pith. sign in

arxiv: 2606.11038 · v1 · pith:BWN4K6T6new · submitted 2026-06-09 · ❄️ cond-mat.mtrl-sci · physics.comp-ph

Synthetic pre-training of graph-network models for predicting solid-state NMR parameters

Chiheb Ben Mahmoud , Carlos Bornes , Christopher J. Heard , Luk\'a\v{s} Grajciar , Jonathan R. Yates , Volker L. Deringer This is my paper

Pith reviewed 2026-06-27 12:26 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci physics.comp-ph
keywords solid-state NMRgraph neural networkspre-trainingmachine learningtensorial propertiesdata efficiencymaterials modelingfine-tuning
0
0 comments X

The pith

Graph models for solid-state NMR parameters gain data efficiency from synthetic pre-training before fine-tuning on real calculations.

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

The paper shows that graph-based machine-learning models for predicting tensorial NMR parameters can be pre-trained on inexpensive synthetic data generated by an earlier model and then fine-tuned on smaller sets of accurate first-principles calculations. The largest gains in data efficiency appear when the pre-training and fine-tuning datasets cover the same range of compositions and atomic arrangements. This approach is tested for chemical transferability across different material spaces. The method therefore reduces the number of costly quantum-mechanical calculations needed to reach useful predictive accuracy.

Core claim

A synthetic pre-training and fine-tuning protocol for graph-network models improves data efficiency for solid-state NMR tensor predictions when the pre-training data and subsequent ground-truth data share the same compositional and configurational space; the protocol begins with an existing ML model to generate synthetic tensorial supervision and follows with targeted refinement on first-principles data.

What carries the argument

Two-stage synthetic pre-training and fine-tuning protocol on graph networks, where models first learn from ML-generated synthetic NMR tensors and then adapt to first-principles reference data.

If this is right

  • Fewer first-principles calculations are required to reach a target accuracy level when pre-training and fine-tuning spaces overlap.
  • The protocol supports initial exploration of chemical transferability between different material families.
  • Training workflows for other tensorial properties can combine cheap synthetic labels with sparse accurate labels.
  • High-throughput screening of solid-state NMR signatures becomes more feasible with reduced computational cost.

Where Pith is reading between the lines

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

  • The same pre-training logic could be applied to other expensive tensorial properties such as electric-field gradients or magnetic susceptibilities.
  • If the quality of the synthetic generator improves over time, the amount of required ground-truth data could shrink further.
  • The approach may help build larger, more diverse training sets for NMR models by bootstrapping from existing cheaper predictions.

Load-bearing premise

Synthetic tensor data produced by the existing ML model must contain enough transferable features to serve as useful supervision for models that will later see real first-principles data.

What would settle it

Train two otherwise identical graph models on the same small set of ground-truth NMR data, one with and one without the synthetic pre-training stage, and check whether the pre-trained version shows clearly higher accuracy on held-out structures.

Figures

Figures reproduced from arXiv: 2606.11038 by Carlos Bornes, Chiheb Ben Mahmoud, Christopher J. Heard, Jonathan R. Yates, Luk\'a\v{s} Grajciar, Volker L. Deringer.

Figure 1
Figure 1. Figure 1: Synthetic pre-training for graph-based models of tensorial NMR parameters. The [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Evolution of errors in synthetically-pre-trained-and-fine-tuned and directly-trained [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Evolution of errors in TensorMACE-5000-F(L) pre-trained using synthetic labels of different quality in a-SiO2. We show the prediction errors for isotropic and anisotropic components of the MS tensor for silicon and oxygen as a function of the QM NMR training structures. Blue circles and oranges crosses correspond to models pre-trained with higher- and lower-quality synthetic labels, respectively; empty bla… view at source ↗
Figure 4
Figure 4. Figure 4: Evolution of errors in synthetically-pre-trained-and-fine-tuned and directly-trained small [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Orientation-resolved deviations of the predicted [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Transferability of synthetically pre-trained models from a-SiO [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Transferability of synthetically pre-trained models from a-SiO [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
read the original abstract

Nuclear magnetic resonance (NMR) is a powerful probe of atomic structure, but accurate quantum-mechanical predictions of tensorial NMR parameters are computationally demanding. This creates a bottleneck both for direct quantum-mechanical studies and for collecting high-quality training data for machine-learning (ML) models. Here, we introduce a synthetic pre-training and fine-tuning protocol for graph-based ML models of solid-state NMR parameters. We first pre-train models on synthetic tensorial data, as obtained using an existing ML model, and subsequently fine-tune those models on new ground-truth data. We observe a pronounced improvement in data efficiency when pre-training and fine-tuning span the same compositional and configurational space, and we carry out initial experiments regarding chemical transferability. Our work outlines a route toward future data-efficient training workflows for tensorial ML models for solid-state NMR, combining inexpensive synthetic supervision with targeted first-principles refinement.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 1 minor

Summary. The manuscript introduces a synthetic pre-training and fine-tuning protocol for graph-based ML models predicting solid-state NMR tensorial parameters. Models are first pre-trained on synthetic tensors generated by an existing ML model, then fine-tuned on ground-truth first-principles data. The central observation is a pronounced improvement in data efficiency when pre-training and fine-tuning occur in the same compositional and configurational space, accompanied by initial experiments on chemical transferability.

Significance. If the central claim holds with appropriate controls, the work could provide a practical route to more data-efficient training of tensorial ML models in materials science by combining inexpensive synthetic supervision with targeted first-principles refinement, addressing the computational bottleneck in generating high-quality NMR training data.

major comments (2)
  1. [Abstract] Abstract: the claim of 'pronounced improvement in data efficiency' is presented without any quantitative details on error metrics, baselines, dataset sizes, error bars, or experimental protocols, preventing assessment of whether the observation is supported by evidence.
  2. [Methods / Data description] The manuscript does not specify the training corpus of the existing ML model used to generate the synthetic tensors. This is load-bearing for the transfer claim, because overlap between that corpus and the fine-tuning sets could render the observed efficiency gain an artifact of distilling the existing model's training distribution rather than a general benefit of synthetic supervision.
minor comments (1)
  1. The abstract and title could more explicitly name the graph-network architecture and the specific NMR parameters (e.g., chemical shielding tensors) under study.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive feedback. We address the two major comments point by point below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim of 'pronounced improvement in data efficiency' is presented without any quantitative details on error metrics, baselines, dataset sizes, error bars, or experimental protocols, preventing assessment of whether the observation is supported by evidence.

    Authors: We agree that the abstract would benefit from quantitative support for the central claim. In the revised manuscript we will expand the abstract to include specific error metrics (MAE values before/after fine-tuning), baseline comparisons, the range of dataset sizes examined, and explicit reference to error bars and protocols reported in the results section. revision: yes

  2. Referee: [Methods / Data description] The manuscript does not specify the training corpus of the existing ML model used to generate the synthetic tensors. This is load-bearing for the transfer claim, because overlap between that corpus and the fine-tuning sets could render the observed efficiency gain an artifact of distilling the existing model's training distribution rather than a general benefit of synthetic supervision.

    Authors: We acknowledge that the training corpus of the pre-existing model was not described in sufficient detail. We will add an explicit subsection in the Methods that states the composition, size, and source of that corpus together with a direct comparison of its chemical space to the fine-tuning sets used in our experiments. This addition will allow readers to evaluate possible distributional overlap. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper describes an empirical pre-training protocol that generates synthetic tensorial NMR data from an existing ML model and then fine-tunes on independent ground-truth first-principles labels. The central observations concern data-efficiency gains when compositional/configurational spaces overlap; these are presented as experimental results rather than derivations that reduce to fitted parameters or self-citations by construction. No load-bearing step equates a prediction to its own input via definition, renaming, or an unverified self-citation chain. The workflow remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides insufficient detail to identify specific free parameters, axioms, or invented entities in the method.

pith-pipeline@v0.9.1-grok · 5707 in / 1203 out tokens · 25363 ms · 2026-06-27T12:26:19.894708+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

61 extracted references · 51 canonical work pages

  1. [1]

    Kuhn , author R

    author author S. Kuhn , author R. P. \ De Jesus ,\ and\ author R. M. \ Borges ,\ https://doi.org/10.3390/encyclopedia4040102 journal journal Encyclopedia \ volume 4 ,\ pages 1568 ( year 2024 ) NoStop

    work page doi:10.3390/encyclopedia4040102 2024

  2. [2]

    Das \ and\ author K

    author author S. Das \ and\ author K. M. \ Merz ,\ https://doi.org/10.1021/acs.chemrev.5c00259 journal journal Chemical Reviews \ volume 125 ,\ pages 9256 ( year 2025 ) NoStop

    work page doi:10.1021/acs.chemrev.5c00259 2025

  3. [3]

    Rupp , author R

    author author M. Rupp , author R. Ramakrishnan ,\ and\ author O. A. \ Von Lilienfeld ,\ https://doi.org/10.1021/acs.jpclett.5b01456 journal journal The Journal of Physical Chemistry Letters \ volume 6 ,\ pages 3309 ( year 2015 ) NoStop

    work page doi:10.1021/acs.jpclett.5b01456 2015

  4. [4]

    Cuny , author Y

    author author J. Cuny , author Y. Xie , author C. J. \ Pickard ,\ and\ author A. A. \ Hassanali ,\ https://doi.org/10.1021/acs.jctc.5b01006 journal journal Journal of Chemical Theory and Computation \ volume 12 ,\ pages 765 ( year 2016 ) NoStop

    work page doi:10.1021/acs.jctc.5b01006 2016

  5. [5]

    author author F. M. \ Paruzzo , author A. Hofstetter , author F. Musil , author S. De , author M. Ceriotti ,\ and\ author L. Emsley ,\ https://doi.org/10.1038/s41467-018-06972-x journal journal Nature Communications \ volume 9 ,\ pages 4501 ( year 2018 ) NoStop

    work page doi:10.1038/s41467-018-06972-x 2018

  6. [6]

    Chaker , author M

    author author Z. Chaker , author M. Salanne , author J.-M. \ Delaye ,\ and\ author T. Charpentier ,\ https://doi.org/10.1039/C9CP02803J journal journal Physical Chemistry Chemical Physics \ volume 21 ,\ pages 21709 ( year 2019 ) NoStop

    work page doi:10.1039/c9cp02803j 2019

  7. [7]

    Gerrard , author L

    author author W. Gerrard , author L. A. \ Bratholm , author M. J. \ Packer , author A. J. \ Mulholland , author D. R. \ Glowacki ,\ and\ author C. P. \ Butts ,\ https://doi.org/10.1039/C9SC03854J journal journal Chemical Science \ volume 11 ,\ pages 508 ( year 2020 ) NoStop

    work page doi:10.1039/c9sc03854j 2020

  8. [8]

    Kwon , author D

    author author Y. Kwon , author D. Lee , author Y.-S. \ Choi , author M. Kang ,\ and\ author S. Kang ,\ https://doi.org/10.1021/acs.jcim.0c00195 journal journal Journal of Chemical Information and Modeling \ volume 60 ,\ pages 2024 ( year 2020 ) NoStop

    work page doi:10.1021/acs.jcim.0c00195 2024

  9. [9]

    Han , author H

    author author J. Han , author H. Kang , author S. Kang , author Y. Kwon , author D. Lee ,\ and\ author Y.-S. \ Choi ,\ https://doi.org/10.1039/D2CP04542G journal journal Physical Chemistry Chemical Physics \ volume 24 ,\ pages 26870 ( year 2022 ) NoStop

    work page doi:10.1039/d2cp04542g 2022

  10. [10]

    author author M. C. \ Venetos , author M. Wen ,\ and\ author K. A. \ Persson ,\ https://doi.org/10.1021/acs.jpca.2c07530 journal journal The Journal of Physical Chemistry A \ volume 127 ,\ pages 2388 ( year 2023 ) NoStop

    work page doi:10.1021/acs.jpca.2c07530 2023

  11. [11]

    Bånkestad , author K

    author author M. Bånkestad , author K. M. \ Dorst , author G. Widmalm ,\ and\ author J. Rönnols ,\ https://doi.org/10.1039/D4RA03428G journal journal RSC Advances \ volume 14 ,\ pages 26585 ( year 2024 ) NoStop

    work page doi:10.1039/d4ra03428g 2024

  12. [12]

    Charpentier ,\ https://doi.org/10.1039/D4FD00129J journal journal Faraday Discussions \ volume 255 ,\ pages 370 ( year 2025 ) NoStop

    author author T. Charpentier ,\ https://doi.org/10.1039/D4FD00129J journal journal Faraday Discussions \ volume 255 ,\ pages 370 ( year 2025 ) NoStop

    work page doi:10.1039/d4fd00129j 2025

  13. [13]

    author author A. F. \ Harper , author S. S. \ Köcher , author K. Reuter ,\ and\ author C. Scheurer ,\ https://doi.org/10.1039/D5TA05090A journal journal Journal of Materials Chemistry A \ volume 13 ,\ pages 35389 ( year 2025 ) NoStop

    work page doi:10.1039/d5ta05090a 2025

  14. [14]

    Grisafi , author D

    author author A. Grisafi , author D. M. \ Wilkins , author G. Csányi ,\ and\ author M. Ceriotti ,\ https://doi.org/10.1103/PhysRevLett.120.036002 journal journal Physical Review Letters \ volume 120 ,\ pages 036002 ( year 2018 ) NoStop

    work page doi:10.1103/physrevlett.120.036002 2018

  15. [15]

    Nature Communications 13(1) (2022) https://doi.org/10.1038/s41467-022-29939-5

    author author S. Batzner , author A. Musaelian , author L. Sun , author M. Geiger , author J. P. \ Mailoa , author M. Kornbluth , author N. Molinari , author T. E. \ Smidt ,\ and\ author B. Kozinsky ,\ https://doi.org/10.1038/s41467-022-29939-5 journal journal Nature Communications \ volume 13 ,\ pages 2453 ( year 2022 ) NoStop

    work page doi:10.1038/s41467-022-29939-5 2022

  16. [16]

    Batatia , author D

    author author I. Batatia , author D. P. \ Kovacs , author G. Simm , author C. Ortner ,\ and\ author G. Csanyi ,\ in\ https://proceedings.neurips.cc/paper_files/paper/2022/file/4a36c3c51af11ed9f34615b81edb5bbc-Paper-Conference.pdf booktitle Advances in neural information processing systems ,\ Vol. volume 35 ,\ editor edited by\ editor S. Koyejo , editor S....

    2022

  17. [17]

    Duval , author S

    author author A. Duval , author S. V. \ Mathis , author C. K. \ Joshi , author V. Schmidt , author S. Miret , author F. D. \ Malliaros , author T. Cohen , author P. Liò , author Y. Bengio ,\ and\ author M. Bronstein ,\ http://arxiv.org/abs/2312.07511 title A Hitchhiker 's Guide to Geometric GNNs for 3D Atomic Systems , \ ( year 2024 ),\ note arXiv:2312.07...

    arXiv 2024

  18. [18]

    Ben Mahmoud , author L

    author author C. Ben Mahmoud , author L. A. M. \ Rosset , author J. R. \ Yates ,\ and\ author V. L. \ Deringer ,\ https://doi.org/10.1063/5.0274240 journal journal The Journal of Chemical Physics \ volume 163 ,\ pages 024118 ( year 2025 a ) NoStop

    work page doi:10.1063/5.0274240 2025

  19. [19]

    Kellner , author J

    author author M. Kellner , author J. B. \ Holmes , author R. Rodriguez-Madrid , author F. Viscosi , author Y. Zhang , author L. Emsley ,\ and\ author M. Ceriotti ,\ https://doi.org/10.1021/acs.jpclett.5c01819 journal journal The Journal of Physical Chemistry Letters \ volume 16 ,\ pages 8714 ( year 2025 ) NoStop

    work page doi:10.1021/acs.jpclett.5c01819 2025

  20. [20]

    Ditchfield ,\ https://doi.org/10.1080/00268977400100711 journal journal Molecular Physics \ volume 27 ,\ pages 789 ( year 1974 ) NoStop

    author author R. Ditchfield ,\ https://doi.org/10.1080/00268977400100711 journal journal Molecular Physics \ volume 27 ,\ pages 789 ( year 1974 ) NoStop

    work page doi:10.1080/00268977400100711 1974

  21. [21]

    Wolinski , author J

    author author K. Wolinski , author J. F. \ Hinton ,\ and\ author P. Pulay ,\ https://doi.org/10.1021/ja00179a005 journal journal Journal of the American Chemical Society \ volume 112 ,\ pages 8251 ( year 1990 ) NoStop

    work page doi:10.1021/ja00179a005 1990

  22. [22]

    author author C. J. \ Pickard \ and\ author F. Mauri ,\ https://doi.org/10.1103/PhysRevB.63.245101 journal journal Physical Review B \ volume 63 ,\ pages 245101 ( year 2001 ) NoStop

    work page doi:10.1103/physrevb.63.245101 2001

  23. [23]

    author author J. R. \ Yates , author C. J. \ Pickard ,\ and\ author F. Mauri ,\ https://doi.org/10.1103/PhysRevB.76.024401 journal journal Physical Review B \ volume 76 ,\ pages 024401 ( year 2007 ) NoStop

    work page doi:10.1103/physrevb.76.024401 2007

  24. [24]

    Charpentier ,\ https://doi.org/10.1016/j.ssnmr.2011.04.006 journal journal Solid State Nuclear Magnetic Resonance \ volume 40 ,\ pages 1 ( year 2011 ) NoStop

    author author T. Charpentier ,\ https://doi.org/10.1016/j.ssnmr.2011.04.006 journal journal Solid State Nuclear Magnetic Resonance \ volume 40 ,\ pages 1 ( year 2011 ) NoStop

    work page doi:10.1016/j.ssnmr.2011.04.006 2011

  25. [25]

    Bonhomme , author C

    author author C. Bonhomme , author C. Gervais , author F. Babonneau , author C. Coelho , author F. Pourpoint , author T. Azaïs , author S. E. \ Ashbrook , author J. M. \ Griffin , author J. R. \ Yates , author F. Mauri ,\ and\ author C. J. \ Pickard ,\ https://doi.org/10.1021/cr300108a journal journal Chemical Reviews \ volume 112 ,\ pages 5733 ( year 201...

    work page doi:10.1021/cr300108a 2012

  26. [26]

    Deng , author P

    author author B. Deng , author P. Zhong , author K. Jun , author J. Riebesell , author K. Han , author C. J. \ Bartel ,\ and\ author G. Ceder ,\ https://doi.org/10.1038/s42256-023-00716-3 journal journal Nature Machine Intelligence \ volume 5 ,\ pages 1031 ( year 2023 ) NoStop

    work page doi:10.1038/s42256-023-00716-3 2023

  27. [27]

    Neumann , author J

    author author M. Neumann , author J. Gin , author B. Rhodes , author S. Bennett , author Z. Li , author H. Choubisa , author A. Hussey ,\ and\ author J. Godwin ,\ https://doi.org/10.48550/ARXIV.2410.22570 title Orb: A Fast , Scalable Neural Network Potential , \ ( year 2024 ),\ note version Number: 1 NoStop

    work page doi:10.48550/arxiv.2410.22570 2024

  28. [28]

    Batatia , author P

    author author I. Batatia , author P. Benner , author Y. Chiang , author A. M. \ Elena , author D. P. \ Kovács , author J. Riebesell , author X. R. \ Advincula , author M. Asta , author M. Avaylon , author W. J. \ Baldwin , author F. Berger , author N. Bernstein , author A. Bhowmik , author F. Bigi , author S. M. \ Blau , author V. Cărare , author M. Cerio...

    work page doi:10.1063/5.0297006 2025

  29. [29]

    Mazitov , author F

    author author A. Mazitov , author F. Bigi , author M. Kellner , author P. Pegolo , author D. Tisi , author G. Fraux , author S. Pozdnyakov , author P. Loche ,\ and\ author M. Ceriotti ,\ https://doi.org/10.1038/s41467-025-65662-7 journal journal Nature Communications \ volume 16 ,\ pages 10653 ( year 2025 ) NoStop

    work page doi:10.1038/s41467-025-65662-7 2025

  30. [30]

    Bornes , author C

    author author C. Bornes , author C. B. \ Mahmoud , author V. L. \ Deringer , author C. J. \ Heard ,\ and\ author L. Grajciar ,\ https://doi.org/10.48550/ARXIV.2603.22268 title An Accurate Tensorial Model for Prediction of Full Zeolite NMR Spectra , \ ( year 2026 ),\ note version Number: 1 NoStop

    work page doi:10.48550/arxiv.2603.22268 2026

  31. [31]

    author author J. L. A. \ Gardner , author Z. Faure Beaulieu ,\ and\ author V. L. \ Deringer ,\ https://doi.org/10.1039/D2DD00137C journal journal Digital Discovery \ volume 2 ,\ pages 651 ( year 2023 ) NoStop

    work page doi:10.1039/d2dd00137c 2023

  32. [32]

    Shui , author D

    author author Z. Shui , author D. Karls , author M. Wen , author i. Nikiforov , author E. Tadmor ,\ and\ author G. Karypis ,\ in\ https://proceedings.neurips.cc/paper_files/paper/2022/file/5ef1df239d6640a27dd6ed9a59f518c9-Paper-Conference.pdf booktitle Advances in neural information processing systems ,\ Vol. volume 35 ,\ editor edited by\ editor S. Koyej...

    2022

  33. [33]

    author author J. L. A. \ Gardner , author K. T. \ Baker ,\ and\ author V. L. \ Deringer ,\ https://doi.org/10.1088/2632-2153/ad1626 journal journal Machine Learning: Science and Technology \ volume 5 ,\ pages 015003 ( year 2024 ) NoStop

    work page doi:10.1088/2632-2153/ad1626 2024

  34. [34]

    Zaverkin , author D

    author author V. Zaverkin , author D. Holzmüller , author L. Bonfirraro ,\ and\ author J. Kästner ,\ https://doi.org/10.1039/D2CP05793J journal journal Physical Chemistry Chemical Physics \ volume 25 ,\ pages 5383 ( year 2023 ) NoStop

    work page doi:10.1039/d2cp05793j 2023

  35. [35]

    Zhang , author X

    author author D. Zhang , author X. Liu , author X. Zhang , author C. Zhang , author C. Cai , author H. Bi , author Y. Du , author X. Qin , author A. Peng , author J. Huang , author B. Li , author Y. Shan , author J. Zeng , author Y. Zhang , author S. Liu , author Y. Li , author J. Chang , author X. Wang , author S. Zhou , author J. Liu , author X. Luo , a...

    work page doi:10.1038/s41524-024-01493-2 2024

  36. [36]

    Ben Mahmoud , author J

    author author C. Ben Mahmoud , author J. L. A. \ Gardner ,\ and\ author V. L. \ Deringer ,\ https://doi.org/10.1038/s43588-024-00636-1 journal journal Nature Computational Science \ volume 4 ,\ pages 384 ( year 2024 ) NoStop

    work page doi:10.1038/s43588-024-00636-1 2024

  37. [37]

    Batatia , author S

    author author I. Batatia , author S. Batzner , author D. P. \ Kovács , author A. Musaelian , author G. N. C. \ Simm , author R. Drautz , author C. Ortner , author B. Kozinsky ,\ and\ author G. Csányi ,\ https://doi.org/10.1038/s42256-024-00956-x journal journal Nature Machine Intelligence \ volume 7 ,\ pages 56 ( year 2025 b ) NoStop

    work page doi:10.1038/s42256-024-00956-x 2025

  38. [38]

    HAEBERLEN \ ( publisher Academic Press ,\ year 1976 )\ p

    in\ https://doi.org/10.1016/B978-0-12-025561-0.50001-0 booktitle High Resolution Nmr in Solids Selective Averaging ,\ editor edited by\ editor U. HAEBERLEN \ ( publisher Academic Press ,\ year 1976 )\ p. pages ii NoStop

    work page doi:10.1016/b978-0-12-025561-0.50001-0 1976

  39. [39]

    Batatia ,\ https://github.com/ACEsuit/mace title MACE , \ ( year 2026 ),\ note https://github.com/ACEsuit/mace NoStop

    author author I. Batatia ,\ https://github.com/ACEsuit/mace title MACE , \ ( year 2026 ),\ note https://github.com/ACEsuit/mace NoStop

    2026

  40. [40]

    author author J. L. A. \ Gardner ,\ https://doi.org/https://github.com/vldgroup/graph-pes title Graph PES , \ ( year 2026 ),\ note https://github.com/vldgroup/graph-pes NoStop

    2026

  41. [41]

    author author L. C. \ Erhard , author J. Rohrer , author K. Albe ,\ and\ author V. L. \ Deringer ,\ https://doi.org/10.1038/s41467-024-45840-9 journal journal Nature Communications \ volume 15 ,\ pages 1927 ( year 2024 ) NoStop

    work page doi:10.1038/s41467-024-45840-9 1927

  42. [42]

    Ben Mahmoud , author Z

    author author C. Ben Mahmoud , author Z. El-Machachi , author K. A. \ Gierczak , author J. L. A. \ Gardner ,\ and\ author V. L. \ Deringer ,\ https://doi.org/10.1039/D5DD00103J journal journal Digital Discovery \ volume 4 ,\ pages 3389 ( year 2025 b ) NoStop

    work page doi:10.1039/d5dd00103j 2025

  43. [43]

    Ben Mahmoud , author L

    author author C. Ben Mahmoud , author L. Rosset , author J. Yates ,\ and\ author V. Deringer ,\ https://doi.org/10.5281/ZENODO.15775327 title Graph-neural-network predictions of solid-state NMR parameters in silica from spherical tensor decomposition , \ ( year 2025 c ) NoStop

    work page doi:10.5281/zenodo.15775327 2025

  44. [44]

    Erlebach , author M

    author author A. Erlebach , author M. Šípka , author I. Saha , author P. Nachtigall , author C. J. \ Heard ,\ and\ author L. Grajciar ,\ https://doi.org/10.1038/s41467-024-48609-2 journal journal Nature Communications \ volume 15 ,\ pages 4215 ( year 2024 ) NoStop

    work page doi:10.1038/s41467-024-48609-2 2024

  45. [45]

    Lei , author C

    author author C. Lei , author C. Bornes , author O. Bengtsson , author A. Erlebach , author B. Slater , author L. Grajciar ,\ and\ author C. J. \ Heard ,\ https://doi.org/10.1039/D4FD00100A journal journal Faraday Discussions \ volume 255 ,\ pages 46 ( year 2025 ) NoStop

    work page doi:10.1039/d4fd00100a 2025

  46. [46]

    author author L. B. \ McCusker , author F. Liebau ,\ and\ author G. Engelhardt ,\ https://doi.org/10.1351/pac200173020381 journal journal Pure and Applied Chemistry \ volume 73 ,\ pages 381 ( year 2001 ) NoStop

    work page doi:10.1351/pac200173020381 2001

  47. [47]

    Baerlocher , author D

    author author C. Baerlocher , author D. Brouwer , author B. Marler ,\ and\ author L. McCusker ,\ https://www.iza-structure.org/databases/ title Database of Zeolite Structures , \ NoStop

  48. [48]

    Imbalzano , author A

    author author G. Imbalzano , author A. Anelli , author D. Giofré , author S. Klees , author J. Behler ,\ and\ author M. Ceriotti ,\ https://doi.org/10/gds5hz journal journal The Journal of Chemical Physics \ volume 148 ,\ pages 241730 ( year 2018 ) NoStop

    2018

  49. [49]

    Profeta , author F

    author author M. Profeta , author F. Mauri ,\ and\ author C. J. \ Pickard ,\ https://doi.org/10.1021/ja027124r journal journal Journal of the American Chemical Society \ volume 125 ,\ pages 541 ( year 2003 ) NoStop

    work page doi:10.1021/ja027124r 2003

  50. [50]

    author author S. J. \ Clark , author M. D. \ Segall , author C. J. \ Pickard , author P. J. \ Hasnip , author M. I. J. \ Probert , author K. Refson ,\ and\ author M. C. \ Payne ,\ https://doi.org/10.1524/zkri.220.5.567.65075 journal journal Zeitschrift für Kristallographie - Crystalline Materials \ volume 220 ,\ pages 567 ( year 2005 ) NoStop

    work page doi:10.1524/zkri.220.5.567.65075 2005

  51. [51]

    author author J. P. \ Perdew , author K. Burke ,\ and\ author M. Ernzerhof ,\ https://doi.org/10/bppfwt journal journal Physical Review Letters \ volume 77 ,\ pages 3865 ( year 1996 ) NoStop

    1996

  52. [52]

    author author J. P. \ Perdew , author A. Ruzsinszky , author G. I. \ Csonka , author O. A. \ Vydrov , author G. E. \ Scuseria , author L. A. \ Constantin , author X. Zhou ,\ and\ author K. Burke ,\ https://doi.org/10.1103/PhysRevLett.100.136406 journal journal Physical Review Letters \ volume 100 ,\ pages 136406 ( year 2008 ) NoStop

    work page doi:10.1103/physrevlett.100.136406 2008

  53. [53]

    author author G. I. \ Csonka , author J. P. \ Perdew , author A. Ruzsinszky , author P. H. T. \ Philipsen , author S. Lebègue , author J. Paier , author O. A. \ Vydrov ,\ and\ author J. G. \ Ángyán ,\ https://doi.org/10.1103/PhysRevB.79.155107 journal journal Physical Review B \ volume 79 ,\ pages 155107 ( year 2009 ) NoStop

    work page doi:10.1103/physrevb.79.155107 2009

  54. [54]

    Tran , author J

    author author F. Tran , author J. Stelzl ,\ and\ author P. Blaha ,\ https://doi.org/10.1063/1.4948636 journal journal The Journal of Chemical Physics \ volume 144 ,\ pages 204120 ( year 2016 ) NoStop

    work page doi:10.1063/1.4948636 2016

  55. [55]

    Loshchilov \ and\ author F

    author author I. Loshchilov \ and\ author F. Hutter ,\ in\ https://openreview.net/forum?id=Bkg6RiCqY7 booktitle International conference on learning representations \ ( year 2019 ) NoStop

    2019

  56. [56]

    author author M. J. \ Duer ,\ @noop english title Introduction to solid-state NMR spectroscopy ,\ edition first published \ ed.\ ( publisher Blackwell Publishing ,\ address Oxford Malden Carlton, Victoria ,\ year 2004 ) NoStop

    2004

  57. [57]

    author author A. Stukowski ,\ https://doi.org/10.1088/0965-0393/18/1/015012 journal journal Modelling and Simulation in Materials Science and Engineering \ volume 18 ,\ pages 015012 ( year 2010 ) NoStop

    work page doi:10.1088/0965-0393/18/1/015012 2010

  58. [58]

    Greiser , author M

    author author S. Greiser , author M. Hunger ,\ and\ author C. Jäger ,\ https://doi.org/10.1016/j.ssnmr.2016.10.004 journal journal Solid State Nuclear Magnetic Resonance \ volume 79 ,\ pages 6 ( year 2016 ) NoStop

    work page doi:10.1016/j.ssnmr.2016.10.004 2016

  59. [59]

    Zilka , author S

    author author M. Zilka , author S. Sturniolo , author S. P. \ Brown ,\ and\ author J. R. \ Yates ,\ https://doi.org/10.1063/1.4996750 journal journal The Journal of Chemical Physics \ volume 147 ,\ pages 144203 ( year 2017 ) NoStop

    work page doi:10.1063/1.4996750 2017

  60. [60]

    Han , author Y

    author author B. Han , author Y. Liu , author S. W. \ Ginzinger ,\ and\ author D. S. \ Wishart ,\ https://doi.org/10.1007/s10858-011-9478-4 journal journal Journal of Biomolecular NMR \ volume 50 ,\ pages 43 ( year 2011 ) NoStop

    work page doi:10.1007/s10858-011-9478-4 2011

  61. [61]

    Kuryla , author F

    author author D. Kuryla , author F. Berger , author G. Csányi ,\ and\ author A. Michaelides ,\ https://doi.org/10.1063/5.0296997 journal journal The Journal of Chemical Physics \ volume 163 ,\ pages 224313 ( year 2025 ) NoStop

    work page doi:10.1063/5.0296997 2025