Graph neural networks for models of active matter

Abstract

Cell migration is a critical process that plays a fundamental role in a wide range of biological phenomena, including embryogenesis, wound healing, and immune responses. Despite its significance, accurately modeling the complex dynamics of cellular migration remains challenging.

This thesis investigates the use of graph neural networks (GNNs) to simulate the dynamics of cellular migration systems. In particular, this work explores the efficacy of several message-passing GNN architectures including a GaT and classical message-passing GNNs. The performances of these models are evaluated with a focus on their ability to replicate ground truth statistics.

Beyond simulation, a key objective of this thesis is to enhance the interpretability of the models by recovering the underlying mathematical expressions of the interactions governing cellular interactions. To achieve this, L1 sparsity regularization is applied to the messages of a single-layer GNN. This results in a dimension reduction that enables the use of genetic programming for symbolic regression.

Recognizing the challenges posed by symbolic regression in the context of highly complex analytic equations, this thesis introduces a methodology that decomposes the messages into a weighted sum of basis functions. Applying symbolic regression on the weights provides a more tractable means to recover the governing equations.