Partial Differential Equations and Machine Learning

The goal of this project is to explore the intersection between Partial Differential Equations (PDEs) and Machine Learning.
The project will focus on two strands of research.

On one hand, we will work on non-linear non-local aggregation diffusion equations. In particular, we will study the inverse problem of learning the coefficients of this PDE from noisy observations of its trajectory.
We will explore theoretical and numerical aspects of this problem. On the theoretical side, we will derive new stability estimates à la Dobrushin controlling the error of the trajectory predictions in terms of the estimation error of the coefficients. On the numerical side, we will analyse the results of the learning procedure using numerical solutions and compare them with our theoretical results.

On the other hand, we will study evolution PDEs on graphs. This is a novel area of research with great potential for applications since it combines the power of PDEs to model dynamics with the capability of graphs to encode interesting geometric structures. In particular, we will study evolution PDEs on co-evolving graphs, that is, the graph also evolves in time and its dynamics depend on the dynamics of the PDE defined on it. We will explore theoretical aspects of this problem as well as its applications in machine learning.

This project falls within the EPSRC Mathematical Analysis research area.