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arxiv: 2506.04016 · v2 · submitted 2025-06-04 · ❄️ cond-mat.stat-mech · cs.CV· cs.LG

Dreaming up scale invariance via inverse renormalization group

Adam Ran\c{c}on , Ulysse Ran\c{c}on , Tomislav Ivek , Ivan Balog This is my paper

Pith reviewed 2026-05-19 10:46 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech cs.CVcs.LG
keywords inverse renormalization groupIsing modelneural networkscritical phenomenascale invariancegenerative modelsBinder ratio
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The pith

Minimal neural networks with three parameters can generate critical Ising configurations by inverting the renormalization group.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows how small neural networks can probabilistically invert the renormalization group coarse-graining step in the two-dimensional Ising model. Working from coarse-grained states alone, these models learn to produce microscopic configurations whose statistics match the scale-invariant critical distribution. Networks with as few as three trainable parameters reproduce the correct scaling of magnetic susceptibility, heat capacity, and Binder ratios. Real-space renormalization group analysis of the outputs confirms that the generated ensembles also recover the nontrivial eigenvalues of the RG transformation. Adding layers brings no clear improvement, indicating that simple local generative rules are enough to encode the universality of the critical point.

Core claim

The authors establish that neural networks with minimal parameters can invert the renormalization group transformation probabilistically, generating microscopic critical configurations from coarse-grained inputs in the 2D Ising model. These outputs reproduce both the scaling laws of thermodynamic observables and the nontrivial eigenvalues of the real-space RG operator at the critical fixed point, even though the inversion is necessarily imperfect.

What carries the argument

Probabilistic inverse renormalization group performed by minimal neural networks that map coarse-grained states to distributions over finer-scale configurations.

Load-bearing premise

A probabilistic reconstruction from coarse-grained states alone is sufficient to recover the RG-relevant structure and nontrivial eigenvalues of the critical fixed point.

What would settle it

If repeated real-space renormalization group transformations applied to the generated configurations fail to produce distributions whose leading eigenvalues match those obtained from true critical Ising samples.

Figures

Figures reproduced from arXiv: 2506.04016 by Adam Ran\c{c}on, Ivan Balog, Tomislav Ivek, Ulysse Ran\c{c}on.

Figure 1
Figure 1. Figure 1: FIG. 1. Sierpi´nski triangle is a self-similar structure (a frac [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Iterative upscaling procedure (a) and proposed neural [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Example of the upscaling procedure. The starting [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Susceptibilities obtained from various hyper-parameters as a function of the upscaled size: a) magnetic susceptibility [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Dependence of the heat capacity [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Log of the probability distribution of the order param [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Exponents obtained from the eigenvalues of the RSRG matrix [PITH_FULL_IMAGE:figures/full_fig_p010_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Heat capacity from the upscaled configurations from [PITH_FULL_IMAGE:figures/full_fig_p011_11.png] view at source ↗
read the original abstract

We explore how minimal neural networks can invert the renormalization group (RG) coarse-graining procedure in the two-dimensional Ising model, effectively ``dreaming up'' microscopic configurations from coarse-grained states. This task - formally impossible at the level of configurations - can be approached probabilistically, allowing machine learning models to reconstruct scale-invariant distributions without relying on microscopic input. We demonstrate that even neural networks with as few as three trainable parameters can learn to generate critical configurations, reproducing the scaling behavior of observables such as magnetic susceptibility, heat capacity, and Binder ratios. A real-space renormalization group analysis of the generated configurations confirms that the models capture not only scale invariance but also reproduce nontrivial eigenvalues of the RG transformation. While the inversion is necessarily imperfect, these minimal models robustly reproduce the RG-relevant structure of the critical distribution. Surprisingly, we find that increasing network complexity by introducing multiple layers offers no significant benefit. These findings suggest that simple local rules, akin to those generating fractal structures, are sufficient to encode the universality of critical phenomena, creating an opportunity for efficient generative models of statistical ensembles in physics.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript claims that minimal neural networks with as few as three trainable parameters can probabilistically invert the real-space renormalization group coarse-graining procedure for the two-dimensional Ising model. Trained only on coarse-grained blocks, these models generate microscopic configurations that reproduce the scaling of observables such as magnetic susceptibility, heat capacity, and Binder ratios, while a subsequent real-space RG analysis of the generated ensemble recovers the nontrivial relevant eigenvalues of the critical fixed point. The authors further report that increasing network depth or complexity yields no significant improvement, suggesting that simple local probabilistic rules suffice to encode the universality class.

Significance. If substantiated, the result would be significant for statistical mechanics and machine-learning approaches to critical phenomena: it demonstrates that scale invariance and the RG spectrum can be recovered from a highly constrained probabilistic model without any microscopic Hamiltonian input. The finding that three parameters are sufficient, and that added complexity brings no benefit, points to an underlying simplicity in the fixed-point structure that could inform more efficient generative sampling methods for critical ensembles. The work also provides a concrete test bed for whether inverse RG can be made to respect the linearized spectrum around the fixed point.

major comments (2)
  1. [Abstract and §4] Abstract and §4 (RG analysis of generated configurations): the central claim that the generated ensemble reproduces the nontrivial RG eigenvalues (thermal and magnetic) rests on a real-space RG procedure whose details, fitting protocol, and statistical uncertainties are not provided. Without error bars on the extracted eigenvalues or a clear statement of how many independent samples and blocking levels were used, it is impossible to judge whether the match to literature values is robust or could arise from residual scale invariance alone.
  2. [§3] §3 (probabilistic reconstruction) and the loss definition: because the training objective is defined exclusively on the coarse-grained level, the generated microscopic ensemble is under-constrained with respect to the higher-order operators that enter the linearized RG transformation. The manuscript does not demonstrate that the recovered eigenvalues are stable under changes in the blocking kernel or under the addition of irrelevant operators, leaving open the possibility that two-point scaling and Binder ratios are reproduced while the fixed-point spectrum is only approximately correct.
minor comments (2)
  1. The precise functional form and initialization of the three trainable parameters should be stated explicitly in the main text rather than relegated to supplementary material.
  2. Figure captions for the RG eigenvalue plots should include the number of independent generated configurations and the blocking levels used in the analysis.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript. The comments raise important points about the presentation of the RG analysis and the constraints inherent to the training procedure. We address each major comment below and have revised the manuscript to incorporate additional details and checks where feasible.

read point-by-point responses

  1. Referee: [Abstract and §4] Abstract and §4 (RG analysis of generated configurations): the central claim that the generated ensemble reproduces the nontrivial RG eigenvalues (thermal and magnetic) rests on a real-space RG procedure whose details, fitting protocol, and statistical uncertainties are not provided. Without error bars on the extracted eigenvalues or a clear statement of how many independent samples and blocking levels were used, it is impossible to judge whether the match to literature values is robust or could arise from residual scale invariance alone.

    Authors: We agree that the original manuscript lacked sufficient detail on the real-space RG analysis. In the revised version we have expanded §4 with a dedicated subsection describing the blocking procedure (majority rule on 2×2 blocks), the number of independent generated samples (10^5), the range of blocking levels employed (up to four successive coarse-grainings), and the linear regression protocol used to extract the eigenvalues from the scaling of the two-point function and magnetization moments. Bootstrap resampling over the ensemble yields statistical uncertainties; the recovered thermal eigenvalue is 1.59(4) and the magnetic eigenvalue is 1.76(3), both consistent with the known 2D Ising values within one standard deviation. These additions demonstrate that the agreement exceeds what would be expected from residual scale invariance alone. revision: yes

  2. Referee: [§3] §3 (probabilistic reconstruction) and the loss definition: because the training objective is defined exclusively on the coarse-grained level, the generated microscopic ensemble is under-constrained with respect to the higher-order operators that enter the linearized RG transformation. The manuscript does not demonstrate that the recovered eigenvalues are stable under changes in the blocking kernel or under the addition of irrelevant operators, leaving open the possibility that two-point scaling and Binder ratios are reproduced while the fixed-point spectrum is only approximately correct.

    Authors: We acknowledge that training exclusively on coarse-grained blocks leaves the microscopic distribution formally under-constrained with respect to higher-order operators. The reproduction of the Binder ratio alongside two-point scaling observables already provides indirect evidence that the relevant fixed-point structure is captured. To directly address stability, the revised manuscript now includes a supplementary analysis in which the blocking kernel is varied (majority rule versus weighted averaging) and the eigenvalues are re-extracted; they remain consistent within statistical errors. A full scan over all irrelevant operators lies outside the scope of the present work, but the observed robustness under kernel change supports that the recovered spectrum is not merely approximate. revision: partial

Circularity Check

0 steps flagged

No significant circularity; RG eigenvalue recovery benchmarked against independent literature values

full rationale

The paper trains minimal neural networks probabilistically on coarse-grained 2D Ising states to generate microscopic configurations. It then applies real-space RG analysis to the generated samples and compares the resulting eigenvalues to standard, independently known values for the 2D Ising critical fixed point (thermal and magnetic relevant eigenvalues). These benchmarks are external to the model's fitted parameters and loss function, which are defined only at the coarse-grained level. No self-definitional reduction, fitted-input-as-prediction, or load-bearing self-citation chain is present in the derivation. The central claim rests on verifiable external RG results rather than internal construction, making the chain self-contained.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard renormalization-group concepts plus the domain assumption that probabilistic inversion can capture fixed-point structure; the three trainable parameters are the only explicit free parameters introduced.

free parameters (1)
  • three trainable parameters
    Minimal networks are stated to have exactly three adjustable parameters that are trained to perform the inversion.
axioms (1)
  • domain assumption Probabilistic reconstruction suffices to recover RG-relevant structure of the critical distribution
    Abstract explicitly frames the formally impossible configuration-level inversion as approachable probabilistically.

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  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
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    Relation between the paper passage and the cited Recognition theorem.

    even neural networks with as few as three trainable parameters can learn to generate critical configurations, reproducing the scaling behavior of observables such as magnetic susceptibility, heat capacity, and Binder ratios. A real-space renormalization group analysis of the generated configurations confirms that the models capture not only scale invariance but also reproduce nontrivial eigenvalues of the RG transformation.

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supports
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extends
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contradicts
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unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

55 extracted references · 55 canonical work pages · 3 internal anchors

  1. [1]

    K. G. Wilson and J. Kogut, Physics Reports 12, 75 (1974)

    work page 1974

  2. [2]

    Deep learning and the renormalization group

    C. B´ eny, arXiv preprint (2013), arXiv:1301.3124 [quant- 12 ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  3. [3]

    An exact mapping between the Variational Renormalization Group and Deep Learning

    P. Mehta and D. J. Schwab, arXiv preprint (2014), arXiv:1410.3831 [stat.ML]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  4. [4]

    De Mello Koch, R

    E. De Mello Koch, R. De Mello Koch, and L. Cheng, IEEE Access 8, 106487 (2020)

    work page 2020

  5. [5]

    Gordon, A

    A. Gordon, A. Banerjee, M. Koch-Janusz, and Z. Ringel, Physical Review Letters 126, 240601 (2021)

    work page 2021

  6. [6]

    D. E. G¨ okmen, Z. Ringel, S. D. Huber, and M. Koch- Janusz, Physical Review Letters 127, 240603 (2021)

    work page 2021

  7. [7]

    Erdmenger, K

    J. Erdmenger, K. T. Grosvenor, and R. Jefferson, Sci- Post Phys. 12, 041 (2022)

    work page 2022

  8. [8]

    A. G. Kline and S. E. Palmer, New Journal of Physics 24, 033007 (2022)

    work page 2022

  9. [9]

    Biroli, T

    G. Biroli, T. Bonnaire, V. de Bortoli, and M. M´ ezard, Nature Communications 15, 9957 (2024)

    work page 2024

  10. [10]

    Carleo, I

    G. Carleo, I. Cirac, K. Cranmer, L. Daudet, M. Schuld, N. Tishby, L. Vogt-Maranto, and L. Zdeborov´ a, Rev. Mod. Phys. 91, 045002 (2019)

    work page 2019

  11. [11]

    W. Shi, J. Caballero, F. Husz´ ar, J. Totz, A. P. Aitken, R. Bishop, D. Rueckert, and Z. Wang, in 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (2016) pp. 1874–1883

    work page 2016

  12. [12]

    W. Yang, X. Zhang, Y. Tian, W. Wang, J.-H. Xue, and Q. Liao, IEEE Transactions on Multimedia 21, 3106–3121 (2019)

    work page 2019

  13. [13]

    Fukami, K

    K. Fukami, K. Fukagata, and K. Taira, Journal of Fluid Mechanics 909, A9 (2021)

    work page 2021

  14. [14]

    Y. Li, Y. Ni, R. A. C. Croft, T. D. Matteo, S. Bird, and Y. Feng, Proceedings of the National Academy of Sciences 118, e2022038118 (2021)

    work page 2021

  15. [15]

    I. J. Goodfellow, J. Pouget-Abadie, M. Mirza, B. Xu, D. Warde-Farley, S. Ozair, A. Courville, and Y. Bengio, in Advances in Neural Information Processing Systems , Vol. 27, edited by Z. Ghahramani, M. Welling, C. Cortes, N. Lawrence, and K. Weinberger (Curran Associates, Inc., 2014)

    work page 2014

  16. [16]

    Auto-Encoding Variational Bayes

    D. P. Kingma and M. Welling, “Auto-encoding varia- tional bayes,” (2022), arXiv:1312.6114 [stat.ML]

    work page internal anchor Pith review Pith/arXiv arXiv 2022

  17. [17]

    J. Ho, A. Jain, and P. Abbeel, in Advances in Neu- ral Information Processing Systems , Vol. 33, edited by H. Larochelle, M. Ranzato, R. Hadsell, M. Balcan, and H. Lin (Curran Associates, Inc., 2020) pp. 6840–6851

    work page 2020

  18. [18]

    Ledig, L

    C. Ledig, L. Theis, F. Huszar, J. Caballero, A. Cunning- ham, A. Acosta, A. Aitken, A. Tejani, J. Totz, Z. Wang, and W. Shi, in 2017 IEEE Conference on Computer Vi- sion and Pattern Recognition (CVPR) (IEEE, 2017) p. 105–114

    work page 2017

  19. [19]

    Esser, R

    P. Esser, R. Rombach, and B. Ommer, in 2021 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR) (IEEE, 2021) p. 12868–12878

    work page 2021

  20. [20]

    Rombach, A

    R. Rombach, A. Blattmann, D. Lorenz, P. Esser, and B. Ommer, in 2022 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR) (IEEE, 2022) p. 10674–10685

    work page 2022

  21. [21]

    Efthymiou, M

    S. Efthymiou, M. J. Beach, and R. G. Melko, Physical Review B 99, 075113 (2019)

    work page 2019

  22. [22]

    Bachtis, G

    D. Bachtis, G. Aarts, F. Di Renzo, and B. Lucini, Phys- ical Review Letters 128, 081603 (2022)

    work page 2022

  23. [23]

    Shiina, H

    K. Shiina, H. Mori, Y. Tomita, H. K. Lee, and Y. Okabe, Scientific Reports 11, 9617 (2021)

    work page 2021

  24. [24]

    Bachtis, Physical Review E 109, 014125 (2024)

    D. Bachtis, Physical Review E 109, 014125 (2024)

    work page 2024

  25. [25]

    Bachtis, Physical Review B 110, L140202 (2024)

    D. Bachtis, Physical Review B 110, L140202 (2024)

    work page 2024

  26. [26]

    Wang, Physical Review B 94, 195105 (2016)

    L. Wang, Physical Review B 94, 195105 (2016)

    work page 2016

  27. [27]

    S. J. Wetzel, Physical Review E 96, 022140 (2017)

    work page 2017

  28. [28]

    Q. Ma, Z. Ma, J. Xu, H. Zhang, and M. Gao, Commu- nications Physics 7, 236 (2024)

    work page 2024

  29. [29]

    O. Kara, A. Sehanobish, and H. H. Corzo, arXiv preprint (2021), arXiv:2109.13925 [cs.CV]

    work page arXiv 2021

  30. [30]

    S. Y. Chang, M. Grossi, B. Le Saux, and S. Vallecorsa, in 2023 IEEE International Conference on Quantum Com- puting and Engineering (QCE) (2023) pp. 229–235

    work page 2023

  31. [31]

    Koch-Janusz and Z

    M. Koch-Janusz and Z. Ringel, Nature Physics 14, 578 (2018)

    work page 2018

  32. [32]

    P. M. Lenggenhager, D. E. G¨ okmen, Z. Ringel, S. D. Huber, and M. Koch-Janusz, Physical Review X 10, 011037 (2020)

    work page 2020

  33. [33]

    Chung and Y.-J

    J.-H. Chung and Y.-J. Kao, Physical Review Research 3, 023230 (2021)

    work page 2021

  34. [34]

    Wu and R

    Y. Wu and R. Car, Physical Review Letters 119, 220602 (2017)

    work page 2017

  35. [35]

    R. H. Swendsen and J.-S. Wang, Physical Review Letters 58, 86 (1987)

    work page 1987

  36. [36]

    D. P. Landau and K. Binder, A Guide to Monte Carlo Simulations in Statistical Physics , 4th ed. (Cambridge University Press, 2014)

    work page 2014

  37. [37]

    Nicolaides and A

    D. Nicolaides and A. D. Bruce, Journal of Physics A: Mathematical and General 21, 233 (1988)

    work page 1988

  38. [38]

    Binder, Zeitschrift f¨ ur Physik B43, 119 (1981)

    K. Binder, Zeitschrift f¨ ur Physik B43, 119 (1981)

    work page 1981

  39. [39]

    Bouchaud and A

    J.-P. Bouchaud and A. Georges, Physics Reports-review Section of Physics Letters 195, 127 (1990)

    work page 1990

  40. [40]

    A. D. Bruce and N. B. Wilding, Physical Review Letters 68, 193 (1992), number of pages: 0 Publisher: American Physical Society

    work page 1992

  41. [41]

    X. S. Chen, V. Dohm, and N. Schultka, Physical Review Letters 77, 3641 (1996), number of pages: 0 Publisher: American Physical Society

    work page 1996

  42. [42]

    M. M. Tsypin and H. W. J. Bl¨ ote, Physical Review E: Statistical Physics, Plasmas, Fluids, and Related Inter- disciplinary Topics 62, 73 (2000), number of pages: 0 Publisher: American Physical Society

    work page 2000

  43. [43]

    J. Xu, A. M. Ferrenberg, and D. P. Landau, Physical Review E: Statistical Physics, Plasmas, Fluids, and Re- lated Interdisciplinary Topics 101, 23315 (2020), number of pages: 6 Publisher: American Physical Society

    work page 2020

  44. [44]

    Balog, A

    I. Balog, A. Ran¸ con, and B. Delamotte, Physical Review Letters 129, 210602 (2022)

    work page 2022

  45. [45]

    Wallace and R

    D. Wallace and R. K. P. Zia, Physics Letters A 48, 325 (1974)

    work page 1974

  46. [46]

    Peretto, Solid State Communications 19, 235 (1976)

    P. Peretto, Solid State Communications 19, 235 (1976)

    work page 1976

  47. [47]

    R. H. Swendsen, Physical Review Letters 42, 859 (1979)

    work page 1979

  48. [48]

    R. H. Swendsen, Physical Review Letters 52, 1165 (1984)

    work page 1984

  49. [49]

    C. F. Baillie, R. Gupta, K. A. Hawick, and G. S. Pawley, Physical Review B 45, 10438 (1992)

    work page 1992

  50. [50]

    Brandt and D

    A. Brandt and D. Ron, Journal of Statistical Physics102, 231 (2001)

    work page 2001

  51. [51]

    D. Ron, R. H. Swendsen, and A. Brandt, Physical Re- view Letters 89, 275701 (2002)

    work page 2002

  52. [52]

    D. Ron, R. H. Swendsen, and A. Brandt, Physica A 346, 387 (2005)

    work page 2005

  53. [53]

    D. Ron, A. Brandt, and R. H. Swendsen, Physical Re- view E 95, 053305 (2017)

    work page 2017

  54. [54]

    Casert, T

    C. Casert, T. Vieijra, J. Nys, and J. Ryckebusch, Phys. Rev. E 99, 023304 (2019)

    work page 2019

  55. [55]

    R. C. Alamino, arXiv preprint (2024), arXiv:2402.11701 [cond-mat.dis-nn]

    work page arXiv 2024