Rigor with Machine Learning from Field Theory to the Poincaré Conjecture

TL;DR

Rigor with Machine Learning investigates how ML can be integrated into rigor-focused disciplines like theoretical physics and pure mathematics. It advocates two complementary strategies: using ML to generate conjectures and obtain rigorous verification via reinforcement learning, and developing ML-theory frameworks such as the NN-FT correspondence and neural-network metric flows that recast field theory and geometric flows in ML terms. The survey covers concrete applications in string theory, algebraic geometry, knot theory, and four-dimensional Poincaré conjecture contexts, including a neural-network formulation of a $\phi^4$ theory and a gradient-flow view of Ricci flow. By connecting ML theory with fundamental mathematical and physical structures, the work outlines a blueprint for obtaining rigorous, interpretable, and potentially non-perturbative insights from ML in high-energy theory and geometry.

Abstract

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 Poincaré 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 Poincaré conjecture.

Figures (2)

• Figure 1: (a) The three Reidemeister moves and the band move. The band move needs to preserve orientations on the link. (b) After applying a band move to the square knot, the result can be deformed (via Reidemeister moves) into the unlink with $2$ components. (c) The kind of intersection allowed in a ribbon disk. (d) A ribbon disk for the square knot. Figure taken from ribbons.
• Figure 2: Performance comparison of the TRPO, A3C and RW algorithms on the UNKNOT problem. Left: Fraction of unknots whose braid words could be reduced to the empty braid word as a function of initial braid word length $\ell$. Right: Average number of actions necessary to reduce the input braid word to the empty braid word as a function of $\ell$.