# Towards a Theory of Quantum Gravity from Neural Networks

^{1}

^{2}

## Abstract

**:**

## 1. Introduction

## 2. Neural Networks

- $\mathbf{x}$, column state vector of boundary (i.e., input/output) and bulk (i.e., hidden) neurons,
- $\widehat{P}$, boundary projection operator to subspace spanned by boundary neurons,
- ${p}_{\partial}\left(\widehat{P}\mathbf{x}\right)$, probability distribution which describes the training dataset,
- $\widehat{w}$, weight matrix which describes connections and interactions between neurons,
- $\mathbf{b}$, column bias vector which describes bias in the inputs of individual neurons,
- $\mathbf{f}\left(\mathbf{y}\right)$, activation map which describes a non-linear part of the dynamics,
- $H(\mathbf{x},\mathbf{b},\widehat{w})=H(\mathbf{x},\mathbf{q})$, loss function where $\mathbf{q}$ denotes collectively both $\mathbf{b}$ and $\widehat{w}$.

## 3. Madelung Equations

## 4. Schrodinger Equation

## 5. Lorentz Symmetry

## 6. Emergent Space-Time

## 7. Geodesic Equation

## 8. Einstein Equations

## 9. Discussion

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Everett, H. Relative State Formulation of Quantum Mechanics. Rev. Mod. Phys.
**1957**, 29, 454–462. [Google Scholar] [CrossRef][Green Version] - Adler, S. Quantum Theory as an Emergent Phenomenon; Cambridge UP: Cambridge, UK, 2004. [Google Scholar]
- Hooft, G.’t. Emergent Quantum Mechanics and Emergent Symmetries. AIP Conf. Proc.
**2007**, 957, 154–163. [Google Scholar] - Blasone, M.; Jizba, P.; Scardigli, F. Can quantum mechanics be an emergent phenomenon? J. Phys. Conf. Ser.
**2009**, 174, 012034. [Google Scholar] [CrossRef][Green Version] - Grossing, G.; Fussy, S.; Mesa Pascasio, J.; Schwabl, H. The Quantum as an Emergent System. J. Phys. Conf. Ser.
**2012**, 361, 012008. [Google Scholar] [CrossRef][Green Version] - Acosta, D.; de Cordoba, P.F.; Isidro, J.M.; Santander, J.L.G. Emergent quantum mechanics as a classical, irreversible thermodynamics. Int. J. Geom. Methods Mod. Phys.
**2013**, 10, 1350007. [Google Scholar] [CrossRef] - Fernandez De Cordoba, P.; Isidro, J.M.; Perea, M.H. Emergent quantum mechanics as a thermal ensemble. Int. J. Geom. Methods Mod. Phys.
**2014**, 11, 1450068. [Google Scholar] [CrossRef][Green Version] - Caticha, A. Entropic Dynamics: Quantum Mechanics from Entropy and Information Geometry. Ann. Phys.
**2019**, 531, 1700408. [Google Scholar] [CrossRef][Green Version] - Vanchurin, V. Entropic mechanics: Towards a stochastic description of quantum mechanics. Found. Phys.
**2019**, 50, 40–53. [Google Scholar] [CrossRef][Green Version] - Bohm, D. A Suggested Interpretation of the Quantum Theory in Terms of ’Hidden Variables’ I. Phys. Rev.
**1952**, 85, 166–179. [Google Scholar] [CrossRef] - Bell, J. On the Einstein Podolsky Rosen Paradox. Physics Physique Fizika
**1964**, 1, 195–200. [Google Scholar] [CrossRef][Green Version] - Vanchurin, V. The World as a Neural Network. Entropy
**2020**, 22, 1210. [Google Scholar] [CrossRef] - Katsnelson, M.I.; Vanchurin, V. Emergent Quantumness in Neural Networks. Found. Phys.
**2021**, 51, 94. [Google Scholar] [CrossRef] - Vanchurin, V.; Vilenkin, A.; Winitzki, S. Predictability crisis in inflationary cosmology and its resolution. Phys. Rev. D.
**2000**, 61, 083507. [Google Scholar] [CrossRef][Green Version] - Witten, E. Anti-de Sitter space and holography. Adv. Theor. Math. Phys.
**1998**, 2, 253. [Google Scholar] [CrossRef] - Susskind, L. The World as a hologram. J. Math. Phys.
**1995**, 36, 6377. [Google Scholar] [CrossRef][Green Version] - Maldacena, J.M. The Large N limit of superconformal field theories and supergravity. Int. J. Theor. Phys.
**1999**, 38, 1113. [Google Scholar] [CrossRef][Green Version] - Ashtekar, A. New Variables for Classical and Quantum Gravity. Phys. Rev. Lett.
**1986**, 57, 2244–2247. [Google Scholar] [CrossRef] - Rovelli, C.; Smolin, L. Loop Space Representation of Quantum General Relativity. Nucl. Phys.
**1990**, 80, B331. [Google Scholar] [CrossRef] - Ashtekar, A.; Bojowald, M.; Lewandowski, J. Mathematical structure of loop quantum cosmology. Adv. Theor. Math. Phys.
**2003**, 7, 233–268. [Google Scholar] [CrossRef] - Jacobson, T. Thermodynamics of space-time: The Einstein equation of state. Phys. Rev. Lett.
**1995**, 75, 1260. [Google Scholar] [CrossRef][Green Version] - Padmanabhan, T. Thermodynamical Aspects of Gravity: New insights. Rep. Prog. Phys.
**2010**, 73, 046901. [Google Scholar] [CrossRef][Green Version] - Verlinde, E.P. On the Origin of Gravity and the Laws of Newton. J. High Energy Phys.
**2011**, 1104, 029. [Google Scholar] [CrossRef][Green Version] - Vanchurin, V. Covariant Information Theory and Emergent Gravity. Int. J. Mod. Phys. A
**2018**, 33, 1845019. [Google Scholar] [CrossRef] - Dvali, G. Black Holes as Brains: Neural Networks with Area Law Entropy. Fortschritte der Physik
**2018**, 66, 1800007. [Google Scholar] [CrossRef][Green Version] - Alexander, S.; Cunningham, W.J.; Lanier, J.; Smolin, L.; Stanojevic, S.; Toomey, M.W.; Wecker, D. The Autodidactic Universe. arXiv
**2021**, arXiv:2104.03902. [Google Scholar] - Vanchurin, V. Toward a theory of machine learning. Mach. Learn. Sci. Technol.
**2021**, 2, 035012. [Google Scholar] [CrossRef] - Jaynes, E.T. Information Theory and Statistical Mechanics. Phys. Rev. Ser. II
**1957**, 106, 620–630. [Google Scholar] [CrossRef] - Jaynes, E.T. Information Theory and Statistical Mechanics II. Phys. Rev. Ser. II.
**1957**, 108, 171–190. [Google Scholar] [CrossRef] - Madelung, E. Quantentheorie in hydrodynamischer Form. Z. Phys.
**1927**, 40, 322–326. (In German) [Google Scholar] [CrossRef] - Wallstrom, T.C. Inequivalence between the Schrödinger equation and the Madelung hydrodynamic equations. Phys. Rev. A
**1994**, 49, 1613–1617. [Google Scholar] [CrossRef] - Carroll, S.M. Spacetime and Geometry: An Introduction to General Relativity; Addison-Wesley: San Francisco, CA, USA, 2004. [Google Scholar]
- Vanchurin, V.; Wolf, Y.I.; Katsnelson, M.O.; Koonin, E.V. Towards a Theory of Evolution as Multilevel Learning. arXiv
**2021**, arXiv:2110.14602. [Google Scholar] - Vanchurin, V.; Wolf, Y.I.; Katsnelson, M.O.; Koonin, E.V. Thermodynamics of Evolution and the Origin of Life. arXiv
**2021**, arXiv:2110.15066. [Google Scholar] - Katsnelson, M.I.; Vanchurin, V.; Westerhout, T. Self-organized criticality in Neural Networks. arXiv
**2021**, arXiv:2107.03402. [Google Scholar]

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Vanchurin, V. Towards a Theory of Quantum Gravity from Neural Networks. *Entropy* **2022**, *24*, 7.
https://doi.org/10.3390/e24010007

**AMA Style**

Vanchurin V. Towards a Theory of Quantum Gravity from Neural Networks. *Entropy*. 2022; 24(1):7.
https://doi.org/10.3390/e24010007

**Chicago/Turabian Style**

Vanchurin, Vitaly. 2022. "Towards a Theory of Quantum Gravity from Neural Networks" *Entropy* 24, no. 1: 7.
https://doi.org/10.3390/e24010007