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.

There are no known nontrivial compact Calabi-Yau metrics, objects of central importance in string theory and algebraic geometry, despite decades of study. The essence of the problem is that theorems by Calabi [1] and Yau [2, 3] guarantee the existence of a Ricci-flat Kähler metric (Calabi-Yau metric) when certain criteria are satisfied, but Yau's proof is non-constructive. It is not for lack of examples satisfying the criteria, since topological constructions ensure the existence of an exponentially large number of examples [4-6]. The problem also does not prevent certain types of progress in string theory, since aspects of Calabi-Yau manifolds can be studied without knowing the metric. For instance, much is known about volumes of calibrated submanifolds [7], an artifact of supersymmetry and the existence of BPS objects, as well a metric deformations the preserve Ricci-flatness, the (in)famous moduli spaces [8].