Machine learning the Ising transition: A comparison between discriminative and generative approaches

Difei Zhang; Frank Sch\"afer; Julian Arnold

REVIEW 2 major objections 3 minor 23 references

Machine learning the Ising transition: A comparison between discriminative and generative approaches

T0 review · 2 major / 3 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The best phase detector on the Ising model is not one model class, but a ranking that shifts with the indicator and with available physics knowledge.

desk verdict A transparent, reproducible benchmark on the 4x4 Ising model, but the data-driven headline ranking is built on an asymmetric hyperparameter search, so the generative-vs-discriminative conclusion should not be taken as settled. read the letter →

arxiv 2411.19370 v1 pith:FXFO4N6F submitted 2024-11-28 cond-mat.dis-nn cond-mat.stat-mechcs.LG

Difei Zhang , Frank Schäfer , Julian Arnold This is my paper · ORCID

classification cond-mat.dis-nncond-mat.stat-mechcs.LG

keywords Isingmodelphasetransitiondetectiondiscriminativevsgenerativemodelssupervisedlearningbyconfusionprediction-basedmethodPixelCNNFisherinformation

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This numerical case study asks a practical design question: when machine learning is used to locate a phase transition, should the classifier be discriminative (learn $P(y|x)$ directly) or generative (model $P(x|y)$ and use Bayes' rule)? The paper answers the question for the thermal transition of the $4\times4$ square-lattice ferromagnetic Ising model, using three indicator methods — supervised learning (SL), learning by confusion (LBC), and the prediction-based method (PBM). Its central finding is that the best choice is not one fixed approach: in the data-driven setting, neural-network discriminative models are favored for SL and PBM, the NN generative model is favored for LBC, while in the knowledge-driven setting a simple histogram over the energy sufficient statistic wins at almost every time budget. If the pattern generalizes, it gives practical guidance for automating phase-diagram mapping in other classical lattice models.

What carries the argument

The argument runs through scalar phase-transition indicators that are underapproximators of the Fisher information: $I_{\mathrm{SL}}(\gamma)$ from supervised-learning posteriors, $I_{\mathrm{LBC}}(\gamma)$ from learning-by-confusion error probabilities, and $I_{\mathrm{PBM}}(\gamma)$ from a predictive model's output gradient and variance. Each indicator reduces phase detection to a classification task whose posterior $P(y|x)$ can be estimated either directly (discriminative, here a CNN or MLP) or through a learned density $P(x|y)$ combined with Bayes' rule (generative, here a PixelCNN or histogram binning). The decisive simplification in the knowledge-driven setting is that the energy $E$ is a minimal sufficient statistic for the inverse temperature, shrinking the state space from $2^{L^2}$ configurations to $L^2-1$ energy levels and making the nonparametric histogram accurate with few samples. Computation-time comparisons use envelopes over hyperparameter runs, with data-generation time excluded in the data-driven setting and included in the knowledge-driven setting.

What would settle it

Run the same three indicators on an $L=8$ Ising lattice with raw spin snapshots, counting all hyperparameter-search time in the budget and using a deeper convolutional architecture; if histogram binning matches or beats the CNN for SL or PBM at any shared computation budget in the data-driven setting, the paper's central ranking is not robust. A cheaper check: on the $4\times4$ system, include the time spent on the hyperparameter sweeps (the scattered points in Fig. 3) in the reported budgets and see whether the favored model for LBC remains the PixelCNN.

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Extended reading notes

Core claim

The paper claims that, for locating the Ising transition from raw spin configurations of a $4\times4$ lattice, the relative efficiency of discriminative versus generative modeling is controlled by the indicator being computed and by how much physical knowledge is available. In the data-driven setting, comparing mean-squared error of the learned indicator versus the ground-truth indicator at fixed dataset size and fixed computation time, the discriminative CNN is the best choice for SL and PBM, whereas the generative PixelCNN wins for LBC. In the knowledge-driven setting, where the Hamiltonian is known and the energy is used as a minimal sufficient statistic, the nonparametric histogram-based generative approach outperforms both NN approaches in computation time for all but the smallest sample counts and largest error tolerances. The paper presents these as empirical findings on one model, with the expectation that the qualitative ranking extends to larger systems and other classical lattice models.

Load-bearing premise

The rankings depend on the chosen CNN and PixelCNN architectures, fixed optimizers, and small hyperparameter grids being fair stand-ins for discriminative and generative modeling, and on excluding hyperparameter-tuning cost from reported runtimes; change those choices and the ordering could change.

Editorial extensions

If this is right

When raw spin snapshots are the only data and the goal is SL or PBM phase detection on the $4\times4$ Ising model, a discriminative CNN reaches a given indicator accuracy faster than histogram binning or PixelCNN.
For learning-by-confusion on the same data, the generative PixelCNN is the most compute-efficient of the three model families.
When the Hamiltonian is known, histogram binning on energies outperforms both NN approaches for all but the smallest datasets because the energy sufficient statistic collapses the state space.
In the data-driven setting, NN-based methods stop beating the histogram once dataset size approaches the state-space size $|\mathcal{X}|=2^{16}$; internal biases then dominate.
The qualitative ordering is expected by the authors to persist for larger systems, with the state-space size setting the threshold dataset size.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A broader reading is that no universal 'generative beats discriminative' answer exists for phase detection; the winning model is selected by the indicator's information geometry, with LBC needing the fuller conditional distribution while SL and PBM hinge on one-dimensional sufficient statistics.
A testable extension would be replacing the PixelCNN with an energy-based model or normalizing flow to see if the LBC advantage for generative models remains, or moving to the $L=8$ lattice where the histogram's state-space collapse weakens.
If hyperparameter-search cost were included in the time budgets, the discriminative NN ranking for SL/PBM could shift at short budgets, since the reported envelopes hide tuning overhead; this is an implicit limitation of the comparison as presented.
The sufficient-statistic argument suggests that in knowledge-driven settings, investing in learned or exact dimension reduction before classification may confer more benefit than choosing between discriminative and generative classifiers.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this is a useful benchmark paper, not a general answer. The cleanest result is the knowledge-driven one: once you know the Hamiltonian, histogram binning on the energy (the sufficient statistic) beats the NNs at almost every time budget on this 4x4 Ising model. The data-driven rankings are interesting but less solid. The PixelCNN gets multiple hyperparameter configurations (epochs 10 to 10000 and two learning rates), while the discriminative CNN gets one fixed configuration. The bold envelopes in Fig. 3 are lower envelopes over those runs. An envelope over many curves will look better than a single curve even if the methods are equally good. The authors state that tuning cost is not included, but the deeper problem is that search effort is not matched. This specifically hits the LBC-generative-favored conclusion in Sec. 5. The SL/PBM discriminative-favored parts may survive, but the comparison as plotted cannot establish the ordering as stated.

Credit where it is earned: ground-truth indicators are computed exactly for L=4, errors are averaged over five runs, hyperparameters are fully tabulated, and the code is openly available. That is the kind of transparency that makes a numerical comparison reproducible and worth engaging with. The paper also correctly places itself relative to the sufficient-statistic reduction and the Fisher-information connection, though those are self-cited from the same group.

Soft spots, in proportion: the single system size L=4 is the main limitation. The conclusion that the trends hold for larger systems and other models is explicitly an expectation, not a demonstrated result. The choice of CNN and PixelCNN as representatives of their respective approaches is reasonable but not defended against alternatives. Minor: computation times come from a V-100 GPU with limited reporting on run-to-run timing variance, though the five-seed error bands help.

Who this is for: practitioners choosing methods for automated phase-transition detection in small classical lattice models. It is not a general theory of when generative or discriminative models win. It deserves a serious referee, but the authors should be asked to match the number of hyperparameter trials across methods, report tuning-inclusive times or justify the envelope comparison, and either add evidence at larger L or soften the generalization claims. I would not cite it as evidence for the data-driven ranking until that is fixed, but the knowledge-driven comparison and the reproducible setup make it a legitimate paper to send to review.

Referee Report

2 major / 3 minor

Summary. The paper presents a numerical case study on the 2D square-lattice ferromagnetic Ising model, comparing discriminative and generative approaches for the classification tasks underlying three phase-transition indicators: supervised learning (SL), learning by confusion (LBC), and the prediction-based method (PBM). For L=4, where ground-truth indicators are computed exactly, the authors compare histogram-based nonparametric generative models, MLP/CNN discriminative models, and PixelCNN generative models in both a data-driven setting (raw spin configurations, no Hamiltonian knowledge) and a knowledge-driven setting (Hamiltonian known, energy used as a sufficient statistic). They report mean-square errors as functions of dataset size and computation time, concluding that in the data-driven setting NN-based methods are favored over histogram binning, with the generative approach favored for LBC and the discriminative approach favored for SL and PBM, while in the knowledge-driven setting the nonparametric energy-histogram approach dominates except at very small time budgets. The code is publicly available.

Significance. If the rankings are robust, the paper would give concrete practical guidance for choosing between discriminative and generative models in automated phase-transition detection. The study has notable strengths: exact ground-truth indicators for L=4, errors averaged over five independent runs, public code, a clear reduction via the energy sufficient statistic in the knowledge-driven setting, and a connection to Fisher-information underapproximators. However, the central computational comparison rests on a single small system size and on evaluation envelopes built from unequal hyperparameter search budgets, so the contribution is currently a conditional case study rather than a general answer to the question posed in the abstract.

major comments (2)

[§5, Fig. 3, Tables A1 and B1] The central data-driven ranking is derived from envelopes over unequal hyperparameter search budgets. Table B1 gives the PixelCNN eight data-driven configurations (learning rates 0.001 and 0.0001, epochs 10/100/1000/10000), while Table A1 gives the CNN a single configuration (100 epochs, learning rate 0.001). The bold envelope in Fig. 3 is therefore a lower envelope over eight PixelCNN runs versus one CNN run; an envelope over more runs cannot do worse, so the claim that generative modeling is most favored for LBC may be an artifact of search effort rather than of the modeling paradigm. Please match the search budgets across methods, report the full set of individual curves for the discriminative models, and state or bound the cost of hyperparameter selection, which is currently excluded from the reported computation times.
[§5, Secs. 3 and 4] All numerical evidence is for a single 4×4 lattice, where the transition is finite-size rounded and the state space has 65,536 configurations. The conclusion extrapolates to larger system sizes and to other thermal lattice models, but no larger-L experiment or scaling argument is provided. To support the practical recommendation stated in the abstract, the authors should either restrict the claims to the L=4 case study or add results for at least one or two larger system sizes to show that the rankings do not reverse.

minor comments (3)

[§2.2, Eqs. (9)-(10)] The KL-divergence labels appear to be reversed: Eq. (9) as written is D_KL(\tilde P_θ || P), not KL(P || \tilde P_θ), and Eq. (10) is D_KL(P || \tilde P_θ), not KL(\tilde P_θ || P). The subsequent forward/reverse KL discussion follows the standard formulas, so this is a notation issue rather than an error in the training losses, but it should be corrected for clarity.
[Fig. C4 caption] In the caption for Fig. C4(a), the text says the small MLP has 610 parameters, but the legend and the data-driven setting use CNN models; this should read "small CNN".
[Fig. 4 caption] The caption contains an extra parenthesis in "prediction-based method (PBM))".

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the central discriminative-vs-generative comparison is an empirical benchmark; self-citations supply background theorems only.

full rationale

The paper's central claim, the ranking of discriminative versus generative approaches for the SL, LBC, and PBM indicators, is an empirical result obtained by computing mean-square errors between learned and ground-truth indicators as functions of dataset size and computation time (Figs. 2-4). No step in this comparison is defined in terms of its own output: the indicator formulas in Sec. 2.1 are fixed mathematical quantities whose ground-truth values are computed exactly for the L=4 Ising model, and the learned approximations are evaluated against those independent targets. The self-cited background results, namely the energy as a minimum sufficient statistic (Refs. [6,7]) and the Fisher-information underapproximation (Ref. [8]), are parameter-free theorems with stated assumptions that do not include the paper's performance ranking; using them to justify the histogram-on-energies baseline or to motivate the indicators does not make the headline ranking reduce to an input. The PixelCNN is introduced as a generative architecture via external prior work (Refs. [17-19]) and is validated against reference physical quantities in Fig. C1, so no ansatz is smuggled in solely by self-citation. One caveat, but not circularity: Fig. 3's envelopes compare four PixelCNN hyperparameter settings against a single CNN setting (Tabs. A1/B1), so the 'generative favored for LBC' ordering may partly reflect unequal search effort; this is a fairness and robustness concern, not a self-referential derivation. The stated limitation to L=4 is acknowledged by the authors and does not affect the internal logic of the comparison. Overall, the derivation chain is self-contained with respect to the paper's central empirical claims.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new physical entities. It relies on standard Boltzmann statistics, sufficient-statistic reduction from prior self-cited work, and a set of hand-chosen model and training parameters whose values are detailed in the appendices. The central comparison is empirical, so the ledger is dominated by methodological choices rather than new postulates.

free parameters (6)

LBC label-window size l = 5
Chosen by hand for the learning-by-confusion indicator; the indicator and its error depend on this window (Section 2.1, Fig. 2).
PixelCNN reverse-KL annealing rate = 0.9, 0.98, 0.99, 0.998 (per run)
Hand-tuned; Appendix B states careful design of the annealing rate was essential for the knowledge-driven generative model.
Learning rates = 0.001, 0.0001, 0.01
Chosen per model and setting (Tables A1, B1); affects training convergence and the computation-time tradeoff.
Number of training epochs = 100, 1000, 10000
Chosen per model and setting; directly sets computation time and accuracy in the tradeoff curves.
MCMC thermalization and sampling parameters = 1e5 thermalization steps, sample every 10 steps
Chosen for data generation; the paper assumes stationarity after this period (Appendix A).
Neural network architectures = CNN 3894 params, MLP 2242, PixelCNN 2017 (small variants 787/610)
Hand-selected to be comparable across approaches; all results depend on these architecture choices (Tabs. A1, B1).

assumptions (5)

domain assumption Spin configurations are drawn from the exact Boltzmann distribution at each temperature after MCMC thermalization.
Section 2.3 and Appendix A; if the chains have not converged, all learned indicators are biased.
domain assumption The energy is a minimum sufficient statistic for the inverse temperature in the Ising model.
Invoked in Section 2.3 to justify compressing spin configurations to energies; the proof is cited to the authors' prior work Refs. [6,7], not re-derived here.
domain assumption The three indicators (SL, LBC, PBM) are valid underapproximators of the Fisher information.
Section 2.1 states 'It can be shown...' with citation to Ref. [8] by the same group; used to motivate the indicator framework.
domain assumption Any target density can be modeled as a Boltzmann distribution with a suitable Hamiltonian, enabling reverse-KL training from the known Hamiltonian.
Section 2.2, citing Refs. [14,15]; this underpins the knowledge-driven PixelCNN training.
ad hoc to paper The PixelCNN autoregressive model class is expressive enough to capture the Ising Boltzmann distribution.
Appendix C shows it reproduces energy and magnetization for the tested settings, but no guarantee for all temperatures; the comparison relies on this approximation.

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Cite this review

Pith. "Pith review of Machine learning the Ising transition: A comparison between discriminative and generative approaches." pith.science (2026). https://pith.science/paper/FXFO4N6F

@misc{pith2026241119370,
  author       = {Pith},
  title        = {Pith review of: Machine learning the Ising transition: A comparison between discriminative and generative approaches},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FXFO4N6F}},
  note         = {Machine review of arXiv:2411.19370}
}

read the original abstract

The detection of phase transitions is a central task in many-body physics. To automate this process, the task can be phrased as a classification problem. Classification problems can be approached in two fundamentally distinct ways: through either a discriminative or a generative method. In general, it is unclear which of these two approaches is most suitable for a given problem. The choice is expected to depend on factors such as the availability of system knowledge, dataset size, desired accuracy, computational resources, and other considerations. In this work, we answer the question of how one should approach the solution of phase-classification problems by performing a numerical case study on the thermal phase transition in the classical two-dimensional square-lattice ferromagnetic Ising model.

Figures

Figures reproduced from arXiv: 2411.19370 by the authors.

**Figure 2.** Mean-square-error between machine-learned indicator [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** Mean-square-error between machine-learned indicator [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Mean-square-error between machine-learned indicator [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

23 extracted references · 18 canonical work pages

[1]

Sachdev, Quantum Phase Transitions

S. Sachdev, Quantum Phase Transitions. Cambridge University Press, 2011

work page 2011
[2]

Goldenfeld, Lectures On Phase Transitions And The Renormalization Group

N. Goldenfeld, Lectures On Phase Transitions And The Renormalization Group. CRC Press, 2018

work page 2018
[3]

Machine learning and the physical sciences,

G. Carleo, I. Cirac, K. Cranmer, L. Daudet, M. Schuld, N. Tishby, L. Vogt-Maranto, and L. Zdeborov´ a, “Machine learning and the physical sciences,”Rev. Mod. Phys., vol. 91, p. 045002, Dec 2019

work page 2019
[4]

How To Use Neural Networks To Investigate Quantum Many-Body Physics,

J. Carrasquilla and G. Torlai, “How To Use Neural Networks To Investigate Quantum Many-Body Physics,” PRX Quantum, vol. 2, p. 040201, Nov 2021

work page 2021
[5]

Modern applications of machine learning in quantum sciences,

A. Dawid, J. Arnold, B. Requena, A. Gresch, M. P lodzie´ n, K. Donatella, K. Nicoli, P. Stornati, R. Koch, M. B¨ uttner,et al., “Modern applications of machine learning in quantum sciences,” arXiv:2204.04198, 2022

arXiv 2022
[6]

Replacing neural networks by optimal analytical predictors for the detection of phase transitions,

J. Arnold and F. Sch¨ afer, “Replacing neural networks by optimal analytical predictors for the detection of phase transitions,” Phys. Rev. X, vol. 12, p. 031044, Sep 2022

work page 2022
[7]

Mapping out phase diagrams with generative classifiers,

J. Arnold, F. Sch¨ afer, A. Edelman, and C. Bruder, “Mapping out phase diagrams with generative classifiers,” Phys. Rev. Lett., vol. 132, p. 207301, May 2024

work page 2024
[8]

Machine learning phase transitions: Connections to the Fisher information,

J. Arnold, N. L¨ orch, F. Holtorf, and F. Sch¨ afer, “Machine learning phase transitions: Connections to the Fisher information,” arXiv:2311.10710, 2023

arXiv 2023

Show all 23 references

[9]

Machine learning phases of matter,

J. Carrasquilla and R. G. Melko, “Machine learning phases of matter,” Nat. Phys., vol. 13, no. 5, pp. 431–434, 2017

2017
[10]

Learning phase transitions by confusion,

E. P. Van Nieuwenburg, Y.-H. Liu, and S. D. Huber, “Learning phase transitions by confusion,” Nat. Phys., vol. 13, no. 5, pp. 435–439, 2017

2017
[11]

Vector field divergence of predictive model output as indication of phase transitions,

F. Sch¨ afer and N. L¨ orch, “Vector field divergence of predictive model output as indication of phase transitions,” Phys. Rev. E, vol. 99, p. 062107, Jun 2019

2019
[12]

Interpretable and unsupervised phase classification,

J. Arnold, F. Sch¨ afer, M. ˇZonda, and A. U. J. Lode, “Interpretable and unsupervised phase classification,” Phys. Rev. Res., vol. 3, p. 033052, Jul 2021

2021
[13]

Crystal Statistics. i. A Two-Dimensional Model with an Order-Disorder Transition,

L. Onsager, “Crystal Statistics. i. A Two-Dimensional Model with an Order-Disorder Transition,” Phys. Rev., vol. 65, pp. 117–149, Feb 1944

1944
[14]

The physics of energy-based models,

P. Huembeli, J. M. Arrazola, N. Killoran, M. Mohseni, and P. Wittek, “The physics of energy-based models,” Quantum Machine Intelligence, vol. 4, no. 1, p. 1, 2022

2022
[15]

Introduction to latent variable energy-based models: a path toward autonomous machine intelligence,

A. Dawid and Y. LeCun, “Introduction to latent variable energy-based models: a path toward autonomous machine intelligence,” Journal of Statistical Mechanics: Theory and Experiment, vol. 2024, no. 10, p. 104011, 2024

2024
[16]

Pixel recurrent neural networks,

A. Van Den Oord, N. Kalchbrenner, and K. Kavukcuoglu, “Pixel recurrent neural networks,” in International conference on machine learning, pp. 1747–1756, PMLR, 2016

2016
[17]

Solving statistical mechanics using variational autoregressive networks,

D. Wu, L. Wang, and P. Zhang, “Solving statistical mechanics using variational autoregressive networks,” Phys. Rev. Lett., vol. 122, p. 080602, Feb 2019

2019
[18]

Asymptotically unbiased estimation of physical observables with neural samplers,

K. A. Nicoli, S. Nakajima, N. Strodthoff, W. Samek, K.-R. M¨ uller, and P. Kessel, “Asymptotically unbiased estimation of physical observables with neural samplers,” Phys. Rev. E, vol. 101, p. 023304, Feb 2020

2020
[19]

Unbiased Monte Carlo cluster updates with autoregressive neural networks,

D. Wu, R. Rossi, and G. Carleo, “Unbiased Monte Carlo cluster updates with autoregressive neural networks,” Phys. Rev. Research, vol. 3, p. L042024, Nov 2021

2021
[20]

Fast detection of phase transitions with multi-task learning- by-confusion,

J. Arnold, F. Sch¨ afer, and N. L¨ orch, “Fast detection of phase transitions with multi-task learning- by-confusion,” arXiv:2311.09128, 2023

2023 arXiv
[21]

Fine-tuning neural network quantum states,

R. Rende, S. Goldt, F. Becca, and L. L. Viteritti, “Fine-tuning neural network quantum states,” arXiv:2403.07795, 2024

2024 arXiv
[22]

Enhancing variational Monte Carlo simulations using a programmable quantum simulator,

M. S. Moss, S. Ebadi, T. T. Wang, G. Semeghini, A. Bohrdt, M. D. Lukin, and R. G. Melko, “Enhancing variational Monte Carlo simulations using a programmable quantum simulator,” Phys. Rev. A, vol. 109, p. 032410, Mar 2024

2024
[23]

Github repository

D. Zhang, F. Sch¨ afer, and J. Arnold, “Github repository.” https://github.com/difeizhang/ machine-learning-the-Ising-transition , 2024

2024

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Sachdev, Quantum Phase Transitions

S. Sachdev, Quantum Phase Transitions. Cambridge University Press, 2011

work page 2011

[2] [2]

Goldenfeld, Lectures On Phase Transitions And The Renormalization Group

N. Goldenfeld, Lectures On Phase Transitions And The Renormalization Group. CRC Press, 2018

work page 2018

[3] [3]

Machine learning and the physical sciences,

G. Carleo, I. Cirac, K. Cranmer, L. Daudet, M. Schuld, N. Tishby, L. Vogt-Maranto, and L. Zdeborov´ a, “Machine learning and the physical sciences,”Rev. Mod. Phys., vol. 91, p. 045002, Dec 2019

work page 2019

[4] [4]

How To Use Neural Networks To Investigate Quantum Many-Body Physics,

J. Carrasquilla and G. Torlai, “How To Use Neural Networks To Investigate Quantum Many-Body Physics,” PRX Quantum, vol. 2, p. 040201, Nov 2021

work page 2021

[5] [5]

Modern applications of machine learning in quantum sciences,

A. Dawid, J. Arnold, B. Requena, A. Gresch, M. P lodzie´ n, K. Donatella, K. Nicoli, P. Stornati, R. Koch, M. B¨ uttner,et al., “Modern applications of machine learning in quantum sciences,” arXiv:2204.04198, 2022

arXiv 2022

[6] [6]

Replacing neural networks by optimal analytical predictors for the detection of phase transitions,

J. Arnold and F. Sch¨ afer, “Replacing neural networks by optimal analytical predictors for the detection of phase transitions,” Phys. Rev. X, vol. 12, p. 031044, Sep 2022

work page 2022

[7] [7]

Mapping out phase diagrams with generative classifiers,

J. Arnold, F. Sch¨ afer, A. Edelman, and C. Bruder, “Mapping out phase diagrams with generative classifiers,” Phys. Rev. Lett., vol. 132, p. 207301, May 2024

work page 2024

[8] [8]

Machine learning phase transitions: Connections to the Fisher information,

J. Arnold, N. L¨ orch, F. Holtorf, and F. Sch¨ afer, “Machine learning phase transitions: Connections to the Fisher information,” arXiv:2311.10710, 2023

arXiv 2023