# Rigor with Machine Learning from Field Theory to the Poincar'e Conjecture

arXiv: https://arxiv.org/abs/2402.13321 · HTML: https://mlsanity.com/paper/2402.13321/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 36 of its 85 references that are in the map.
- First posted on arXiv 20 Feb 2024.

## Builds on (most cited first, 30 of 36)

- [Axiomatic Attribution for Deep Networks](https://mlsanity.com/paper/1703.01365/) (2017)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Obfuscated Gradients Give a False Sense of Security: Circumventing Defenses to Adversarial Examples](https://mlsanity.com/paper/1802.00420/) (2018)
- [Wide Neural Networks of Any Depth Evolve as Linear Models Under Gradient Descent](https://mlsanity.com/paper/1902.06720/) (2019)
- [Mastering Chess and Shogi by Self-Play with a General Reinforcement Learning Algorithm](https://mlsanity.com/paper/1712.01815/) (2017)
- [Discovering Symbolic Models from Deep Learning with Inductive Biases](https://mlsanity.com/paper/2006.11287/) (2020)
- [Autoformalization with Large Language Models](https://mlsanity.com/paper/2205.12615/) (2022)
- [HyperTree Proof Search for Neural Theorem Proving](https://mlsanity.com/paper/2205.11491/) (2022)
- [The Principles of Deep Learning Theory](https://mlsanity.com/paper/2106.10165/) (2021)
- [Tensor Programs I: Wide Feedforward or Recurrent Neural Networks of Any Architecture are Gaussian Processes](https://mlsanity.com/paper/1910.12478/) (2019)
- [Tensor Programs II: Neural Tangent Kernel for Any Architecture](https://mlsanity.com/paper/2006.14548/) (2020)
- [Neural Networks and Quantum Field Theory](https://mlsanity.com/paper/2008.08601/) (2020)
- [TacticToe: Learning to Prove with Tactics](https://mlsanity.com/paper/1804.00596/) (2018)
- [Learning to Unknot](https://mlsanity.com/paper/2010.16263/) (2020)
- [Neural Network Approximations for Calabi-Yau Metrics](https://mlsanity.com/paper/2012.15821/) (2020)
- [Symmetry-via-Duality: Invariant Neural Network Densities from Parameter-Space Correlators](https://mlsanity.com/paper/2106.00694/) (2021)
- [Learning Size and Shape of Calabi-Yau Spaces](https://mlsanity.com/paper/2111.01436/) (2021)
- [Building Quantum Field Theories Out of Neurons](https://mlsanity.com/paper/2112.04527/) (2021)
- [Nonperturbative renormalization for the neural network-QFT correspondence](https://mlsanity.com/paper/2108.01403/) (2021)
- [The edge of chaos: quantum field theory and deep neural networks](https://mlsanity.com/paper/2109.13247/) (2021)
- [Neural Network Field Theories: Non-Gaussianity, Actions, and Locality](https://mlsanity.com/paper/2307.03223/) (2023)
- [Deep-Learning the Landscape](https://mlsanity.com/paper/1706.02714/) (2017)
- [Bayesian Renormalization](https://mlsanity.com/paper/2305.10491/) (2023)
- [Structures of Neural Network Effective Theories](https://mlsanity.com/paper/2305.02334/) (2023)
- [Disentangling a Deep Learned Volume Formula](https://mlsanity.com/paper/2012.03955/) (2020)
- [Learning knot invariants across dimensions](https://mlsanity.com/paper/2112.00016/) (2021)
- [The Inverse of Exact Renormalization Group Flows as Statistical Inference](https://mlsanity.com/paper/2212.11379/) (2022)
- [Searching for ribbons with machine learning](https://mlsanity.com/paper/2304.09304/) (2023)
- [Heterotic String Model Building with Monad Bundles and Reinforcement Learning](https://mlsanity.com/paper/2108.07316/) (2021)
- [A duality connecting neural network and cosmological dynamics](https://mlsanity.com/paper/2202.11104/) (2022)

## Built on by (most cited first, 12 of 12)

- [KAN: Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2404.19756/) (2024)
- [Data for Mathematical Copilots: Better Ways of Presenting Proofs for Machine Learning](https://mlsanity.com/paper/2412.15184/) (2024)
- [A Triumvirate of AI Driven Theoretical Discovery](https://mlsanity.com/paper/2405.19973/) (2024)
- [TASI Lectures on Physics for Machine Learning](https://mlsanity.com/paper/2408.00082/) (2024)
- [Deep Learning as Ricci Flow](https://mlsanity.com/paper/2404.14265/) (2024)
- [Learning BPS Spectra and the Gap Conjecture](https://mlsanity.com/paper/2405.09993/) (2024)
- [On knot detection via picture recognition](https://mlsanity.com/paper/2510.06284/) (2025)
- [Learning 3-Manifold Triangulations](https://mlsanity.com/paper/2405.09610/) (2024)
- [On the Learnability of Knot Invariants: Representation, Predictability, and Neural Similarity](https://mlsanity.com/paper/2502.12243/) (2025)
- [Colored Jones Polynomials and the Volume Conjecture](https://mlsanity.com/paper/2502.18575/) (2025)
- [Deep Learning based discovery of Integrable Systems](https://mlsanity.com/paper/2503.10469/) (2025)
- [Computations and ML for surjective rational maps](https://mlsanity.com/paper/2510.08093/) (2025)

## Abstract (arXiv)

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 4d 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 3d Poincar'e conjecture.