%AGukov, Sergei%AHalverson, James%ARuehle, Fabian%BJournal Name: Nature Reviews Physics; Journal Volume: 6; Journal Issue: 5
%D2024%INature Reviews Physics
%JJournal Name: Nature Reviews Physics; Journal Volume: 6; Journal Issue: 5
%K
%MOSTI ID: 10510080
%PMedium: X
%TRigor with machine learning from field theory to the Poincaré conjecture
%XDespite their successes, machine learning techniques are often stochastic, error-prone and blackbox. How could they then be used in fields such as theoretical physics and pure mathematics for which error-free results and deep understanding are a must? In this Perspective, we discuss techniques for obtaining zero-error results with machine learning, with a focus on theoretical physics and pure mathematics. Non-rigorous methods can enable 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 4D Poincaré conjecture in low-dimensional topology. We also discuss connections between machine learning theory and mathematics or theoretical physics such as 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 used to solve the 3D Poincaré conjecture.
%0Journal Article