标题： 从场论到庞加莱猜想的机器学习严谨性 显示英文标题

标题： Rigor with Machine Learning from Field Theory to the Poincaré Conjecture

摘要： 机器学习技术越来越强大，导致自然科学领域取得了许多突破，但它们通常是随机的、容易出错的，并且是黑箱的。 那么，在理论物理和纯粹数学等高度重视严谨性和理解力的领域中，应该如何利用它们呢？ 在本文中，我们讨论了在自然科学中使用机器学习获得严谨性的技术。 非严谨的方法可以通过猜想生成或通过强化学习进行验证，从而得到严谨的结果。 我们综述了这些用于严谨性的技术应用，从弦理论到低维拓扑中的光滑$4$维庞加莱猜想。 也可以想象在机器学习理论与数学或理论物理之间建立直接的桥梁。 作为例子，我们描述了一种受神经网络理论启发的新场论方法，以及一种由神经网络梯度下降引起的黎曼度量流理论，该理论涵盖了佩雷尔曼提出的里奇流理论，该理论被用来解决$3$维庞加莱猜想。 显示英文摘要

摘要： Machine learning techniques are increasingly powerful, leading to many breakthroughs in the natural sciences, but they are often stochastic, error-prone, and blackbox. How, then, should they be utilized in fields such as theoretical physics and pure mathematics that place a premium on rigor and understanding? In this Perspective we discuss techniques for obtaining rigor in the natural sciences with machine learning. Non-rigorous methods may lead to rigorous results via conjecture generation or verification by reinforcement learning. We survey applications of these techniques-for-rigor ranging from string theory to the smooth $4$d Poincar\'e conjecture in low-dimensional topology. One can also imagine building direct bridges between machine learning theory and either mathematics or theoretical physics. As examples, we describe a new approach to field theory motivated by neural network theory, and a theory of Riemannian metric flows induced by neural network gradient descent, which encompasses Perelman's formulation of the Ricci flow that was utilized to resolve the $3$d Poincar\'e conjecture.