The world as a neural network

TL;DR

This paper explores the idea that the universe functions as a neural network, with its dynamics potentially exhibiting behaviors akin to quantum mechanics and general relativity, suggesting a deep connection between neural networks and fundamental physics.

Contribution

It proposes a novel framework where neural network dynamics can emulate classical, quantum, and relativistic physics, bridging neural networks with fundamental theories of the universe.

Findings

Trainable variables follow Madelung and Hamilton-Jacobi equations near and far from equilibrium.

Emergent spacetime can be described as relativistic strings or curved geometry depending on subsystem interactions.

Entropy production relates to Einstein-Hilbert action, hinting at a holographic duality.

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.