the successful applicant the opportunity to work as part of a larger team of researchers and to travel to the ATI in London as often as needed. The main
applications we will look at involve climate models and models for detecting black holes. This latter application is part of a collaboration between the primary supervisor and Professor Derek Bingham at Simon Fraser University, Vancouver and, as such, there will be an opportunity to spend some time in Vancouver working with Professor Bingham.

This PhD project will appeal to those interested in statistics and machine learning and who want to work on the interface between these two exciting areas in data science to help solve problems in the real world.