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The world as a neural network (2008.01540v1)

Published 4 Aug 2020 in physics.gen-ph and cs.LG

Abstract: We discuss a possibility that the entire universe on its most fundamental level is a neural network. We identify two different types of dynamical degrees of freedom: "trainable" variables (e.g. bias vector or weight matrix) and "hidden" variables (e.g. state vector of neurons). We first consider stochastic evolution of the trainable variables to argue that near equilibrium their dynamics is well approximated by Madelung equations (with free energy representing the phase) and further away from the equilibrium by Hamilton-Jacobi equations (with free energy representing the Hamilton's principal function). This shows that the trainable variables can indeed exhibit classical and quantum behaviors with the state vector of neurons representing the hidden variables. We then study stochastic evolution of the hidden variables by considering $D$ non-interacting subsystems with average state vectors, $\bar{\bf x}{1}$,..., $\bar{\bf x}{D}$ and an overall average state vector $\bar{\bf x}{0}$. In the limit when the weight matrix is a permutation matrix, the dynamics of $\bar{\bf x}{\mu}$ can be described in terms of relativistic strings in an emergent $D+1$ dimensional Minkowski space-time. If the subsystems are minimally interacting, with interactions described by a metric tensor, then the emergent space-time becomes curved. We argue that the entropy production in such a system is a local function of the metric tensor which should be determined by the symmetries of the Onsager tensor. It turns out that a very simple and highly symmetric Onsager tensor leads to the entropy production described by the Einstein-Hilbert term. This shows that the learning dynamics of a neural network can indeed exhibit approximate behaviors described by both quantum mechanics and general relativity. We also discuss a possibility that the two descriptions are holographic duals of each other.

Citations (51)

Summary

  • The paper demonstrates that the universe’s dynamics may emerge from neural network principles, bridging quantum mechanics and classical physics.
  • It models trainable variables as exhibiting quantum-like behavior near equilibrium and Hamilton-Jacobi dynamics far from equilibrium.
  • The work offers a speculative framework linking emergent space-time, gravity, and holographic duality to neural network behavior.

Overview of "The World as a Neural Network" by Vitaly Vanchurin

Vitaly Vanchurin's paper, "The World as a Neural Network," posits a compelling hypothesis that the universe fundamentally operates as a neural network. This hypothesis interrelates concepts from quantum mechanics, general relativity, and machine learning, suggesting that these frameworks may emerge from the underlying dynamics of such a neural network. Vanchurin proposes a novel perspective on the dynamics of the universe, challenging traditional understandings of physical laws.

Main Concepts

The paper introduces two central types of variables within this universal neural network:

  • Trainable Variables: Represented by elements such as bias vectors and weight matrices, whose stochastic dynamics Vanchurin connects to classical and quantum mechanical behaviors.
  • Hidden Variables: Denoted by state vectors of neurons, these variables influence the trainable variables implicitly through the dynamics of the free energy.

Vanchurin explores the dynamics of these variables across different regimes:

  1. Near Equilibrium: The dynamics of trainable variables near equilibrium can be approximated by the Madelung equations, suggesting emergent quantum mechanical properties where free energy plays the role analogous to quantum phase.
  2. Far from Equilibrium: The Hamilton-Jacobi equations describe systems further from equilibrium, illustrating dynamics akin to classical mechanics where free energy serves as Hamilton's principal function.
  3. Emergent Space-Time and Gravity: By considering multiple non-interacting subsystems, Vanchurin shows how the dynamics of hidden variables can be analogized to relativistic strings in an emergent space-time. When these subsystems interact minimally through a metric tensor, emergent gravitational behaviors, similar to those described by general relativity, are observed.

Implications and Theoretical Insights

Vanchurin's hypothesis holds several implications for the theoretical and practical development of AI and our understanding of fundamental physics:

  • Quantum Mechanics as Emergent: The paper contributes to discussions in emergent quantum mechanics, suggesting that quantum behaviors might originate from the stochastic dynamics of neural network-like structures.
  • Emergent Gravity: Relevant to the field of emergent gravity, this work suggests that what we perceive as gravitational interactions could originate from the microscopic neural network's low-complexity structures evolving through learning dynamics.
  • Observer Problem: The paper touches on the unresolved issue of constructing a self-consistent description of observers within a quantum framework. By implying that observers might emerge within a neural network universe through natural selection-like processes, Vanchurin offers a potential path forward for addressing this problem.
  • Holographic Duality: The idea of a holographic duality, where bulk neural network features can be mapped onto boundary elements of a different network, is especially intriguing. It suggests a deeper connection between dense, shallow networks and sparse, deep networks in terms of their learning dynamics and underlying structures.

Speculations on Future Developments

Future explorations might investigate:

  • Experimental Viability: Developing empirical methodologies to detect neural network-like behaviors at fundamental levels of physical laws.
  • AI-Driven Insights: Employing advances in AI to simulate complex neural dynamics that mirror those postulated in Vanchurin's model, potentially revealing underlying mechanisms of physical phenomena.
  • Broader Theoretical Frameworks: Extending the model to incorporate more comprehensive interactions among hidden and trainable variables, potentially uncovering new theoretical underpinnings for other fields of physics and cosmology.

In conclusion, "The World as a Neural Network" provides a thought-provoking framework that might align disparate areas of physics under a united theory of learning dynamics and neural structures. While still speculative, Vanchurin's paper outlines a bold direction for future theoretical and empirical inquiries into the nature of reality.

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