Research Exchanges in the Mathematics of Deep Learning with Applications

深度学习数学及其应用研究交流

• 批准号: EP/Y037286/1
• 负责人: Matthias Ehrhardt
• 金额: $ 16.88万
• 依托单位: University of Bath
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2024
• 资助国家: 英国
• 起止时间: 2024 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FY037286%2F1
• 关键词: Research; Exchanges; Mathematics; Deep; Learning

The subject of this proposal is "mathematical aspects of deep learning algorithms and their applications". We will address several questions related to the mathematical foundations of neural networks and set up an interdisciplinary team to aidthe design of test problems and validate the research results obtained. The impact of neural networks and deep learning in recent years has been profound and unprecedented. But in the wake of the vast progress in this area, several questions and concerns have been raised about the robustness, reliability, accuracy, reproducibility and feasibility of neural networks. It is widely recognised that the mathematical sciences, are a key enabling technology in many aspects of machine learning, not the least to resolve some of the above mentioned concerns. Mathematical language and formalism can bring more rigour and precision to the understanding of the deep learning methodology. Recently, deep learning methods have been applied to physical simulations, and to discover the underlying mathematical model. Most of the work in this area has been limited to proof-of-concept and has not been applied to practical problems. An alternative approach is to make use of reduced order modelling, and this can also be combined with machine learning methods. The aim of this project is to understand, study, prove, and test the properties of deep learning algorithms using ideas from dynamical systems, geometry and optimisation. The research objectives are three-fold. The first pertains to understanding the general properties of neural networks and their impact on a range of applications. The second is about the use of neural networks for investigating dynamical systems, and their applications to physical models. Finally we establish a new and complementary network of mathematicians from European and third countries for studying neural networks and the methods of deep learning with connections to a range of application areas through staff exchanges.

该建议的主题是“深度学习算法及其应用的数学方面”。我们将解决与神经网络的数学基础有关的几个问题，并建立一个跨学科团队，以帮助设计问题并验证获得的研究结果。近年来，神经网络和深度学习的影响是史无前例的。但是，在这一领域取得了巨大进步之后，关于神经网络的鲁棒性，可靠性，准确性，可重复性和可行性，已经提出了一些问题和关注点。人们普遍认识到，数学科学是机器学习许多方面的关键促成技术，而不是解决上述某些问题。数学语言和形式主义可以使对深度学习方法的理解更加严格和精确。最近，深度学习方法已应用于物理模拟，并发现潜在的数学模型。该领域的大多数工作都仅限于概念验证，并且尚未应用于实际问题。另一种方法是利用减少的订单建模，这也可以与机器学习方法结合使用。该项目的目的是使用动态系统，几何和优化的思想来理解，研究，证明和测试深度学习算法的特性。研究目标是三倍。第一个与理解神经网络的一般特性及其对一系列应用的影响有关。第二个是关于使用神经网络来研究动态系统及其在物理模型中的应用。最后，我们建立了一个来自欧洲和第三国的数学家的新的互补网络，用于研究神经网络以及通过员工交流与一系列应用程序领域的联系的深度学习方法。