Research Exchanges in the Mathematics of Deep Learning with Applications

本次提案的主题是“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数学基础相关的几个问题，并建立一个跨学科团队来帮助设计测试问题并验证所获得的研究结果。近年来，神经网络和深度学习的影响是前所未有的。但随着这一领域的巨大进步，人们对神经网络的鲁棒性、可靠性、准确性、可重复性和可行性提出了一些问题和担忧。人们普遍认为，数学科学是机器学习许多方面的关键使能技术，尤其是解决上述一些问题。数学语言和形式主义可以为深度学习方法的理解带来更多的严谨性和精确性。最近，深度学习方法已被应用于物理模拟，并发现潜在的数学模型。这一领域的大部分工作都局限于概念验证，尚未应用于实际问题。另一种方法是使用降阶建模，这也可以与机器学习方法相结合。这个项目的目的是理解、研究、证明和测试深度学习算法的性质，使用动态系统、几何和优化的思想。研究目标有三个方面。第一个是理解神经网络的一般特性及其对一系列应用的影响。第二部分是关于神经网络在动态系统研究中的应用，以及它们在物理模型中的应用。最后，我们通过人员交流，建立了一个由来自欧洲和第三国的数学家组成的新的互补网络，以研究神经网络和深度学习方法，并与一系列应用领域建立联系。

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.