Rigor with Machine Learning from Field Theory to the Poincaré
  Conjecture
Sergei Gukov James Halverson Fabian Ruehle
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.