Rigor with Machine Learning from Field Theory to the Poincar\'e Conjecture
<|reference_start|>Rigor with Machine Learning from Field Theory to the Poincar\'e Conjecture: 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.<|reference_end|>
@article{gukov2024rigor,
    title={Rigor with Machine Learning from Field Theory to the Poincar\'e
  Conjecture},
    author={Sergei Gukov, James Halverson, Fabian Ruehle},
    journal={arXiv preprint arXiv:2402.13321},
    year={2024},
    archivePrefix={arXiv},
    eprint={2402.13321},
    primaryClass={hep-th cs.LG}
}