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Emergent Quantumness in Neural Networks

Foundations of Physics volume 51, Article number: 94 (2021) Cite this article

Abstract

It was recently shown that the Madelung equations, that is, a hydrodynamic form of the Schrödinger equation, can be derived from a canonical ensemble of neural networks where the quantum phase was identified with the free energy of hidden variables. We consider instead a grand canonical ensemble of neural networks, by allowing an exchange of neurons with an auxiliary subsystem, to show that the free energy must also be multivalued. By imposing the multivaluedness condition on the free energy we derive the Schrödinger equation with “Planck’s constant” determined by the chemical potential of hidden variables. This shows that quantum mechanics provides a correct statistical description of the dynamics of the grand canonical ensemble of neural networks at the learning equilibrium. We also discuss implications of the results for machine learning, fundamental physics and, in a more speculative way, evolutionary biology.

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Notes

  1. MIK thanks Grigory Volovik for emphasizing this connection at our old discussions of the logical inference approach.

  2. We thank Nikolay Mikhailovsky for pointing out the connection to Occam’s razor principle.

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Acknowledgements

VV was supported in part by the Foundational Questions Institute (FQXi). MIK acknowledges a support by NWO via Spinoza Prize.

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Authors and Affiliations

  1. Institute for Molecules and Materials, Radboud University, Heyendaalseweg 135, 6525 AJ, Nijmegen, The Netherlands

    Mikhail I. Katsnelson

  2. Department of Physics, University of Minnesota, Duluth, MN, 55812, USA

    Vitaly Vanchurin

  3. Duluth Institute for Advanced Study, Duluth, MN, 55804, USA

    Vitaly Vanchurin

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  1. Mikhail I. Katsnelson
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  2. Vitaly Vanchurin
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Correspondence to Vitaly Vanchurin.

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Katsnelson, M.I., Vanchurin, V. Emergent Quantumness in Neural Networks. Found Phys 51, 94 (2021). https://doi.org/10.1007/s10701-021-00503-3

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  • DOI: https://doi.org/10.1007/s10701-021-00503-3