# The World as a Neural Network

## Abstract

**:**

## 1. Introduction

## 2. Neural Networks

## 3. Thermodynamics of Learning

**Second Law of Learning:**

**First Law of Learning:**

## 4. Entropic Mechanics

**Principle of Stationary Entropy Production:**

## 5. Quantum Mechanics

- (1)
- $-4D\frac{{\partial}^{2}}{\partial {q}_{k}^{2}}$, entropy production due to stochastic dynamics of ${q}_{k}$’s,
- (2)
- $\gamma \frac{{\partial}^{2}F}{\partial {q}_{k}^{2}}$, entropy production due to learning dynamics of ${q}_{k}$’s,
- (3)
- $\mu \frac{\partial F}{\partial t}$, free energy production due to dynamics of ${x}_{i}$’s,
- (4)
- $\mu \gamma {\left(\frac{\partial F}{\partial {q}_{k}}\right)}^{2}$, free energy production due to learning dynamics of ${q}_{k}$’s, and
- (5)
- $\mu V$, the (negative of) total time-averaged free energy production.

## 6. Hamiltonian Mechanics

## 7. Hidden Variables

## 8. Relativistic Strings

## 9. Emergent Gravity

## 10. Holography

## 11. Discussion

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- 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] - Everett, H. Relative State Formulation of Quantum Mechanics. Rev. Mod. Phys.
**1957**, 29, 454–462. [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
**1964**, 1, 195–200. [Google Scholar] [CrossRef] [Green Version] - 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] - Dvali, G. Black Holes as Brains: Neural Networks with Area Law Entropy. Fortsch. Phys.
**2018**, 66, 1800007. [Google Scholar] [CrossRef] [Green Version] - Hashimoto, K.; Sugishita, S.; Tanaka, A.; Tomiya, A. Deep learning and the AdS/CFT correspondence. Phys. Rev. D
**2018**, 98, 046019. [Google Scholar] [CrossRef] [Green Version] - Vanchurin, V. Towards a theory of machine learning. arXiv
**2020**, arXiv:2004.09280. [Google Scholar] - Vanchurin, V. Information Graph Flow: A geometric approximation of quantum and statistical systems. Found. Phys.
**2018**, 48, 636. [Google Scholar] [CrossRef] [Green Version] - 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] - Vanchurin, V. Entropic Mechanics: Towards a stochastic description of quantum mechanics. Found. Phys.
**2019**, 50, 40. [Google Scholar] [CrossRef] [Green Version] - Prigogine, I. Etude Thermodynamique des phénoménes irréversibles. Bull. Acad. Roy. Blg. Cl. Sci.
**1945**, 31, 600–606. [Google Scholar] - Klein, M.J.; Meijer, P.H.E. Principle of minimum entropy production. Phys. Rev.
**1954**, 96, 250–255. [Google Scholar] [CrossRef] - Onsager, L. Reciprocal relations in irreversible processes, I. Phys. Rev.
**1931**, 37, 405–426. [Google Scholar] [CrossRef] - Vanchurin, V. Covariant Information Theory and Emergent Gravity. Int. J. Mod. Phys. A
**2018**, 33, 1845019. [Google Scholar] [CrossRef] - Carleo, G.; Cirac, I.; Cranmer, K.; Daudet, L.; Schuld, M.; Tishby, N.; Vogt-Maranto, L.; Zdeborova, L. Machine learning and the physical sciences. Rev. Mod. Phys.
**2019**, 91, 045002. [Google Scholar] [CrossRef] [Green Version] - Wu, T.; Tegmark, M. Toward an artificial intelligence physicist for unsupervised learning. Phys. Rev. E.
**2019**, 100, 033311. [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. Meth. 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. Meth. Mod. Phys.
**2014**, 11, 1450068. [Google Scholar] [CrossRef] [Green Version] - Caticha, A. Entropic Dynamics: Quantum Mechanics from Entropy and Information Geometry. Annalen Phys.
**2019**, 531, 1700408. [Google Scholar] [CrossRef] [Green Version] - Swingle, B. Entanglement Renormalization and Holography. Phys. Rev. D
**2012**, 86, 065007. [Google Scholar] [CrossRef] - Almheiri, A.; Dong, X.; Harlow, D. Bulk Locality and Quantum Error Correction in AdS/CFT. J. High Energy Phys.
**2015**, 1504, 163. [Google Scholar] [CrossRef] [Green Version] - Cao, C.; Carroll, S.M.; Michalakis, S. Space from Hilbert Space: Recovering Geometry from Bulk Entanglement? Phys. Rev. D
**2017**, 95, 024031. [Google Scholar] [CrossRef] [Green Version] - Laughlin, R.B. Emergent relativity. Int. J. Mod. Phys. A
**2003**, 18, 831–854. [Google Scholar] [CrossRef] [Green Version] - Bednik, G.; Pujolas, O.; Sibiryakov, S. Emergent Lorentz invariance from Strong Dynamics: Holographic examples. J. High Energy Phys.
**2013**, 11, 064. [Google Scholar] [CrossRef] [Green Version] - Vanchurin, V. A quantum-classical duality and emergent space-time. 10th Math. Phys. Meet.
**2019**, 347–366. [Google Scholar] - Vanchurin, V. Differential equation for partition functions and a duality pseudo-forest. arXiv
**2019**, arXiv:1910.11268. [Google Scholar] - Vanchurin, V. Dual Path Integral: A non-perturbative approach to strong coupling. arXiv
**2019**, arXiv:1912.09265. [Google Scholar] - Barcelo, C.; Visser, M.; Liberati, S. Einstein gravity as an emergent phenomenon? Int. J. Mod. Phys. D
**2001**, 10, 799–806. [Google Scholar] [CrossRef] [Green Version] - Cao, C.; Carroll, S.M. Bulk entanglement gravity without a boundary: Towards finding Einstein?s equation in Hilbert space. Phys. Rev. D
**2018**, 97, 086003. [Google Scholar] [CrossRef] [Green Version] - Smolin, L. Did the Universe Evolve? Class. Quantum Gravity
**1992**, 9, 173–191. [Google Scholar] [CrossRef] - Darwin, C. On the Origin of Species by Means of Natural Selection, or the Preservation of Favored Races in the Struggle for Life; Harvard University Press Cambridge: Cambridge, MA, USA, 1859. [Google Scholar]

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Vanchurin, V.
The World as a Neural Network. *Entropy* **2020**, *22*, 1210.
https://doi.org/10.3390/e22111210

**AMA Style**

Vanchurin V.
The World as a Neural Network. *Entropy*. 2020; 22(11):1210.
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**Chicago/Turabian Style**

Vanchurin, Vitaly.
2020. "The World as a Neural Network" *Entropy* 22, no. 11: 1210.
https://doi.org/10.3390/e22111210