Research Exchanges in the Mathematics of Deep Learning with Applications

这项提议的主题是“深度学习算法的数学方面及其应用”。我们将解决与神经网络的数学基础有关的几个问题，并成立一个跨学科的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深远的和前所未有的。但随着这一领域的巨大进步，人们对神经网络的稳健性、可靠性、准确性、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的关键技术。数学语言和形式主义可以给深度学习方法论的理解带来更多的严谨和精确。最近，深度学习方法被应用于物理模拟，并发现潜在的数学模型。这方面的大部分工作仅限于概念验证，并未应用于实际问题。另一种方法是利用降阶建模，这也可以与机器学习方法相结合。这个项目的目的是利用动力系统、几何学和最优化的思想来理解、研究、证明和测试深度学习算法的特性。研究目标有三个方面。第一个是关于了解神经网络的一般特性及其对一系列应用程序的影响。第二个是关于神经网络在研究动力系统中的使用，以及它们在物理模型中的应用。最后，我们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，用于研究神经网络和深度学习方法，并通过人员交流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.