Using Gaussian Processes to Approximate Solutions to Differential Equations

Differential equations have a lot of applications, but often do not have known solutions, meaning they can only be approximated using numerical methods. In a lot of cases, especially with PDEs, numerical methods are extremely computationally expensive. Therefore, given a small set of numerical approximations, we want to interpolate between them. One way of doing this is by using a statistical object known as a Gaussian process.

 

For this project, I will explore what a Gaussian process is, and how it can be used to approximate functions. I will then explore the function spaces that these approximations live in, and discuss how this can be used to characterise the convergence and accuracy of interpolation using Gaussian processes.

Elizabeth Mabbutt

University of Wollongong

Elizabeth (Liz) Mabbutt is a 2nd year Bachelor of Mathematics student at the University of Wollongong. She has been interested in statistics since taking an introduction to statistics course in first year, and has also developed an interest in pure mathematics and an enjoyment of writing proofs since taking a real analysis course in second year. She is looking towards starting honours in 2025 and a PhD afterwards.

Liz participated in a 2-week research scholarship in the winter of 2023 focusing on metric spaces and a proof of Hutchinson’s Theorem, where she enjoyed collaborating with other people and learning new concepts. She is looking forward to doing more research in other areas of mathematics she is interested in.

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