Identifying PDEs in Interacting Particle Systems using Data-Driven Techniques

Abstract

Partial differential equations (PDEs) are powerful tools to describe large-scale dynamics of microscopic systems. However, deriving a PDE from first principles becomes significantly more involved when systems increase in complexity. In this thesis research, we will consider a data-driven approach to determine (new) PDEs for interacting particle models on a lattice, using the sparse regression based algorithm develop by Brunton et al. We will focus on the average behaviour of the Eden process (this describes growth processes seen in bacteria) and the contact process (CP), which also incorporates death. When the Eden process is combined with the symmetric simple exclusion process (SSEP), \textit{i.e.}, diffusive spreading and hard-core exclusion, the motion obeys the Fisher equation, which is a PDE that models the spread of growing populations. The CP is more complex as it possesses a phase transition with criticality. As of yet, the emergent dynamics have not been captured in the form of a coarse-grained PDE, which makes it an interesting case to study from both a biological and mathematical perspective. At the critical point of the CP, we find that the emergent behaviour can be captured by the Fisher equation with an interface broadening term, which explains the observed length scales at the boundary of the population. Our results highlight the strength of data-driven methods for discovering PDEs in complex particle systems. Applying these techniques to experimental results for bacterial colonies presents new opportunities to study their collective behaviour, which will be the focus of future research.