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Partial Differential Equations and Machine Learning

偏微分方程和机器学习

基本信息

批准号：
2592678
负责人：
金额：
--
依托单位：
University of Oxford
依托单位国家：
英国
项目类别：
Studentship
财政年份：
2020
资助国家：
英国
起止时间：
2020 至无数据
项目状态：
未结题

来源：
https://gtr.ukri.org/projects?ref=studentship-2592678
关键词：
Partial Differential Equations Machine Learning

项目摘要

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.

这个项目的目标是探索偏微分方程（PDE）和机器学习之间的交叉点。该项目将侧重于两个方面的研究。一方面，我们将研究非线性非局部聚集扩散方程。特别是，我们将研究的逆问题，学习这个偏微分方程的系数从噪声观测其轨迹。我们将探讨这个问题的理论和数值方面。在理论方面，我们将得到新的稳定性估计à la Dobrushin控制误差的轨迹预测的系数的估计误差。在数值方面，我们将使用数值解分析学习过程的结果，并将其与我们的理论结果进行比较。另一方面，我们将研究图上的演化偏微分方程。这是一个新的研究领域，具有很大的应用潜力，因为它结合了偏微分方程的动力学模型的能力与图形编码有趣的几何结构。特别是，我们将研究共同进化图上的进化偏微分方程，也就是说，图也在时间上进化，它的动力学依赖于定义在它上面的偏微分方程的动力学。我们将探索这个问题的理论方面，以及它在机器学习中的应用。这个项目属于EPSRC数学分析研究领域的福尔斯。