Partial Differential Equations and Machine Learning

这个项目的目标是探索偏微分方程（PDEs）和机器学习之间的交集。该项目将侧重于两方面的研究。一方面，我们将研究非线性非局部聚集扩散方程。特别是，我们将研究从其轨迹的噪声观测中学习该PDE系数的逆问题。我们将探讨这个问题的理论和数值方面。在理论方面，我们将推导出新的稳定性估计（la Dobrushin），根据系数的估计误差来控制轨迹预测的误差。在数值方面，我们将使用数值解分析学习过程的结果，并将其与理论结果进行比较。另一方面，我们将研究图上的演化偏微分方程。这是一个具有巨大应用潜力的新研究领域，因为它结合了偏微分方程建模动力学的能力和图形编码有趣几何结构的能力。特别是，我们将研究协同演化图上的演化偏微分方程，即图也随时间演化，其动力学取决于图上定义的偏微分方程的动力学。我们将探讨这个问题的理论方面以及它在机器学习中的应用。该项目属于EPSRC数学分析研究领域。

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.