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arxiv: 2604.10209 · v1 · submitted 2026-04-11 · ✦ hep-lat · cond-mat.stat-mech· physics.comp-ph

Recognition: 2 theorem links

· Lean Theorem

Scalable Generative Sampling and Multilevel Estimation for Lattice Field Theories Near Criticality

A. Singha , J. Kauffmann , E. Cellini , K. Jansen , S. Nakajima
Authors on Pith no claims yet

Pith reviewed 2026-05-10 15:57 UTC · model grok-4.3

classification ✦ hep-lat cond-mat.stat-mechphysics.comp-ph
keywords lattice field theorycritical slowing downgenerative samplingmultilevel Monte Carlonormalizing flowsphi^4 theoryrenormalization group
0
0 comments X

The pith

A multiscale generative sampler models the Boltzmann distribution of lattice field theories near criticality through a coarse-to-fine hierarchy of conditional distributions.

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

The paper addresses critical slowing down that makes standard Markov chain sampling inefficient for lattice field theories on large volumes near their critical points. It constructs a generative model that samples from coarse to fine scales, using a conditional Gaussian mixture at each level to capture the main local dependence of new variables on the already-sampled coarse field and a masked continuous normalizing flow to refine the remaining conditional structure. Because the architecture preserves coarse fields exactly during refinement, it supplies exact restriction maps at no extra cost and directly supports unbiased Multilevel Monte Carlo variance reduction. For the two-dimensional scalar phi-four theory at criticality the method produces integrated autocorrelation times orders of magnitude smaller than Hybrid Monte Carlo on large volumes while maintaining high importance-sampling efficiency and reproducing unbiased physical observables in statistical agreement with long HMC runs.

Core claim

The authors establish that a hierarchy of conditional distributions across length scales, with each refinement level employing a conditional Gaussian mixture for dominant local dependence and a masked continuous normalizing flow for residual structure, generates unbiased samples from the critical Boltzmann distribution, yields integrated autocorrelation times orders of magnitude below those of Hybrid Monte Carlo on large volumes, and furnishes exact restriction maps for unbiased Multilevel Monte Carlo at negligible additional cost.

What carries the argument

The coarse-to-fine generative hierarchy that exactly preserves sampled coarse fields at each refinement level, thereby providing exact restriction maps for multilevel Monte Carlo without extra computation.

If this is right

  • Integrated autocorrelation times become orders of magnitude smaller than Hybrid Monte Carlo on large volumes near criticality.
  • Exact restriction maps are obtained at no additional computational cost, enabling direct unbiased Multilevel Monte Carlo variance reduction.
  • Importance-sampling efficiency remains high relative to other generative baselines.
  • Physical observables are reproduced without bias and agree statistically with long Hybrid Monte Carlo simulations.
  • The multiscale construction scales sampling efficiency with lattice volume by isolating long-wavelength modes at coarse levels.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same coarse-to-fine architecture with exact restriction maps could be tested on three-dimensional scalar theories or gauge theories where critical slowing down is more severe.
  • The method's preservation of coarse fields suggests straightforward combination with other variance-reduction techniques such as multilevel importance sampling.
  • Because the conditional models are learned independently at each scale, the approach may extend to dynamical simulations that require consistent multiscale updates.
  • Direct measurement of how the learned conditional distributions deviate from the true ones on small test lattices would quantify the approximation error before scaling to production volumes.

Load-bearing premise

The learned conditional Gaussian mixture plus masked continuous normalizing flow at every refinement level must accurately represent the true conditional distribution of the fine field given the coarse field.

What would settle it

A statistically significant discrepancy in any unbiased observable, such as susceptibility or correlation length, between the generative samples and independent long Hybrid Monte Carlo runs on the same large critical lattices would falsify the claim of unbiased sampling.

Figures

Figures reproduced from arXiv: 2604.10209 by A. Singha, E. Cellini, J. Kauffmann, K. Jansen, S. Nakajima.

Figure 1
Figure 1. Figure 1: FIG. 1. Kadanoff-inspired multilevel partition of a periodic [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Binder cumulant [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Relative deviation of the connected susceptibility [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. MLMC vs plain IS variance at equal budget [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. HMC integrated autocorrelation time [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: shows the IS efficiency ESS/N at N = 105 samples for all four methods. Our method maintains ESS/N ≥ 0.67 up to L = 64 and 0.19 at L = 128, while SR-NF drops to 0.06% at L = 128 and colapses to 0.010% at L = 256. Dense CNF is competitive up to L = 64 (ESS/N = 0.68) but cannot be trained at larger volumes. Hutch CNF achieves near-zero efficiency even at small L (0.5% at L = 16), reflecting mode collapses and… view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Training ESS [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
read the original abstract

Sampling lattice field theories near criticality is severely hindered by critical slowing down, which makes standard Markov chain methods increasingly inefficient at large lattice volumes. We introduce a multiscale generative sampler, inspired by renormalization-group ideas, that models the Boltzmann distribution through a coarse-to-fine hierarchy across length scales. At each level, a conditional Gaussian mixture model captures the main local dependence of newly introduced variables on the already-sampled coarse field, while a masked continuous normalizing flow refines the remaining conditional structure. Coarse levels encode the dominant long-wavelength modes, and finer levels progressively add short-distance fluctuations. In addition, because the architecture preserves coarse fields exactly during refinement, it provides exact restriction maps at no additional computational cost and directly enables unbiased Multilevel Monte Carlo (MLMC) variance reduction. For the two-dimensional scalar $\phi^4$ theory at criticality, the method achieves integrated autocorrelation times orders of magnitude smaller than Hybrid Monte Carlo (HMC) on large volumes, maintains high importance-sampling efficiency relative to other generative baselines, and reproduces unbiased physical observables in statistical agreement with long HMC simulations.

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

1 major / 2 minor

Summary. The paper introduces a multiscale generative sampler for lattice field theories near criticality, modeling the Boltzmann distribution via a coarse-to-fine hierarchy. At each level, a conditional Gaussian mixture captures local dependence on the coarse field, refined by a masked continuous normalizing flow. The architecture preserves coarse fields exactly, enabling unbiased Multilevel Monte Carlo (MLMC) variance reduction. For the 2D scalar φ⁴ theory at criticality, it claims integrated autocorrelation times orders of magnitude smaller than HMC on large volumes, high importance-sampling efficiency relative to other generative baselines, and reproduction of unbiased physical observables in statistical agreement with long HMC runs.

Significance. If the unbiasedness and efficiency claims hold, this represents a meaningful advance in mitigating critical slowing down for lattice simulations. The exact restriction maps and direct enablement of MLMC are technical strengths that could reduce computational costs for large-volume studies, with potential applicability beyond φ⁴ to other critical theories. The approach combines generative modeling with multilevel estimation in a way that, if validated, offers a scalable alternative to standard MCMC methods.

major comments (1)
  1. [Multilevel estimation and generative hierarchy] The unbiasedness claim for physical observables (Abstract) depends on the learned conditional distributions at each level inducing marginals that are consistent with the coarse-level model after exact restriction. Independent training of the conditional Gaussian mixture + masked CNF at each refinement level does not automatically enforce this cross-level consistency; any approximation error or mode mismatch would bias the telescoping MLMC estimator even before importance weights. The manuscript must provide explicit verification (e.g., moment or distribution comparisons between restricted fine-level samples and independently generated coarse samples) to support the central unbiasedness assertion.
minor comments (2)
  1. The abstract reports statistical agreement with HMC and large autocorrelation reduction but omits quantitative details on training convergence, effective sample size, specific autocorrelation time ratios, or importance-sampling efficiency metrics; these should be added with supporting tables or figures.
  2. Clarify implementation specifics of the masked continuous normalizing flow and the exact form of the conditional Gaussian mixture (e.g., number of components, masking strategy) to aid reproducibility.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thorough review and for identifying this key technical point on cross-level consistency in the hierarchical generative model. We address the concern directly below and will revise the manuscript to include the requested verification.

read point-by-point responses

  1. Referee: The unbiasedness claim for physical observables (Abstract) depends on the learned conditional distributions at each level inducing marginals that are consistent with the coarse-level model after exact restriction. Independent training of the conditional Gaussian mixture + masked CNF at each refinement level does not automatically enforce this cross-level consistency; any approximation error or mode mismatch would bias the telescoping MLMC estimator even before importance weights. The manuscript must provide explicit verification (e.g., moment or distribution comparisons between restricted fine-level samples and independently generated coarse samples) to support the central unbiasedness assertion.

    Authors: We agree that independent per-level training does not automatically guarantee marginal consistency and that explicit checks are necessary to rigorously support the unbiasedness of the MLMC estimator. The architecture does ensure that coarse fields are preserved exactly under refinement, providing an exact restriction map at no extra cost. However, to confirm that the learned conditionals induce consistent marginals, we will add in the revised manuscript direct comparisons at each hierarchy level: (i) low-order moments (mean, variance, and two-point functions) and (ii) where computationally feasible, binned distribution comparisons or estimated KL divergences between (a) coarse fields obtained by exact restriction of fine-level samples and (b) independent samples generated at the coarse level. These verifications will be presented alongside the existing observable comparisons with HMC, which already indicate that any residual mismatch is statistically negligible for the observables of interest. We believe this addition will fully address the referee's concern while preserving the central claim that the exact restriction enables unbiased MLMC. revision: yes

Circularity Check

0 steps flagged

No significant circularity; claims rest on empirical validation against external HMC baselines

full rationale

The paper's core construction uses exact restriction maps (coarse field preserved by design) to enable MLMC, with each level's conditional Gaussian mixture + masked CNF trained independently on the target action. The reported performance gains (autocorrelation times, efficiency) and unbiased observables are demonstrated via direct numerical comparison to independent long HMC runs and other generative baselines, not by re-deriving quantities from the model's own fitted parameters. No equation reduces a claimed result to an input by construction, no self-citation chain bears the central claim, and the unbiasedness statement is qualified by statistical agreement with external reference simulations rather than asserted as automatic from architecture alone. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard probabilistic modeling assumptions and the existence of a well-defined Boltzmann distribution; no new physical entities or ad-hoc constants are introduced beyond learned model parameters.

axioms (2)
  • domain assumption The target distribution is the Boltzmann distribution exp(-S[φ]) for the lattice action S.
    Invoked in the abstract as the distribution to be sampled.
  • domain assumption Conditional distributions at each scale can be modeled by Gaussian mixtures followed by continuous normalizing flows.
    Core modeling choice stated in the method description.

pith-pipeline@v0.9.0 · 5509 in / 1321 out tokens · 36079 ms · 2026-05-10T15:57:40.208374+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

89 extracted references · 39 canonical work pages · 3 internal anchors

  1. [1]

    K. G. Wilson and J. Kogut, Phys. Rep.12, 75 (1974)

    1974

  2. [2]

    Creutz,Quarks, gluons and lattices(Cambridge Uni- versity Press, 1983)

    M. Creutz,Quarks, gluons and lattices(Cambridge Uni- versity Press, 1983)

    1983

  3. [3]

    Creutz, L

    M. Creutz, L. Jacobs, and C. Rebbi, Physics Reports95, 201 (1983)

    1983

  4. [4]

    Aoki et al., FLAG review 2024, Phys

    Y. Aokiet al.(Flavour Lattice Averaging Group (FLAG)), Phys. Rev. D113, 014508 (2026), arXiv:2411.04268 [hep-lat]

    work page arXiv 2026

  5. [5]

    Aoyamaet al., Phys

    T. Aoyamaet al., Phys. Rept.887, 1 (2020), arXiv:2006.04822 [hep-ph]

    work page arXiv 2020

  6. [6]

    Alibertiet al., The anomalous magnetic moment of the muon in the Standard Model: An update, Physics Reports 1143, 1 (2025), arXiv:2505.21476 [hep-ph]

    R. Alibertiet al., Phys. Rept.1143, 1 (2025), arXiv:2505.21476 [hep-ph]

    work page arXiv 2025

  7. [7]

    Wolff, Nuclear Physics B17, 93 (1990)

    U. Wolff, Nuclear Physics B17, 93 (1990)

    1990

  8. [8]

    Wolff, Comp

    U. Wolff, Comp. Phys. Comm.156, 143 (2004)

    2004

  9. [9]

    Schaefer, R

    S. Schaefer, R. Sommer, and F. Virotta (ALPHA), Nucl. Phys. B845, 93 (2011), arXiv:1009.5228 [hep-lat]

    work page arXiv 2011

  10. [10]

    E. G. Tabak and E. Vanden-Eijnden, Communications in Mathematical Sciences8, 217 (2010)

    2010

  11. [11]

    E. G. Tabak and C. V. Turner, Communications on Pure and Applied Mathematics66, 145 (2013)

    2013

  12. [12]

    Rezende and S

    D. Rezende and S. Mohamed, inProceedings of the 32nd International Conference on Machine Learning, Vol. 37 (2015) pp. 1530–1538

    2015

  13. [13]

    R. T. Chen, Y. Rubanova, J. Bettencourt, and D. K. Du- venaud, Advances in neural information processing sys- tems31(2018), arXiv:1806.07366 [cs.LG]

    work page arXiv 2018

  14. [14]

    M. S. Albergo, G. Kanwar, and P. E. Shanahan, Phys. Rev. D100, 034515 (2019)

    2019

  15. [15]

    Kanwar, M

    G. Kanwar, M. S. Albergo, D. Boyda, K. Cranmer, D. C. Hackett, S. Racani` ere, D. J. Rezende, and P. E. Shanahan, Phys. Rev. Lett.125, 121601 (2020), arXiv:2003.06413 [hep-lat]

    work page arXiv 2020

  16. [16]

    M. S. Albergo, G. Kanwar, S. Racani` ere, D. J. Rezende, J. M. Urban, D. Boyda, K. Cranmer, D. C. Hackett, and P. E. Shanahan, Phys. Rev. D104, 114507 (2021), arXiv:2106.05934 [hep-lat]

    work page arXiv 2021

  17. [17]

    K. A. Nicoli, S. Nakajima, N. Strodthoff, W. Samek, et al., Phys. Rev. E101, 023304 (2020)

    2020

  18. [18]

    K. A. Nicoli, C. J. Anders, L. Funcke, T. Hartung,et al., Phys. Rev. Lett.126, 032001 (2021)

    2021

  19. [19]

    Singha, D

    A. Singha, D. Chakrabarti, and V. Arora, Phys. Rev. D 107, 014512 (2023)

    2023

  20. [20]

    Singha, D

    A. Singha, D. Chakrabarti, and V. Arora, Phys. Rev. D 108, 074518 (2023)

    2023

  21. [21]

    Abbott, A

    R. Abbott, A. Botev, D. Boyda, D. C. Hackett, G. Kan- war, S. Racani` ere, D. J. Rezende, F. Romero-L´ opez, P. E. Shanahan, and J. M. Urban, Phys. Rev. D109, 094514 (2024), arXiv:2401.10874 [hep-lat]

    work page arXiv 2024

  22. [22]

    P. Tuo, Z. Zeng, J. Chen, and B. Cheng, Journal of Chem- ical Theory and Computation21, 11427 (2025)

    2025

  23. [23]

    Abbott, D

    R. Abbott, D. Boyda, Y. Fu, D. C. Hackett, G. Kan- war, F. Romero-L´ opez, P. E. Shanahan, and J. M. Urban, Variance reduction in lattice QCD observables via nor- malizing flows (2026), arXiv preprint, arXiv:2603.02984 [hep-lat]

    work page arXiv 2026

  24. [24]

    Improvement of Heatbath Algorithm in LFT using Generative models

    A. Faraz, A. Singha, D. Chakrabarti, S. Nakajima, and V. Arora, PoSLATTICE2024, 036 (2025), arXiv:2308.08615 [physics.comp-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  25. [25]

    Finkenrath, PoSLATTICE2023, 022 (2024)

    J. Finkenrath, PoSLATTICE2023, 022 (2024)

    2024

  26. [26]

    Kreit, A

    J. Kreit, A. Bulgarelli, L. Funcke, T. Luu, D. Schuh, S. Singh, and L. Verzichelli, in42th International Sym- posium on Lattice Field Theory(2026) arXiv:2601.18273 [cond-mat.str-el]

    work page arXiv 2026

  27. [27]

    Schuh, L

    D. Schuh, L. Funcke, J. Kreit, T. Luu, and S. Singh, arXiv (2026), arXiv:2603.18205 [cond-mat.str-el]

    work page arXiv 2026

  28. [28]

    Kanaujia, M

    V. Kanaujia, M. S. Scheurer, and V. Arora, SciPost Phys. 16, 132 (2024)

    2024

  29. [29]

    Bacchio, P

    S. Bacchio, P. Kessel, S. Schaefer, and L. Vaitl, Phys. Rev. D107, L051504 (2023), arXiv:2212.08469 [hep-lat]

    work page arXiv 2023

  30. [30]

    Gerdes, P

    M. Gerdes, P. de Haan, C. Rainone, R. Bondesan,et al., SciPost Physics15, 238 (2023)

    2023

  31. [31]

    Caselle, E

    M. Caselle, E. Cellini, and A. Nada, JHEP02, 048, arXiv:2307.01107 [hep-lat]

    work page arXiv

  32. [32]

    Gerdes, P

    M. Gerdes, P. de Haan, R. Bondesan, and M. C. N. Cheng, Phys. Rev. D112, 094516 (2025), arXiv:2410.13161 [hep-lat]

    work page arXiv 2025

  33. [33]

    Caselle, E

    M. Caselle, E. Cellini, A. Nada, and M. Panero, JHEP 07, 015, arXiv:2201.08862 [hep-lat]

    work page arXiv

  34. [34]

    Caselle, E

    M. Caselle, E. Cellini, and A. Nada, JHEP02, 090, arXiv:2409.15937 [hep-lat]

    work page arXiv

  35. [35]

    Bulgarelli, E

    A. Bulgarelli, E. Cellini, K. Jansen, S. K¨ uhn, A. Nada, S. Nakajima, K. A. Nicoli, and M. Panero, Phys. Rev. Lett.134, 151601 (2025), arXiv:2410.14466 [quant-ph]

    work page arXiv 2025

  36. [36]

    Bulgarelli, E

    A. Bulgarelli, E. Cellini, and A. Nada, Phys. Rev. D111, 074517 (2025), arXiv:2412.00200 [hep-lat]

    work page arXiv 2025

  37. [37]

    Scaling flow-based approaches for topology sampling in $\mathrm{SU}(3)$ gauge theory

    C. Bonanno, A. Bulgarelli, E. Cellini, A. Nada, D. Pan- falone, D. Vadacchino, and L. Verzichelli, JHEP04, 051, arXiv:2510.25704 [hep-lat]. 10

    work page internal anchor Pith review Pith/arXiv arXiv

  38. [38]

    D. Wu, L. Wang, and P. Zhang, Phys. Rev. Lett.122, 080602 (2019)

    2019

  39. [39]

    Singha, E

    A. Singha, E. Cellini, K. A. Nicoli, K. Jansen, S. K¨ uhn, and S. Nakajima, inThe Thirteenth International Conference on Learning Representations (ICLR 2025) (2025)

    2025

  40. [40]

    Bia las, P

    P. Bia las, P. Korcyl, and T. Stebel, Computer Physics Communications281, 108502 (2022)

    2022

  41. [41]

    X. Du, J. Nam, S. Liu, and R. G´ omez-Bombarelli, Scaling autoregressive models for lattice thermodynamics (2026), arXiv:2603.14695 [cond-mat.stat-mech]

    work page arXiv 2026

  42. [42]

    L. Wang, G. Aarts, and K. Zhou, JHEP05, 060, arXiv:2309.17082 [hep-lat]

    work page arXiv

  43. [43]

    Q. Zhu, G. Aarts, W. Wang, K. Zhou, and L. Wang, arXiv (2025), arXiv:2502.05504 [hep-lat]

    work page arXiv 2025

  44. [44]

    Kanwar and O

    G. Kanwar and O. Vega, in42th International Sympo- sium on Lattice Field Theory(2025) arXiv:2512.19877 [hep-lat]

    work page arXiv 2025

  45. [45]

    O. Vega, J. Komijani, A. El-Khadra, and M. Marinkovic, arXiv preprint arXiv:2510.26081 (2025)

    work page arXiv 2025

  46. [46]

    Kanaujia and V

    V. Kanaujia and V. Arora, arXiv (2025), arXiv:2510.21330 [cs.LG]

    work page arXiv 2025

  47. [47]

    title Diffusion Models for SU(2) Lattice Gauge Theory in Two Dimensions http://arxiv.org/abs/2602.09045 arXiv:2602.09045

    H. Alharazin, J. Y. Panteleeva, and B. D. Sun, arXiv (2026), arXiv:2602.09045 [hep-lat]

    work page arXiv 2026

  48. [48]

    J. M. Pawlowski and J. M. Urban, Machine Learning: Science and Technology1, 045011 (2020)

    2020

  49. [49]

    Singh, M

    J. Singh, M. Scheurer, and V. Arora, SciPost Physics11, 10.21468/scipostphys.11.2.043 (2021)

    work page doi:10.21468/scipostphys.11.2.043 2021

  50. [50]

    Singha, D

    A. Singha, D. Chakrabarti, and V. Arora, SciPost Phys. Core5, 052 (2022)

    2022

  51. [51]

    Del Debbio, J

    L. Del Debbio, J. M. Rossney, and M. Wilson, Phys. Rev. D104, 094507 (2021), arXiv:2105.12481 [hep-lat]

    work page arXiv 2021

  52. [52]

    Abbottet al., Eur

    R. Abbottet al., Eur. Phys. J. A59, 257 (2023), arXiv:2211.07541 [hep-lat]

    work page arXiv 2023

  53. [53]

    Catumba and A

    G. Catumba and A. Ramos, Eur. Phys. J. C85, 1037 (2025), arXiv:2502.15570 [hep-lat]

    work page arXiv 2025

  54. [54]

    Goodman and A

    J. Goodman and A. D. Sokal, Phys. Rev. Lett.56, 1015 (1986)

    1986

  55. [55]

    Goodman and A

    J. Goodman and A. D. Sokal, Phys. Rev. D40, 2035 (1989)

    2035

  56. [56]

    Janke and T

    W. Janke and T. Sauer, Phys. Rev. E49, 3475 (1994), arXiv:hep-lat/9305016

    work page arXiv 1994

  57. [57]

    K. E. Schmidt, Phys. Rev. Lett.51, 2175 (1983)

    1983

  58. [58]

    Faas and H

    M. Faas and H. Hilhorst, Physica A: Statistical Mechan- ics and its Applications135, 571 (1986)

    1986

  59. [59]

    M. G. Endres, R. C. Brower, W. Detmold, K. Orginos, and A. V. Pochinsky, Phys. Rev. D92, 114516 (2015), arXiv:1510.04675 [hep-lat]

    work page arXiv 2015

  60. [60]

    Jansen, E

    K. Jansen, E. H. M¨ uller, and R. Scheichl, Phys. Rev. D 102, 114512 (2020), arXiv:2008.03090 [hep-lat]

    work page arXiv 2020

  61. [61]

    Grabenstein and B

    M. Grabenstein and B. Mikeska, Phys. Rev. D47, R3103 (1993)

    1993

  62. [62]

    Bia las, P

    P. Bia las, P. Czarnota, P. Korcyl, and T. Stebel, Phys. Rev. E107, 054127 (2023), arXiv:2212.04955 [cond- mat.stat-mech]

    work page arXiv 2023

  63. [63]

    Abbott, M

    R. Abbott, M. S. Albergo, D. Boyda, D. C. Hackett, et al., PoSLATTICE2023, 035 (2024), 2404.10819

    work page arXiv 2024

  64. [64]

    Bauer, R

    M. Bauer, R. Kapust, J. M. Pawlowski, and F. L. Tem- men, SciPost Physics19, 077 (2025)

    2025

  65. [65]

    Ihssen, R

    F. Ihssen, R. Kapust, and J. M. Pawlowski, Gener- ative sampling with physics-informed kernels (2025), arXiv:2510.26678 [hep-lat]

    work page arXiv 2025

  66. [66]

    M. B. Giles, Acta Numerica24, 259 (2015)

    2015

  67. [67]

    Duane, A

    S. Duane, A. Kennedy, B. J. Pendleton, and D. Roweth, Physics Letters B195, 216 (1987)

    1987

  68. [68]

    Madras and A

    N. Madras and A. D. Sokal, J. Stat. Phys.50, 109 (1988)

    1988

  69. [70]

    L. P. Kadanoff, Physics Physique Fizika2, 263 (1966)

    1966

  70. [71]

    Binder, Z

    K. Binder, Z. Phys. B43, 119 (1981)

    1981

  71. [72]

    M. F. Hutchinson, Commun. Stat. Simul.19, 433 (1990)

    1990

  72. [73]

    Wolff (ALPHA), Comput

    U. Wolff (ALPHA), Comput. Phys. Commun.156, 143 (2004), [Erratum: Comput.Phys.Commun. 176, 383 (2007)], arXiv:hep-lat/0306017

    work page arXiv 2004

  73. [74]

    D. P. Kingma and J. Ba, Adam: A method for stochastic optimization (2017), arXiv:1412.6980 [cs.LG]. Appendix A: HMC Simulation: Critical Slowing Down and Scaling

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  74. [75]

    All simulations are performed at κc = 0.2705,λ= 0.022

    Simulation Details ForL∈ {8,16,32,64,128}we run HMC chains with molecular-dynamics trajectory lengthτ MD = 1 and nleap = 10 leapfrog steps (step sizeϵ= 0.1), fixed across all lattice sizes. All simulations are performed at κc = 0.2705,λ= 0.022. The acceptance rate decreases withLsince the step size is not re-tuned per volume (Table II). TABLE II. HMC acce...

  75. [76]

    Power-Law Fit and Extrapolation Table III collects the measuredτ int(|m|) values. The dynamical critical exponentz dyn is extracted by fitting the power-law ansatzτ int =A L zdyn in log-log space us- ing an unweighted least-squares fit to the five measured pointsL∈ {8,16,32,64,128}: zdyn = 1.99±0.10, A= 0.30,(A1) where the uncertainty is a bootstrap stand...

  76. [77]

    Overall structure.Our method is a hierarchical coarse-to-fine sampler

    Our Multilevel Method a. Overall structure.Our method is a hierarchical coarse-to-fine sampler. It starts from a coarse 8×8 lat- tice and appliesK= log 2(L/8) successive upsampling steps, each doubling the linear lattice size. The full model therefore consists of a standalone coarse-level CNF fol- lowed byKupsampling modules. b. Coarse-level CNF.The coars...

  77. [78]

    At each upsampling step, the coarse field is naively repeated on a 2×2 block, Gaussian block noise is injected, and a full-lattice (unmasked) CNF refines all L2 ℓ+1 sites jointly

    SR-NF SR-NF follows the coarse-to-fine upsampling architec- ture of [64], replacing the HMC-based coarsest sampler with the same dense CNF used in our method (RK4, 40 steps). At each upsampling step, the coarse field is naively repeated on a 2×2 block, Gaussian block noise is injected, and a full-lattice (unmasked) CNF refines all L2 ℓ+1 sites jointly. We...

  78. [79]

    The vector field is adensesingle-layer FFT-based parameterisation in which each spatial mode 12 interacts with all others, withO(L 4) parameters

    Dense CNF Dense CNF [30] is a single-scale baseline: one CNF acts directly on the fullL×Llattice, starting from an isotropic Gaussian prior. The vector field is adensesingle-layer FFT-based parameterisation in which each spatial mode 12 interacts with all others, withO(L 4) parameters. It is integrated with RK4 forn steps = 50 evaluations. A richer Fourie...

  79. [80]

    Its vector field is a 3-layer convolutional network with 32 hidden channels, 7×7 kernels, and GELU activations

    Hutch CNF Hutch CNF is also a single-scale baseline acting di- rectly on the full lattice. Its vector field is a 3-layer convolutional network with 32 hidden channels, 7×7 kernels, and GELU activations. The divergence is esti- mated with the Hutchinson stochastic trace. The ODE is solved with a fixed-step RK4 integrator atn steps = 40, and gradient checkp...

  80. [81]

    Our method and SR-NF use initial learning rate 10−2; Dense CNF uses 5×10 −3 as in Ref

    Training All methods are trained end-to-end with the reverse- KL objective using the Adam optimizer [74] (β 1 = 0.8, β2 = 0.9). Our method and SR-NF use initial learning rate 10−2; Dense CNF uses 5×10 −3 as in Ref. [30]. All methods apply an exponential learning-rate decay with γ= 0.9994 and minimum 10 −4. Gradient norms are clipped at 0.5 for our method,...

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