Mutual information, neural networks and the renormalization group


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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Fig. 1: The RSMI algorithm.
Fig. 2: The weights of the RSMI network trained on the Ising model.
Fig. 3: The dimer model.
Fig. 4: The weights of the RSMI network trained on dimer model data.
Fig. 5: RG flow for the 2D Ising model.


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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.

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

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Correspondence to Maciej Koch-Janusz.

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

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Koch-Janusz, M., Ringel, Z. Mutual information, neural networks and the renormalization group. Nature Phys 14, 578–582 (2018).

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