ARC Postgraduate Research Scholarship on "Developing error quantification of Koopman operator algorithms"

This PhD scholarship at the University of Sydney is offered as part of the ARC DECRA project "From chaos to clarity: reliable data-driven analysis of dynamical systems" The stipend is $42,754 per annum tax-exempt (3.5 years), with generous additional opportunities for tutorial work. The School of Mathematics and Statistics is a top maths department, with a large and active dynamical systems group. Sydney is a world-class city with unparalleled natural beauty and a really excellent quality of life. For further enquiries, please contact Caroline Wormell caroline.wormell@sydney.edu.au Supervisor: Dr Caroline Wormell, School of Mathematics and Statistics Project title: Developing error quantification of Koopman operator algorithms Project summary: In scientific applications, many interesting dynamical systems are governed by equations that are unknown to use. We would like to predict them and study their emergent behaviour, but we may only have a limited amount of observations to work with. Various tools exist that facilitate this up to some error, most of which involve approximating these systems' Koopman operator. The Koopman operator is a linear operator on functions that encodes composition by the dynamics and whose eigenfunctions reveal emergent patterns. For many chaotic systems, the stability of the approximated Koopman operator can be highly variable depending on what is being measured. This makes error quantification very important in applications. This project will develop mathematically rigorous error quantification for a kernel-based algorithm known as kernel Extended Dynamical Mode Decomposition. We will seek to quantitatively understand how least-squares approximation affects the spectrum of infinite-dimensional operators, and develop indicators and statistical tests to measure this. This will involve applying, and where needed developing, rigorous approximation theory and probability theory in the context of dynamical systems. A strong mathematical background with at least one of functional analysis or probability theory would be desirable

Last updated: 13 March 2026