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Mutual information, neural networks and the renormalization group

Nature Physicsvolume 14pages578582 (2018) | Download Citation


Physical systems differing in their microscopic details often display strikingly similar behaviour when probed at macroscopic scales. Those universal properties, largely determining their physical characteristics, are revealed by the powerful renormalization group (RG) procedure, which systematically retains ‘slow’ degrees of freedom and integrates out the rest. However, the important degrees of freedom may be difficult to identify. Here we demonstrate a machine-learning algorithm capable of identifying the relevant degrees of freedom and executing RG steps iteratively without any prior knowledge about the system. We introduce an artificial neural network based on a model-independent, information-theoretic characterization of a real-space RG procedure, which performs this task. We apply the algorithm to classical statistical physics problems in one and two dimensions. We demonstrate RG flow and extract the Ising critical exponent. Our results demonstrate that machine-learning techniques can extract abstract physical concepts and consequently become an integral part of theory- and model-building.

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We thank S. Huber and P. Fendley for discussions. M.K.-J. gratefully acknowledges the support of the Swiss National Science Foundation. Z.R. was supported by the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement no. 657111.

Author information


  1. Institute for Theoretical Physics, ETH Zurich, Zurich, Switzerland

    • Maciej Koch-Janusz
  2. Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem, Israel

    • Zohar Ringel


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M.K.-J. and Z.R. contributed equally to this work.

Competing interests

The authors declare no competing interests.

Corresponding author

Correspondence to Maciej Koch-Janusz.

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    Methods, Supplementary Figures 1–6, Supplementary References 1–12

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