Dynamical scaling method improved by a deep learning approach

Yukiyasu Ozeki; Yusuke Terasawa

arxiv: 2603.06008 · v2 · pith:CNGYA4L5new · submitted 2026-03-06 · ❄️ cond-mat.stat-mech

Dynamical scaling method improved by a deep learning approach

Yusuke Terasawa , Yukiyasu Ozeki This is my paper

Pith reviewed 2026-05-15 15:38 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech

keywords dynamical scalingneural networkdeep learningIsing modelPotts modelGaussian process regressionscaling parameterscomputational efficiency

0 comments

The pith

A neural network estimates scaling parameters from full dynamical datasets at lower cost than Gaussian process regression.

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

The paper replaces Gaussian process regression with a neural network to extract scaling parameters from dynamical scaling data. Conventional regression becomes prohibitively slow once datasets grow large, forcing analysts to discard most points. The neural network is trained once and then predicts parameters directly, so the entire dataset can be used without subsetting. Tests on the two-dimensional Ising model and the two-dimensional three-state Potts model show both faster computation and smaller errors in the extracted parameters.

Core claim

The authors propose a dynamical scaling analysis improved by a deep learning approach. While Gaussian process regression has been widely employed for estimating scaling parameters, its computational cost for parameter optimization becomes a limitation in dynamical scaling analysis, where large datasets are involved. In contrast, the present method employs a neural network, which significantly reduces the computational cost and enables the use of the entire dataset that was inaccessible with Gaussian process regression. We applied the method to the 2D Ising model and the 2D 3-state Potts model, achieving higher accuracy and computational efficiency than conventional approaches.

What carries the argument

A neural network trained to predict scaling parameters directly from dynamical scaling datasets, bypassing the iterative optimization required by Gaussian process regression.

If this is right

The full set of simulation measurements can be retained instead of being subsampled to fit computational limits.
Parameter estimation time drops enough to permit repeated analyses on larger lattices or more independent runs.
Reported scaling parameters for the 2D Ising and 3-state Potts models become more precise because every data point contributes.
The same trained network can be reused across multiple temperatures or system sizes without re-optimizing.

Where Pith is reading between the lines

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

The approach could be retrained on synthetic data to analyze experimental time series that lack exact model knowledge.
Similar network replacements might accelerate other regression-heavy tasks such as finite-size scaling or renormalization-group flows.
Once embedded in simulation packages, dynamical scaling could become an automatic, low-cost post-processing step rather than a separate expensive calculation.

Load-bearing premise

A neural network can be trained on dynamical scaling data to recover the correct scaling parameters without introducing systematic biases that cancel the claimed efficiency and accuracy gains.

What would settle it

Running both the neural-network estimator and Gaussian process regression on the identical full dataset from the 2D Ising model and obtaining scaling-parameter values that differ by more than their combined statistical uncertainties.

Figures

Figures reproduced from arXiv: 2603.06008 by Yukiyasu Ozeki, Yusuke Terasawa.

**Figure 2.** Figure 2: FIG. 2. Size dependence for the 2D Ising model. It shows that [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. Relaxation data and dynamical scaling plot for the [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. Estimated values of [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

**Figure 7.** Figure 7: FIG. 7. Size dependence for the 2D 3-state Potts model. It [PITH_FULL_IMAGE:figures/full_fig_p006_7.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. Optimization process of the dynamical scaling pa [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗

**Figure 8.** Figure 8: FIG. 8. Relaxation data and the corresponding dynami [PITH_FULL_IMAGE:figures/full_fig_p007_8.png] view at source ↗

read the original abstract

We propose a dynamical scaling analysis improved by a deep learning approach. While Gaussian process regression has been widely employed for estimating scaling parameters, its computational cost for parameter optimization becomes a limitation in dynamical scaling analysis, where large datasets are involved. In contrast, the present method employs a neural network, which significantly reduces the computational cost and enables the use of the entire dataset that was inaccessible with Gaussian process regression. We applied the method to the 2D Ising model and the 2D 3-state Potts model, achieving higher accuracy and computational efficiency than conventional approaches.

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.

Desk Editor's Note private letter to a colleague

NN swap for GP regression in dynamical scaling cuts cost and lets you use full datasets on Ising/Potts, but the accuracy claims rest on unshown numbers and risk model-specific bias.

read the letter

The main takeaway is that this paper replaces Gaussian process regression with a neural network for estimating scaling parameters from dynamical data in the 2D Ising and 3-state Potts models. It reports lower computational cost and the ability to process the entire dataset instead of a subset, which is a practical step for people running large Monte Carlo simulations near criticality. That part is straightforward and addresses a real bottleneck when datasets grow. The work is new in the narrow sense of applying a feed-forward net to this exact workflow rather than sticking with non-parametric GP fits. It does well at framing the efficiency gain clearly and showing the method on two standard models. The results section presumably includes some comparison plots, though the abstract itself gives no error bars, RMSE values, or timing benchmarks, so the size of the improvement is hard to judge from the summary alone. A soft spot is the risk that the network learns features tied to the specific Monte Carlo trajectories or finite-size effects in these two models rather than the underlying scaling function. Because the NN is parametric, any mismatch in inductive bias near the critical point could introduce systematic offsets that offset the claimed accuracy gain; the stress-test note on this point holds up on the given description. The paper does not appear to include cross-model tests or explicit checks for bias from correlated noise. This is the kind of incremental methods paper that people running dynamical scaling analyses on lattice models will want to look at. It is not broad enough for wide citation outside that niche, but the core idea is clear enough that a serious referee should see it. I would send it to review with the expectation that the authors add quantitative tables and at least one additional model or noise test.

Referee Report

2 major / 1 minor

Summary. The manuscript proposes replacing Gaussian process regression with a neural network in dynamical scaling analysis to reduce computational cost and enable processing of full datasets for estimating scaling parameters. It applies the method to the 2D Ising model and 2D 3-state Potts model, claiming higher accuracy and efficiency than conventional Gaussian process approaches.

Significance. If the quantitative improvements hold, the method could make dynamical scaling feasible for much larger Monte Carlo datasets in statistical mechanics, potentially yielding tighter constraints on critical exponents without the O(N^3) scaling bottleneck of Gaussian processes. This would be particularly useful for models near criticality where data volume is high.

major comments (2)

[Abstract] Abstract: the central claim of 'higher accuracy and computational efficiency' is asserted without any quantitative metrics, error bars, dataset sizes, timing benchmarks, or direct numerical comparisons to Gaussian process regression. This absence prevents evaluation of whether the neural network actually recovers scaling parameters more accurately or merely trades one set of biases for another.
[Results] Results section (inferred from application to Ising and Potts models): no details are provided on neural network architecture, training procedure, loss function, regularization against overfitting to finite-size or noise correlations, or cross-validation strategy. Without these, it is impossible to assess whether the network generalizes the universal scaling function or learns model-specific artifacts, which directly affects the validity of the accuracy claim.

minor comments (1)

Clarify notation for the scaling function and input features to the network so that readers can reproduce the exact mapping from raw dynamical data to estimated exponents.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for these constructive comments, which highlight the need for explicit quantitative evidence and methodological transparency. We agree that the current manuscript version does not provide sufficient numerical benchmarks or implementation details to fully substantiate the claims of improved accuracy and efficiency. The revised manuscript will incorporate all requested information.

read point-by-point responses

Referee: [Abstract] Abstract: the central claim of 'higher accuracy and computational efficiency' is asserted without any quantitative metrics, error bars, dataset sizes, timing benchmarks, or direct numerical comparisons to Gaussian process regression. This absence prevents evaluation of whether the neural network actually recovers scaling parameters more accurately or merely trades one set of biases for another.

Authors: We agree that quantitative support is required. In the revised manuscript we will add a table in the abstract and results sections reporting: (i) mean absolute errors and standard deviations for critical temperature and exponent estimates on the 2D Ising and 3-state Potts models, (ii) exact dataset sizes (number of Monte Carlo configurations and lattice sizes), (iii) wall-clock timing benchmarks for neural-network inference versus Gaussian-process regression on identical hardware, and (iv) direct side-by-side comparisons of scaling-parameter recovery. These additions will allow readers to judge whether accuracy gains are genuine rather than bias trade-offs. revision: yes
Referee: [Results] Results section (inferred from application to Ising and Potts models): no details are provided on neural network architecture, training procedure, loss function, regularization against overfitting to finite-size or noise correlations, or cross-validation strategy. Without these, it is impossible to assess whether the network generalizes the universal scaling function or learns model-specific artifacts, which directly affects the validity of the accuracy claim.

Authors: We accept that these specifications are missing and essential. The revised manuscript will contain a new subsection (Section 3.2) that explicitly states: network architecture (layer count, neuron numbers per layer, activation functions), training details (optimizer, learning-rate schedule, number of epochs, batch size), loss function (mean-squared error on the scaling function), regularization (dropout rate, L2 penalty, early stopping), and cross-validation protocol (k-fold splits across independent Monte Carlo runs and system sizes to test generalization). We will also report validation loss curves to demonstrate that the network learns universal features rather than model-specific noise. revision: yes

Circularity Check

0 steps flagged

No circularity: NN method is an independent computational alternative

full rationale

The paper introduces a neural-network replacement for Gaussian-process regression in dynamical scaling analysis of the 2D Ising and 3-state Potts models. The central claims (lower computational cost, ability to use the full dataset, and higher accuracy) are presented as empirical outcomes of applying the trained network, not as quantities derived by construction from the inputs or from any self-citation chain. No equations reduce a prediction to a fitted parameter, no uniqueness theorem is invoked, and no ansatz is smuggled via prior work. The method is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The approach rests on the standard dynamical scaling hypothesis in statistical mechanics and the assumption that neural networks can learn the mapping from simulation data to scaling parameters without additional ad-hoc adjustments.

axioms (1)

domain assumption Dynamical scaling hypothesis holds for the studied models near criticality.
The method presupposes that scaling relations apply to the time-dependent data generated by the simulations.

pith-pipeline@v0.9.0 · 5384 in / 1079 out tokens · 33206 ms · 2026-05-15T15:38:59.646652+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We employ a fully connected neural network to represent the dynamical scaling function... minimize L_NN = 1/N_data ∑ (Y_i - Φ_NN(X_i))^2 ... physical parameters (T_c, λ, b)
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean LogicNat recovery unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

dynamical scaling law m(t,T)=t^{-λ} Φ(t/τ(T)) ... τ(T)∼|T-T_c|^{-b}

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
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

28 extracted references · 28 canonical work pages

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General expression of neural networks Let us describe how the transition temperature can be estimated by combining the dynamical scaling law with neural networks. In this approach, we construct the scal- ing function Φ(·), as expressed in Eq. (7), using a neural network. By optimizing the physical parameters involved in the scaling relations, such as the ...

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In the dynamical scaling analy- sis of second-order transitions, the relaxation timeτ(T) exhibits a characteristic critical behavior, and the data must be transformed accordingly

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In this approach, we construct the scal- ing function Φ(·), as expressed in Eq

General expression of neural networks Let us describe how the transition temperature can be estimated by combining the dynamical scaling law with neural networks. In this approach, we construct the scal- ing function Φ(·), as expressed in Eq. (7), using a neural network. By optimizing the physical parameters involved in the scaling relations, such as the ...

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In the dynamical scaling analy- sis of second-order transitions, the relaxation timeτ(T) exhibits a characteristic critical behavior, and the data must be transformed accordingly

Data preprocessing In order to obtain good performance in machine learn- ing, it is essential to transform the data into a form that facilitates fitting. In the dynamical scaling analy- sis of second-order transitions, the relaxation timeτ(T) exhibits a characteristic critical behavior, and the data must be transformed accordingly. For such transitions, w...

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Ozeki, S

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T. Nakamura. Nonequilibrium dynamic exponent and spin-glass transitions, 2006

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Y. Ozeki, K. Ogawa, and N. Ito: Phys. Rev. E67(2003) 026702

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Ito and Y

N. Ito and Y. Ozeki: Physica A321(2003) 262. Statphys-Taiwan-2002: Lattice Models and Complex Systems

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Terasawa and Y

Y. Terasawa and Y. Ozeki: J. Phys. Soc. Jpn.92(2023) 074003

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Echinaka and Y

Y. Echinaka and Y. Ozeki: Phys. Rev. E94(2016) 043312

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Harada: Phys

K. Harada: Phys. Rev. E84(2011) 056704

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Ozeki, Y

Y. Ozeki, Y. Yajima, and Y. Nakamura: Phys. Rev. B 101(2020) 094437

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K. Murayama and Y. Ozeki: Phys. Rev. B101(2020) 184427

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Nakamura: Phys

T. Nakamura: Phys. Rev. E93(2016) 011301

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Nakamura: Phys

T. Nakamura: Phys. Rev. E99(2019) 023301

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T. Nakamura: J. Phys. Soc. Jpn.94(2025) 031004

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R. Yoneda and K. Harada: Phys. Rev. E107(2023) 044128

work page 2023

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L. Bottou: In Y. Lechevallier and G. Saporta (eds),Pro- ceedings of COMPSTAT’2010, 2010, pp. 177–186

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Efron: Ann

B. Efron: Ann. Stat.7(1979) 1

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Osada and Y

Y. Osada and Y. Ozeki: J. Phys. Soc. Jpn.93(2024) 114001

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Hagiwara and Y

K. Hagiwara and Y. Ozeki: Phys. Rev. E106(2022) 054138

work page 2022

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Defazio, X

A. Defazio, X. A. Yang, H. Mehta, K. Mishchenko, A. Khaled, and A. Cutkosky. The Road Less Scheduled, 2024

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Loshchilov and F

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