Mathematical Foundations of Intelligence: An "Erlangen Programme" for AI

智能的数学基础：人工智能的“埃尔兰根计划”

• 批准号: EP/Y028872/1
• 负责人: Michael Bronstein
• 金额: $ 1091.65万
• 依托单位: University of Oxford
• 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%2FY028872%2F1
• 关键词: Mathematical; Foundations; Intelligence; Erlangen; Programme

In 1872, Felix Klein published his now famous Erlangen Programme, in which he treated geometry as the study of invariants, formalised using group theory. This radically new approach allowed tying together different types of non-Euclidean geometries that had emerged in the nineteenth century and has had a profound methodological and cultural impact on geometry in particular and mathematics in general. New fields of mathematics such as exterior calculus, algebraic topology, the theory of fibre bundles and sheaves, and category theory emerged as a continuation of Klein's blueprint. The Erlangen Programme was also fundamental for the development of physics in the first half of the twentieth century, with Noether's theorem and the notion of gauge invariance successfully providing a unification framework for electromagnetic, weak, and strong interactions, culminating in the Standard Model in the 1970s.Now is the time for an "Erlangen Programme" for AI, based on rigorous mathematical principles that would bring better understanding of existing AI methods as well as a new generation of methods that have guaranteed expressive and generalisation power, better interpretability, scalability, and data- and computational-efficiency. Just as the ideas of Klein's Erlangen Programme spilled into other disciplines and produced new theories in mathematics, physics, and beyond, we will draw inspiration from these analogies in our AI research programme. By resorting to powerful tools from the mathematical and algorithmic fields sometimes considered "exotic" in applied domains, new theoretical insights and computational models can be derived. Our "Erlangen Programme of AI" will study four fundamental questions that underlie modern AI/ML systems, striving to provide rigorous mathematical answers. How can hidden structures in data be discovered and expressed in the language of geometry and topology in order to be exploited by ML models? Can we use geometric and topological tools to characterise ML models in order to understand when and how they work and fail? How can we guarantee learning to benefit from these structures, and use these insights to develop better, more efficient, and safer new models? Finally, how can we use such models in future AI systems that make decisions potentially affecting billions of people? With a centre at Oxford, and broad geographic coverage of the UK, the Hub will bring together leading experts in mathematical, algorithmic, and computational fields underpinning AI/ML systems as well as their applications in scientific and industrial settings. Some of the Hub participants have a track record of previous successful work together, while other collaborations are new. The research programme in the proposed Hub is intended to break barriers between different fields and bring a diverse and geographically-distributed cohort of leading UK experts rarely seen together with the purpose of strong cross-fertilisation. In the fields of AI/ML, our work will contribute to the exploitation of tools from currently underexplored mathematical fields. Conversely, our programme will help attract the attention of theoreticians to new problems and applications.

1872年，费利克斯·克莱因（Felix Klein）发表了他现在著名的埃尔兰根（Erlangen）计划，其中他将几何形状视为对不变的研究，并使用小组理论正式化。这种彻底的新方法允许将不同类型的非欧国人几何形状捆绑在一起，这些几何形状在19世纪出现，并且对几何形状尤其对几何形状产生了深远的方法论和文化影响，通常是数学。数学的新领域，例如外部演算，代数拓扑，纤维束和滑轮理论以及类别理论，是克莱因蓝图的延续。 Erlangen计划对于二十世纪上半叶的物理发展也是至关重要的，Noether的定理和规格不变性的概念成功地提供了一个统一的统一框架，以实现电磁，薄弱和强大的互动，并在1970年的标准模型中实现了一定的理解。 AI方法以及新一代的方法，这些方法可以保证表达能力和泛化能力，更好的可解释性，可伸缩性以及数据和计算效率。正如克莱因（Klein）的erlangen计划的思想蔓延到了其他学科中，并在数学，物理和其他方面产生了新的理论一样，我们将在AI研究计划中从这些类比中汲取灵感。通过从有时在应用领域中被视为“异国”的数学和算法字段中诉诸强大的工具，可以得出新的理论见解和计算模型。 我们的“ AI的Erlangen计划”将研究基于现代AI/ML系统的四个基本问题，努力提供严格的数学答案。如何以几何和拓扑的语言发现和表达数据中的隐藏结构，以便被ML模型利用？我们可以使用几何和拓扑工具来表征ML模型，以了解它们何时以及如何工作和失败？我们如何保证学习从这些结构中受益，并利用这些见解来发展更好，更高效，更安全的新模型？最后，我们如何在未来的AI系统中使用此类模型，从而做出可能影响数十亿人的决策？在牛津的中心以及英国广泛的地理覆盖范围内，该枢纽将汇集基于AI/ML系统的数学，算法和计算领域的领先专家，以及它们在科学和工业环境中的应用。一些集线器参与者拥有以前成功工作的记录，而其他合作是新的。拟议中心的研究计划旨在打破不同领域之间的障碍，并带来多样化和地理分布的英国领先专家队列，很少见到强大的交叉灌输。在AI/ML的领域，我们的工作将有助于从当前未经验证的数学字段中对工具的开发。相反，我们的计划将有助于吸引理论家对新问题和应用的关注。