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

• URL: http://arxiv.org/abs/2402.13321v1
• Date: Tue, 20 Feb 2024 19:00:59 GMT
• Title: Rigor with Machine Learning from Field Theory to the Poincar\'e Conjecture
• Authors: Sergei Gukov, James Halverson, Fabian Ruehle
• Abstract summary: 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. One can also imagine building direct bridges between machine learning theory and either mathematics or theoretical physics.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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 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.

Related papers

• Towards a Categorical Foundation of Deep Learning: A Survey [0.0]
  This thesis is a survey that covers some recent work attempting to study machine learning categorically. acting as a lingua franca of mathematics and science, category theory might be able to give a unifying structure to the field of machine learning.
  arXiv  Detail & Related papers  (2024-10-07T13:11:16Z)
• Machine Learning Toric Duality in Brane Tilings [0.0]
  We apply a variety of machine learning methods to the study of Seiberg duality within 4d $mathcalN=1$ quantum field theories. An intricate network of infrared dualities interconnects the space of such theories and partitions it into universality classes. We train a fully connected neural network to identify classes of Seiberg dual theories realised on $mathbbZ_mtimesmathbbZ_n$ orbifolds of the conifold.
  arXiv  Detail & Related papers  (2024-09-23T17:48:14Z)
• A Geometric Framework for Adversarial Vulnerability in Machine Learning [0.0]
  This work starts with the intention of using mathematics to understand the intriguing vulnerability observed by citetszegedy2013 within artificial neural networks. Along the way, we will develop some novel tools with applications far outside of just the adversarial domain.
  arXiv  Detail & Related papers  (2024-07-03T11:01:15Z)
• Machine Learning of the Prime Distribution [49.84018914962972]
  We provide a theoretical argument explaining the experimental observations of Yang-Hui He about the learnability of primes. We also posit that the ErdHos-Kac law would very unlikely be discovered by current machine learning techniques.
  arXiv  Detail & Related papers  (2024-03-19T09:47:54Z)
• Machine learning and information theory concepts towards an AI Mathematician [77.63761356203105]
  The current state-of-the-art in artificial intelligence is impressive, especially in terms of mastery of language, but not so much in terms of mathematical reasoning. This essay builds on the idea that current deep learning mostly succeeds at system 1 abilities. It takes an information-theoretical posture to ask questions about what constitutes an interesting mathematical statement.
  arXiv  Detail & Related papers  (2024-03-07T15:12:06Z)
• Lecture Notes on Quantum Electrical Circuits [49.86749884231445]
  Theory of quantum electrical circuits goes under the name of circuit quantum electrodynamics or circuit-QED. The goal of the theory is to provide a quantum description of the most relevant degrees of freedom. These lecture notes aim at giving a pedagogical overview of this subject for theoretically-oriented Master or PhD students in physics and electrical engineering.
  arXiv  Detail & Related papers  (2023-12-08T19:26:34Z)
• Unified Field Theory for Deep and Recurrent Neural Networks [56.735884560668985]
  We present a unified and systematic derivation of the mean-field theory for both recurrent and deep networks. We find that convergence towards the mean-field theory is typically slower for recurrent networks than for deep networks. Our method exposes that Gaussian processes are but the lowest order of a systematic expansion in $1/n$.
  arXiv  Detail & Related papers  (2021-12-10T15:06:11Z)
• The Autodidactic Universe [0.8795040582681388]
  We present an approach to cosmology in which the Universe learns its own physical laws. We discover maps that put each of these matrix models in correspondence with both a gauge/gravity theory and a mathematical model of a learning machine. We discuss in detail what it means to say that learning takes place in autodidactic systems, where there is no supervision.
  arXiv  Detail & Related papers  (2021-03-29T02:25:02Z)
• Quantum field-theoretic machine learning [0.0]
  We recast the $phi4$ scalar field theory as a machine learning algorithm within the mathematically rigorous framework of Markov random fields. Neural networks are additionally derived from the $phi4$ theory which can be viewed as generalizations of conventional neural networks.
  arXiv  Detail & Related papers  (2021-02-18T16:12:51Z)
• Topological Quantum Gravity of the Ricci Flow [62.997667081978825]
  We present a family of topological quantum gravity theories associated with the geometric theory of the Ricci flow. First, we use BRST quantization to construct a "primitive" topological Lifshitz-type theory for only the spatial metric. We extend the primitive theory by gauging foliation-preserving spacetime symmetries.
  arXiv  Detail & Related papers  (2020-10-29T06:15:30Z)
• Learning to Prove Theorems by Learning to Generate Theorems [71.46963489866596]
  We learn a neural generator that automatically synthesizes theorems and proofs for the purpose of training a theorem prover. Experiments on real-world tasks demonstrate that synthetic data from our approach improves the theorem prover.
  arXiv  Detail & Related papers  (2020-02-17T16:06:02Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.