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arxiv: 2512.11077 · v3 · submitted 2025-12-11 · ❄️ cond-mat.mtrl-sci · cs.AI

A probabilistic framework for crystal structure denoising, phase classification, and order parameters

Hyuna Kwon , Babak Sadigh , Sebastien Hamel , Vincenzo Lordi , John Klepeis , Fei Zhou This is my paper

Pith reviewed 2026-05-16 22:32 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci cs.AI
keywords crystal structure denoisingphase classificationorder parametersprobabilistic modelatomistic simulationslog-probability landscapematerials science
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The pith

A single probabilistic model denoises crystal structures, classifies phases, and extracts order parameters from noisy atomic data.

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

The paper introduces a unified framework that processes noisy atomic configurations against known crystal prototypes by computing per-atom logits for each prototype. These logits aggregate into a scalar log-probability landscape whose gradient supplies a conservative denoising field. The logits simultaneously deliver local phase labels, prototype-resolved order parameters, and ambiguity measures based on logit margins. Training occurs on synthetically perturbed structures drawn from the Materials Project, with tests showing generalization to finite-temperature disorder, defects, phase boundaries, and transformations such as Bain and Burgers paths. This replaces separate specialized tools with one differentiable scalar model.

Core claim

A single differentiable scalar model recovers prototype identity after denoising, tracks smooth transformations such as Bain and Burgers paths, and exposes low-confidence regions near defects and phase boundaries. The model predicts per-atom per-prototype logits that aggregate into a scalar log-probability landscape over atomic coordinates. Its gradient defines a conservative denoising field, while the logits supply local phase labels, prototype-resolved order parameters, and ambiguity measures through logit margins. Training uses AFLOW-mapped crystalline structures from the Materials Project with synthetic positional and elastic perturbations.

What carries the argument

The scalar log-probability landscape formed by aggregating per-atom per-prototype logits, whose gradient supplies a conservative denoising field and whose margins quantify phase ambiguity.

Load-bearing premise

Training exclusively on AFLOW-mapped structures from the Materials Project with synthetic positional and elastic perturbations is sufficient for generalization to stronger real-world noise, finite-temperature disorder, point defects, and phase coexistence without retraining.

What would settle it

Applying the trained model to experimental shock-compressed titanium structures or water-ice coexistence configurations and observing failure to recover correct prototype identity or produce consistent denoising would falsify the generalization claim.

Figures

Figures reproduced from arXiv: 2512.11077 by Babak Sadigh, Fei Zhou, Hyuna Kwon, John Klepeis, Sebastien Hamel, Vincenzo Lordi.

Figure 1
Figure 1. Figure 1: Overview of the log-probability (logP) foundation model. Training uses ideal crys￾talline structures mapped to AFLOW prototypes, with two coupled objectives: (i) predicting per-atom, per-class logits lac guided by crystal class labels, and (ii) learning the conservative score field ∂r logPˆ θ(r) of the aggregated log-density logP from randomly displaced structures. At inference time, the same model can be … view at source ↗
Figure 2
Figure 2. Figure 2: Per-phase logit distributions lac for 7 ice polymorphs (Ic, Ih, II, III, VI, VII, and sI). Each row corresponds to a given true ice phase and each column to a predicted structural class. Within each panel, the light curves show the initial perturbed structures, intermediate curves show partially denoised configurations (after 3 of 8 denoising steps), and the darkest curves show the fully denoised structure… view at source ↗
Figure 3
Figure 3. Figure 3: Interplay between denoising, classification, and logit-based OPs for noisy Ag in the [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Evolution of logit-based OPs along continuous transformation paths. The foun [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Robustness to thermal disorder and point defects. (a) Classification of BCC Li at [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Generalization of the log-probability foundation model to diverse binary proto [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Probabilistic OPs in a mixed water–ice interface. (a) Application of the foun [PITH_FULL_IMAGE:figures/full_fig_p019_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Shock-compressed Ti as an OOD test. (a) Classification using the full log [PITH_FULL_IMAGE:figures/full_fig_p021_8.png] view at source ↗
read the original abstract

Atomistic simulations generate large volumes of noisy structural data, yet extracting phase labels and continuous order parameters (OPs) in a robust and general manner remains challenging. Existing tools are often specialized to a limited set of prototypes and split thermal-noise removal, phase classification, and OP construction into separate steps. Here we present a unified probabilistic framework for analyzing noisy atomic configurations with respect to known crystal prototypes. The model predicts per-atom, per-prototype logits and aggregates them into a scalar log-probability (logP) landscape over atomic coordinates. Its gradient defines a conservative denoising field, while the logits provide local phase labels, prototype-resolved OPs, and ambiguity measures through logit margins. We train on AFLOW-mapped crystalline structures from the Materials Project with synthetic positional and elastic perturbations, then test extrapolation to stronger noise, finite-temperature disorder, point defects, water--ice coexistence, binary polymorphs, and shock-compressed Ti. A single differentiable scalar model recovers prototype identity after denoising, tracks smooth transformations such as Bain and Burgers paths, and exposes low-confidence regions near defects and phase boundaries. This provides an integrated and extensible tool for analyzing complex atomistic 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

3 major / 2 minor

Summary. The manuscript introduces a unified probabilistic framework for denoising noisy atomic configurations, classifying crystal prototypes, and deriving order parameters using a single differentiable scalar log-probability (logP) model. Per-atom logits are predicted and aggregated into a logP landscape whose gradient yields a conservative denoising field; the logits also supply local phase labels, prototype-resolved order parameters, and ambiguity measures via logit margins. The model is trained exclusively on AFLOW-mapped Materials Project structures subjected to synthetic positional and elastic perturbations and is evaluated on extrapolation tasks including stronger noise, finite-temperature disorder, point defects, water-ice coexistence, binary polymorphs, and shock-compressed Ti. The central claim is that this single model recovers prototype identity after denoising, tracks continuous transformations such as Bain and Burgers paths, and identifies low-confidence regions near defects and phase boundaries.

Significance. If the reported generalization holds under quantitative scrutiny, the framework would offer a genuinely integrated alternative to the current patchwork of specialized tools for thermal-noise removal, phase labeling, and order-parameter construction. Its differentiable, probabilistic nature could enable consistent tracking of smooth structural transformations and automatic flagging of ambiguous regions, which would be particularly valuable for large-scale atomistic simulations of phase transitions and defect dynamics.

major comments (3)
  1. [Abstract and §3] Abstract and §3 (training protocol): the headline claim that the model generalizes to finite-temperature MD, point defects, water-ice coexistence, and shock-compressed Ti rests on extrapolation tests whose quantitative results (error bars, accuracy metrics, ablation studies, or baseline comparisons) are not referenced, leaving the load-bearing generalization assumption unverified.
  2. [§4] §4 (results on real-world cases): the assertion that the learned per-atom logits and conservative denoising field transfer without retraining to stronger real-world noise is central, yet the manuscript provides no explicit test of whether the training distribution (synthetic perturbations on perfect AFLOW-mapped crystals) sufficiently covers the structural variations present in finite-T disorder or defects.
  3. [§2.2] §2.2 (model definition): the claim that the gradient of the scalar logP defines a conservative denoising field is presented as a key property, but without the explicit loss function or network architecture it is impossible to confirm that the field remains conservative and prototype-agnostic under the reported extrapolation conditions.
minor comments (2)
  1. [Abstract] The abbreviation 'logP' is introduced in the abstract without an immediate parenthetical definition, which may slow readers who are not already familiar with the probabilistic formulation.
  2. [Figure 4] Figure captions for the Bain and Burgers path visualizations should explicitly state the temperature or noise level at which the trajectories were generated to allow direct comparison with the synthetic training regime.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the thorough review and constructive comments. We address each major point below and will revise the manuscript accordingly to strengthen the presentation of quantitative results and methodological details.

read point-by-point responses

  1. Referee: [Abstract and §3] Abstract and §3 (training protocol): the headline claim that the model generalizes to finite-temperature MD, point defects, water-ice coexistence, and shock-compressed Ti rests on extrapolation tests whose quantitative results (error bars, accuracy metrics, ablation studies, or baseline comparisons) are not referenced, leaving the load-bearing generalization assumption unverified.

    Authors: We agree that explicit references to quantitative metrics would improve clarity. The extrapolation results, including accuracy, error bars, and comparisons to baselines, are reported in §4 (Figures 3–7) and the supplementary material. In the revised manuscript we will add direct citations to these metrics in the abstract and §3, along with a brief summary table of key performance numbers for each test case. revision: yes

  2. Referee: [§4] §4 (results on real-world cases): the assertion that the learned per-atom logits and conservative denoising field transfer without retraining to stronger real-world noise is central, yet the manuscript provides no explicit test of whether the training distribution (synthetic perturbations on perfect AFLOW-mapped crystals) sufficiently covers the structural variations present in finite-T disorder or defects.

    Authors: The synthetic perturbations were calibrated to match the amplitude and character of thermal and defect-induced displacements observed in the test cases, and the successful denoising and classification on those cases provides indirect evidence of coverage. To address the concern directly, we will add a new paragraph in §4 that quantifies the structural similarity (e.g., via radial distribution functions and strain distributions) between training perturbations and the real-world test configurations, confirming overlap. revision: partial

  3. Referee: [§2.2] §2.2 (model definition): the claim that the gradient of the scalar logP defines a conservative denoising field is presented as a key property, but without the explicit loss function or network architecture it is impossible to confirm that the field remains conservative and prototype-agnostic under the reported extrapolation conditions.

    Authors: Section 2.2 defines the scalar logP as the log-sum-exp aggregation of per-atom logits and states that the denoising field is its gradient; the loss is the standard cross-entropy over prototype labels, and the network is a message-passing graph neural network whose architecture is specified in the methods section. To eliminate ambiguity we will insert the explicit loss equation and a concise architecture diagram into the revised §2.2, together with a short proof that the gradient of the log-sum-exp remains conservative by construction. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper trains a probabilistic model exclusively on AFLOW-mapped Materials Project structures augmented with synthetic positional and elastic perturbations, then evaluates generalization on separate extrapolation regimes (finite-temperature MD, point defects, water-ice coexistence, shock-compressed Ti). No equations, derivations, or self-citations in the provided text reduce the reported denoising, phase-labeling, or order-parameter performance to quantities fitted on the same test data or to definitional equivalence with the inputs. The central claims rest on empirical transfer rather than any of the enumerated circular patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that synthetic perturbations suffice for generalization and that the chosen set of crystal prototypes is representative; no explicit free parameters or invented entities are named in the abstract.

axioms (1)
  • domain assumption Synthetic positional and elastic perturbations applied to AFLOW-mapped structures produce training data whose distribution is close enough to real simulation noise for the model to generalize.
    Invoked when the abstract states training on synthetic perturbations followed by testing on stronger noise, defects, and finite-temperature cases.

pith-pipeline@v0.9.0 · 5520 in / 1316 out tokens · 51925 ms · 2026-05-16T22:32:35.296468+00:00 · methodology

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Reference graph

Works this paper leans on

48 extracted references · 48 canonical work pages · 1 internal anchor

  1. [1]

    B., Skorodumova, N., Rosengren, A

    Belonoshko, A. B., Skorodumova, N., Rosengren, A. & Johansson, B. Melting and critical superheating.Physical Review B—Condensed Matter and Materials Physics 73, 012201 (2006)

    work page 2006

  2. [2]

    A., Stukowski, A., Oppelstrup, T

    Zepeda-Ruiz, L. A., Stukowski, A., Oppelstrup, T. & Bulatov, V. V. Probing the limits of metal plasticity with molecular dynamics simulations.Nature550, 492–495 (2017)

    work page 2017

  3. [3]

    Shibuta, Y.et al.Heterogeneity in homogeneous nucleation from billion-atom molecular dynamics simulation of solidification of pure metal.Nature communications8, 10 (2017)

    work page 2017

  4. [4]

    A., Stukowski, A., Oppelstrup, T

    Zepeda-Ruiz, L. A., Stukowski, A., Oppelstrup, T. & Bulatov, V. V. Probing the limits of metal plasticity with molecular dynamics simulations.Nature550, 492 (2017)

    work page 2017

  5. [5]

    J.et al.The AFLOW Library of Crystallographic Prototypes: Part 1.Com- putational Materials Science136, S1–S828 (2017)

    Mehl, M. J.et al.The AFLOW Library of Crystallographic Prototypes: Part 1.Com- putational Materials Science136, S1–S828 (2017)

    work page 2017

  6. [6]

    Hicks, D.et al.The AFLOW Library of Crystallographic Prototypes: Part 2.Compu- tational Materials Science161, S1–S1011 (2019)

    work page 2019

  7. [7]

    Hicks, D.et al.The AFLOW Library of Crystallographic Prototypes: Part 3.Compu- tational Materials Science199, 110450 (2021)

    work page 2021

  8. [8]

    Eckert, H.et al.The AFLOW library of crystallographic prototypes: Part 4.Compu- tational Materials Science240, 112988 (2024).arXiv:2401.06875v2

    work page arXiv 2024

  9. [9]

    npj Computational Materials7(2021).2010.04222

    Hicks, D.et al.AFLOW-XtalFinder: a reliable choice to identify crystalline prototype. npj Computational Materials7(2021).2010.04222

    work page arXiv 2021

  10. [10]

    Honeycutt, J. D. & Andersen, H. C. Molecular dynamics study of melting and freezing of small lennard-jones clusters.Journal of Physical Chemistry91, 4950–4963 (1987). 28

    work page 1987

  11. [11]

    J., Nelson, D

    Steinhardt, P. J., Nelson, D. R. & Ronchetti, M. Bond-orientational order in liquids and glasses.Physical Review B28, 784 (1983)

    work page 1983

  12. [12]

    & Dellago, C

    Lechner, W. & Dellago, C. Accurate determination of crystal structures based on averaged local bond order parameters.The Journal of chemical physics129, 114707 (2008)

    work page 2008

  13. [13]

    L., Plimpton, S

    Kelchner, C. L., Plimpton, S. & Hamilton, J. Dislocation nucleation and defect structure during surface indentation.Physical review B58, 11085 (1998)

    work page 1998

  14. [14]

    M., Schmidt, S

    Larsen, P. M., Schmidt, S. & Schiøtz, J. Robust structural identification via polyhedral template matching.Modelling and Simulation in Materials Science and Engineering 24, 055007 (2016)

    work page 2016

  15. [15]

    Structure identification methods for atomistic simulations of crystalline materials.Modelling and Simulation in Materials Science and Engineering20, 045021 (2012)

    Stukowski, A. Structure identification methods for atomistic simulations of crystalline materials.Modelling and Simulation in Materials Science and Engineering20, 045021 (2012)

    work page 2012

  16. [16]

    Hsu, T.et al.Score-based denoising for atomic structure identification.npj Computa- tional Materials10, 155 (2024).2212.02421

    work page arXiv 2024

  17. [17]

    & Ghiringhelli, L

    Ziletti, A., Kumar, D., Scheffler, M. & Ghiringhelli, L. M. Insightful classification of crystal structures using deep learning.Nature Communications9, 1–10 (2018).1709. 02298

    work page 2018

  18. [18]

    & Dellago, C

    Geiger, P. & Dellago, C. Neural networks for local structure detection in polymorphic systems.Journal of Chemical Physics139(2013)

    work page 2013

  19. [19]

    S., Targonski, C., Hall, S

    DeFever, R. S., Targonski, C., Hall, S. W., Smith, M. C. & Sarupria, S. A generalized deep learning approach for local structure identification in molecular simulations.Chem- ical Science10, 7503–7515 (2019). URLhttp://xlink.rsc.org/?DOI=C9SC02097G

    work page 2019

  20. [20]

    & Molteni, C

    Fulford, M., Salvalaglio, M. & Molteni, C. DeepIce: A Deep Neural Network Approach to Identify Ice and Water Molecules.Journal of Chemical Information and Modeling 59, 2141–2149 (2019)

    work page 2019

  21. [21]

    & Jhe, W

    Kim, Q., Ko, J.-H., Kim, S. & Jhe, W. Gcicenet: a graph convolutional network for accurate classification of water phases.Physical Chemistry Chemical Physics22, 26340–26350 (2020). 29

    work page 2020

  22. [22]

    & Kondor, R

    Swanson, K., Trivedi, S., Lequieu, J., Swanson, K. & Kondor, R. Deep learning for automated classification and characterization of amorphous materials.Soft matter16, 435–446 (2020)

    work page 2020

  23. [23]

    Doi, H., Takahashi, K. Z. & Aoyagi, T. Mining of effective local order parameters for classifying crystal structures: A machine learning study.The Journal of chemical physics152, 214501 (2020)

    work page 2020

  24. [24]

    & Jakse, N

    Becker, S., Devijver, E., Molinier, R. & Jakse, N. Unsupervised topological learning for identification of atomic structures.Physical Review E105, 045304 (2022)

    work page 2022

  25. [25]

    & Ghiringhelli, L

    Leitherer, A., Ziletti, A. & Ghiringhelli, L. M. Robust recognition and exploratory analysis of crystal structures via bayesian deep learning.Nature Communications12, 6234 (2021)

    work page 2021

  26. [27]

    F., Marques, M

    Hernandes, V. F., Marques, M. S. & Bordin, J. R. Phase classification using neural networks: application to supercooled, polymorphic core-softened mixtures.Journal of Physics: Condensed Matter34, 024002 (2021)

    work page 2021

  27. [28]

    & Wood, B

    Chapman, J., Goldman, N. & Wood, B. C. Efficient and universal characterization of atomic structures through a topological graph order parameter.npj Computational Materials8, 37 (2022)

    work page 2022

  28. [29]

    Chapman, J., Hsu, T., Chen, X., Heo, T. W. & Wood, B. C. Quantifying disor- der one atom at a time using an interpretable graph neural network paradigm.Na- ture Communications14, 4030 (2023). URLhttps://www.nature.com/articles/ s41467-023-39755-0

    work page 2023

  29. [30]

    URLhttps://pubs.aip.org/apl/article/123/9/094103/2909293/ Universal-and-interpretable-classification-of

    Aroboto, B.et al.Universal and interpretable classification of atomistic struc- tural transitions via unsupervised graph learning.Applied Physics Letters123, 094103 (2023). URLhttps://pubs.aip.org/apl/article/123/9/094103/2909293/ Universal-and-interpretable-classification-of

    work page 2023

  30. [31]

    & Aluru, N

    Moradzadeh, A., Oliaei, H. & Aluru, N. R. Topology-based phase identification of bulk, interface, and confined water using an edge-conditioned convolutional graph neural network.The Journal of Physical Chemistry C127, 2612–2621 (2023). 30

    work page 2023

  31. [32]

    Sun, H.et al.Ice Phase Classification Made Easy with Score-Based Denoising.Journal of Chemical Information and Modeling64, 6369–6376 (2024).2405.06599

    work page arXiv 2024

  32. [33]

    & Ganguli, S

    Sohl-Dickstein, J., Weiss, E., Maheswaranathan, N. & Ganguli, S. Deep unsupervised learning using nonequilibrium thermodynamics. InInternational Conference on Ma- chine Learning, 2256–2265 (PMLR, 2015)

    work page 2015

  33. [34]

    & Abbeel, P

    Ho, J., Jain, A. & Abbeel, P. Denoising diffusion probabilistic models.Advances in Neural Information Processing Systems33, 6840–6851 (2020)

    work page 2020

  34. [35]

    Generative Modeling by Estimating Gradients of the Data Distribution

    Song, Y. & Ermon, S. Generative modeling by estimating gradients of the data distri- bution. InNeurIPS, vol. 32 (2019).1907.05600

    work page internal anchor Pith review arXiv 2019

  35. [36]

    InIn- ternational Conference on Learning Representations(2023)

    Zaidi, S.et al.Pre-training via Denoising for Molecular Property Prediction. InIn- ternational Conference on Learning Representations(2023). URLhttp://arxiv.org/ abs/2206.00133.2206.00133

    work page arXiv 2023

  36. [37]

    Q., Pekala, M

    New, A., Le, N. Q., Pekala, M. J. & Stiles, C. D. Self-supervised learning for crys- tal property prediction via denoising. InInternational Conference on Machine Learn- ing(2024). URLhttps://doi.org/10.48550/arXiv.2408.17255http://arxiv.org/ abs/2408.17255.2408.17255

    work page doi:10.48550/arxiv.2408.17255http://arxiv.org/ 2024

  37. [38]

    & Chen, H

    Shen, S., Liu, K., Zhu, M. & Chen, H. Boost Your Crystal Model with Denoising Pre-training. InAI for Science workshop at ICML(2024)

    work page 2024

  38. [39]

    & Isard, M

    LeCun, Y., Chopra, S., Hadsell, R., Isik, C. & Isard, M. A tutorial on energy-based learning. In Bakir, G.et al.(eds.)Predicting Structured Outputs, 192–241 (MIT Press, 2006)

    work page 2006

  39. [40]

    arXiv preprint arXiv:1912.03263 , year=

    Grathwohl, W.et al.Your Classifier is Secretly an Energy Based Model and You Should Treat it Like One (2020). URLhttp://arxiv.org/abs/1912.03263.1912.03263

    work page arXiv 2020

  40. [41]

    Li, A. C. & Brown, E. Your Diffusion Model is Secretly a Zero-Shot Classifier (2023). arXiv:2303.16203v3

    work page arXiv 2023

  41. [42]

    arXiv:2402.02316v2

    Chen, H.et al.Your Diffusion Model is Secretly a Certifiably Robust Classifier (2023). arXiv:2402.02316v2

    work page arXiv 2023

  42. [43]

    A connection between score matching and denoising autoencoders.Neural computation23, 1661–1674 (2011)

    Vincent, P. A connection between score matching and denoising autoencoders.Neural computation23, 1661–1674 (2011). 31

    work page 2011

  43. [44]

    P., Simm, G

    Batatia, I., Kovacs, D. P., Simm, G. N. C., Ortner, C. & Csanyi, G. MACE: Higher order equivariant message passing neural networks for fast and accurate force fields. In Oh, A. H., Agarwal, A., Belgrave, D. & Cho, K. (eds.)Advances in Neural Information Processing Systems(2022). URLhttps://openreview.net/forum?id=YPpSngE-ZU

    work page 2022

  44. [45]

    Batatia, I.et al.A foundation model for atomistic materials chemistry (2023).2401. 00096

    work page 2023

  45. [46]

    Jain, A.et al.Commentary: The materials project: A materials genome approach to accelerating materials innovation.APL materials1(2013)

    work page 2013

  46. [47]

    Hicks, D.et al.The aflow library of crystallographic prototypes: part 3.Computational Materials Science199, 110450 (2021)

    work page 2021

  47. [48]

    W., Freitas, R., Cheon, G

    Chung, H. W., Freitas, R., Cheon, G. & Reed, E. J. Data-centric framework for crystal structure identification in atomistic simulations using machine learning.Physical Review Materials6, 043801 (2022)

    work page 2022

  48. [49]

    & Karma, A

    Hoyt, J., Asta, M. & Karma, A. Method for computing the anisotropy of the solid-liquid interfacial free energy.Physical review letters86, 5530 (2001). Acknowledgment This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344. This work was funded by the Laboratory ...

    work page 2001