Pith. sign in

REVIEW 2 minor 2 cited by

Six Open Questions in Machine-Learned Interatomic Potential Foundation Models

T0 review · 0 major / 2 minor · reviewed 2026-06-27 · grok-4.3

Pith's one-line read Six open questions will shape foundational machine-learned interatomic potentials for years ahead.

desk verdict This is a perspective piece that defines foundational MLIPs and lists six open questions, with no new data or derivations. read the letter →

arxiv 2606.07327 v2 pith:ZMZUDXGB submitted 2026-06-05 cond-mat.mtrl-sci cond-mat.dis-nnphysics.app-phphysics.comp-ph

classification cond-mat.mtrl-scicond-mat.dis-nnphysics.app-phphysics.comp-ph
keywords machine-learnedinteratomicpotentialsfoundationmodelsopenquestionsmolecularmodelingmaterialssimulationMLIPs
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

The paper first offers a working definition of foundational MLIPs as models trained on large, diverse datasets that can handle new systems with little additional training. It then identifies and explores six specific open questions that the authors regard as the most important unresolved issues in this area. A reader would care because MLIPs aim to remove the traditional trade-off between simulation scale and accuracy in molecular and materials modeling. The authors argue that, even with fast model development, these questions remain central and will steer research priorities. The piece frames the questions explicitly around the definition to keep the discussion focused on broad applicability rather than narrow model tweaks.

What carries the argument

The working definition of foundational MLIPs, which frames the selection and discussion of the six open questions.

What would settle it

Future research activity in MLIPs that concentrates overwhelmingly on problems outside the six listed questions would undermine the claim that these questions define the field's direction.

Watch

Extended reading notes

Core claim

The authors develop a working definition of foundational MLIPs and use it to articulate six open questions; they claim that, despite rapid progress and proliferation of models, these questions constitute the fundamental challenges that will continue to define cutting-edge research in the field for years to come.

Load-bearing premise

The authors' choice of exactly these six questions, framed by their working definition, correctly identifies the load-bearing challenges rather than other unlisted issues.

Editorial extensions

If this is right

  • Progress on foundational MLIPs will require systematic attention to the six questions rather than isolated model improvements.
  • Models trained on large diverse datasets will need to demonstrate reliable performance on new systems with minimal retraining to qualify as foundational.
  • The tension between scale and accuracy in simulations will remain unresolved until the listed questions receive answers.
  • The field will continue to produce many models, but only those addressing the core questions will set the research agenda.

Reading between the lines

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

  • Resolving the questions could allow a single pretrained model to replace many specialized potentials across different chemical systems.
  • The emphasis on minimal updates for new systems may push the community toward transfer-learning techniques that are currently underdeveloped for interatomic potentials.
  • If the six questions prove decisive, funding and publication priorities in materials modeling may shift toward broad benchmark suites rather than single-material case studies.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 2 minor

Summary. The manuscript develops a working definition of foundational MLIPs (models trained on large, diverse datasets that generalize to new systems with minimal updates) and uses this definition to identify and discuss six open questions that the authors argue will continue to shape cutting-edge research in the field.

Significance. As a perspective piece, the manuscript provides a structured framing that could help organize community discussion around generalization, data requirements, and architectural choices in MLIP development; its value lies in the clarity of the definitional starting point rather than in new empirical or theoretical results.

minor comments (2)
  1. [Abstract] The abstract states that the authors 'start by developing a working definition' but does not preview the six questions; adding a brief enumerated list would improve immediate readability for readers scanning the piece.
  2. Section headings for the six questions are not numbered in the provided text; consistent numbering (e.g., Question 1, Question 2) would make cross-references within the manuscript easier to follow.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive review and recommendation to accept. Their summary correctly identifies the manuscript as a perspective piece that proposes a working definition of foundational MLIPs and frames six open questions around generalization, data, and architecture.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; purely discursive perspective piece

full rationale

The paper is a perspective article that develops a working definition of foundational MLIPs and lists six open questions. It contains no derivations, equations, predictions, or fitted quantities that could reduce to inputs by construction. The central claim is explicitly subjective framing of future research directions rather than a technical result. No self-citation chains or ansatzes are invoked as load-bearing steps. This is self-contained as a non-derivational discussion and receives the default low circularity score.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The perspective rests on the authors' judgment that the listed questions are the most important; no free parameters, mathematical axioms, or new entities are introduced.

assumptions (1)
  • domain assumption Foundational MLIPs are trained on large diverse datasets and promise to work well for new systems with minimal updates.
    This is the working definition the paper develops to frame the questions.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Six Open Questions in Machine-Learned Interatomic Potential Foundation Models." pith.science (2026). https://pith.science/paper/ZMZUDXGB

@misc{pith2026260607327,
  author       = {Pith},
  title        = {Pith review of: Six Open Questions in Machine-Learned Interatomic Potential Foundation Models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZMZUDXGB}},
  note         = {Machine review of arXiv:2606.07327}
}
read the original abstract

Machine-learned interatomic potentials (MLIPs) have had a profound impact on molecular modelling in recent years, promising to resolve the long-standing tension between the scale and accuracy of simulations. There has been a proliferation of new models and designs, and recently the paradigm of ``foundational'' MLIPs has become prevalent. Broadly speaking, foundation models are trained on large diverse datasets and promise to work well for new systems with minimal updates required. However, in such a new and fast moving field, there are many unanswered questions. In this article, we set out to articulate and explore what we see as the most important among these questions. We start by developing a working definition for foundational MLIPs and use this definition to frame the subsequent open questions. Despite the rapid progress in the field of MLIP models, we believe that these are fundamental questions which will continue to define cutting edge research in MLIPs in the years to come.

Figures

Figures reproduced from arXiv: 2606.07327 by the authors.

Figure 1
Figure 1. The connections between definitions and open questions. The text of the article has been analysed using a cosine similarity of vectorized embeddings. On the left, we show how the definition criteria link to the questions, on the right, the relations between the questions (colour-coded as on the left) are displayed. The line widths reflect the similarity of the embeddings of the content. To ground this comparison, we… view at source ↗
Figure 2
Figure 2. Log-log plot of the predictive error on the water data set from [59] using NequIP with rotation order L ∈ {0, 1, 2, 3} as a function of training set size, measured via the force MAE. Figure from Batzner et al. [32] Mattersim, Alexandria, and OMat24 [55, 61, 62, 63]. In recent work with the MatPES data-efficient sampling scheme [64], the authors show that MLIPs trained on a compact, representative 400K dataset (curat… view at source ↗
Figure 3
Figure 3. Overview of conceptual approaches to accurately capture long range interactions in GNNs while mitigating oversmoothing and oversquashing. Strategies include: Graph Rewiring (top left) for structural optimization; Dual Architectures (top right) for global-local separation; Hierarchical Approaches (bottom right) for multi-scale feature extraction; and incorporation of Physics-Informed Priors that encode the long-range… view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Different resolutions of lipid membranes. All-atom (AA) resolution explicitly considers all atoms. Coarse-grain (CG) resolution considers small atom groups and their associated hydrogens. Supra-CG resolution represents solvents implicitly and proteins and lipids as qua…
Figure 5
Figure 5. Figure 5: Combined performance score against the model size (number of model parameters) obtained with different foundation MLIPs, based on Matbench Discovery [38] benchmark website (https://matbench-discovery.materialsproject.org, June 2026). Large-scale atomistic simulations w…
Figure 6
Figure 6. Figure 6: Distribution of the metric rankings from the leaderboard of the Matbench Discovery [38] benchmark website (https://matbench-discovery.materialsproject.org, June 2026). An overarching question covering all of the subjects we have covered is: how do we know if a particul…

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Dyna-Mat: End-to-end benchmarking of foundation machine learning interatomic potentials in finite-temperature ensembles

    cond-mat.mtrl-sci 2026-07 conditional novelty 6.5 of 10

    Dyna-Mat-v1.0 benchmarks 15 foundation MLIPs on finite-T MD observables, finding average force-error correlation with RDF/VDOS but systematic pressure failures and near-Pareto optimality of latest cross-trained models.

  2. VASP Plugins: Linking the Vienna ab-initio Simulation Package with Python

    cond-mat.mtrl-sci 2026-07 accept novelty 5.5 of 10

    A C++/pybind11 shared-memory plugin layer exposes VASP SCF and ionic data as NumPy arrays so Python can modify structure, forces, local potential, and occupancies in place.

Reference graph

Works this paper leans on

268 extracted references · 215 canonical work pages · cited by 2 Pith papers

  1. [1]

    Metropolis N, Rosenbluth A W, Rosenbluth M N, Teller A H and Teller E 1953The Journal of Chemical Physics211087–1092 ISSN 1089-7690 URL http://dx.doi.org/10.1063/1.1699114

  2. [2]

    Hastings W K 1970Biometrika5797–109 ISSN 0006-3444 URL http://dx.doi.org/10.1093/biomet/57.1.97

  3. [3]

    Rahman A 1964Physical Review136A405–A411 ISSN 0031-899X URL http://dx.doi.org/10.1103/PhysRev.136.A405

  4. [4]

    Verlet L 1967Physical Review15998–103 ISSN 0031-899X URL http://dx.doi.org/10.1103/PhysRev.159.98

  5. [5]

    Chandler D and Wolynes P G 1981The Journal of Chemical Physics744078–4095 ISSN 1089-7690 URLhttp://dx.doi.org/10.1063/1.441588

  6. [6]

    Parrinello M and Rahman A 1981Journal of Applied Physics527182–7190 ISSN 1089-7550 URLhttp://dx.doi.org/10.1063/1.328693

  7. [7]

    Ceperley D M 1995Reviews of Modern Physics67279–355 ISSN 1539-0756 URL http://dx.doi.org/10.1103/RevModPhys.67.279

  8. [8]

    Series A, Containing Papers of a Mathematical and Physical Character106463–477 ISSN 2053-9150 URL http://dx.doi.org/10.1098/rspa.1924.0082

    Jones J E 1924Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character106463–477 ISSN 2053-9150 URL http://dx.doi.org/10.1098/rspa.1924.0082

Show all 268 references
  1. [9]

    Alder B J and Wainwright T E 1957The Journal of Chemical Physics271208–1209 ISSN 1089-7690 URLhttp://dx.doi.org/10.1063/1.1743957

  2. [10]

    Behler J and Parrinello M 2007Phys. Rev. Lett.98146401 URL https://link.aps.org/doi/10.1103/PhysRevLett.98.146401

  3. [11]

    Bart´ ok A P, Kondor R and Cs´ anyi G 2013Phys. Rev. B87184115 URL https://link.aps.org/doi/10.1103/PhysRevB.87.184115

  4. [12]

    Drautz R 2019Phys. Rev. B99014104 URL https://link.aps.org/doi/10.1103/PhysRevB.99.014104

  5. [13]

    Batatia I, Kov´ acs D P, Simm G N C, Ortner C and Cs´ anyi G 2022NeurIPS3511423–11436 URLhttps://proceedings.neurips.cc/paper_files/paper/2022/hash/ 4a36c3c51af11ed9f34615b81edb5bbc-Abstract-Conference.html

  6. [14]

    Thomas N, Smidt T, Kearnes S, Yang L, Li L, Kohlhoff K and Riley P 2018 Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds eprint: 1802.08219 URLhttps://arxiv.org/abs/1802.08219

  7. [15]

    Geiger M, Smidt T, M A, Miller B K, Boomsma W, Dice B, Lapchevskyi K, Weiler M, Tyszkiewicz M, Batzner S, Madisetti D, Uhrin M, Frellsen J, Jung N, Sanborn S, Wen M, Rackers J, Rød M and Bailey M 2022 Euclidean neural networks: e3nn URL https://doi.org/10.5281/zenodo.6459381

  8. [16]

    Deng B, Zhong P, Jun K, Riebesell J, Han K, Bartel C J and Ceder G 2023Nature Machine Intelligence51031–1041 URLhttps://www.nature.com/articles/s42256-023-00716-3

  9. [17]

    Bommasani R, Hudson D A, Adeli E, Altman R, Arora S, Arx S v, Bernstein M S, Bohg J, Bosselut A, Brunskill E, Brynjolfsson E, Buch S, Card D, Castellon R, Chatterji N, Chen A, Creel K, Davis J Q, Demszky D, Donahue C, Doumbouya M, Durmus E, Ermon S, Etchemendy J, Ethayarajh K,...

  10. [18]

    Lysogorskiy Y, Bochkarev A and Drautz R 2026npj Computational Materials

  11. [19]

    Kreiman T, Bai Y, Atieh F, Weaver E, Qu E and Krishnapriyan A S 2025arXiv preprint arXiv:2510.02259

  12. [20]

    Vaswani A, Shazeer N, Parmar N, Uszkoreit J, Jones L, Gomez A N, Kaiser L and Polosukhin I 2017Advances in neural information processing systems30

  13. [21]

    Mater.10300 URL https://doi.org/10.1038/s41524-024-01486-1

    Devi R, Butler K T and Sai Gautam G 2024npj Comput. Mater.10300 URL https://doi.org/10.1038/s41524-024-01486-1

  14. [22]

    Chen C, Zuo Y, Ye W, Li X and Ong S P 2021Nat. Comput. Sci.146–53 URL https://doi.org/10.1038/s43588-020-00002-x

  15. [23]

    Kim J, Kim J, Kim J, Lee J, Park Y, Kang Y and Han S 2024J. Am. Chem. Soc.URL https://doi.org/10.1021/jacs.4c14455

  16. [24]

    Batatia I, Lin C, Hart J, Kasoar E, Elena A M, Norwood S W, Wolf T and Cs´ anyi G 2025 ArXiv:2510.25380 URLhttps://arxiv.org/abs/2510.25380

  17. [25]

    Gumber S, Alzate-Vargas L, Nebgen B T, Veelen A v, Kadvani S, Gibson T and Messerly R ArXiv:2506.10211 URLhttp://arxiv.org/abs/2506.10211

  18. [26]

    Mazitov A, Chorna S, Fraux G, Bercx M, Pizzi G, De S and Ceriotti M ArXiv:2506.19674 URLhttp://arxiv.org/abs/2506.19674

  19. [27]

    Park S, Seong K, Yang S, G´ omez-Bombarelli R and Ahn S Learning Collective Variables from BioEmu with Time-Lagged Generation (Preprint2507.07390)

  20. [28]

    Chorna S, Tisi D, Malosso C, How W B, Ceriotti M and Chong S Comparing the latent features of universal machine-learning interatomic potentials (Preprint2512.05717)

  21. [29]

    Edamadaka S, Yang S, Li J and G´ omez-Bombarelli R Universally Converging Representations of Matter Across Scientific Foundation Models (Preprint2512.03750)

  22. [30]

    Sutton R 2019Incomplete Ideas (blog)URL http://www.incompleteideas.net/IncIdeas/BitterLesson.html

  23. [31]

    Kranmer K 2025Theory and practice (blog)URL https://theoryandpractice.org/2025/09/The%20Bittersweet%20Lesson/

  24. [32]

    Commun.132453 URL https://doi.org/10.1038/s41467-022-29939-5

    Batzner S, Musaelian A, Sun L, Geiger M, Mailoa J P, Kornbluth M, Molinari N, Smidt T E and Kozinsky B 2022Nat. Commun.132453 URL https://doi.org/10.1038/s41467-022-29939-5

  25. [33]

    Qu E and Krishnapriyan A S 2024 The importance of being scalable: Improving the speed and accuracy of neural network interatomic potentials across chemical domainsAdvances in Neural Information Processing Systemsvol 37 ed Globerson A, Mackey L, Belgrave D, Fan A, Paquet U, Tom...

  26. [34]

    Smidt T E, Geiger M and Miller B K 2021Phys. Rev. Res.3L012002 URL https://link.aps.org/doi/10.1103/PhysRevResearch.3.L012002

  27. [35]

    Xie Y and Smidt T 2024 ArXiv:2402.02681 URLhttps://arxiv.org/abs/2402.02681 22 ML4AtomsSix Open Questions for MLIPsAuthoret al

  28. [36]

    Hofgard E, Wang R, Walters R and Smidt T 2024 ArXiv:2407.20471 URL https://arxiv.org/abs/2407.20471

  29. [37]

    Rhodes B, Vandenhaute S, ˇSimkus V, Gin J, Godwin J, Duignan T and Neumann M 2025 ArXiv:2504.06231 URLhttps://arxiv.org/abs/2504.06231

  30. [38]

    Riebesell J, Goodall R E A, Benner P, Chiang Y, Deng B, Ceder G, Asta M, Lee A A, Jain A and Persson K A 2025Nat. Mach. Intell.7836–847 URL https://www.nature.com/articles/s42256-025-01055-1

  31. [39]

    Bigi F, Langer M F and Ceriotti M 2025 The dark side of the forces: assessing non-conservative force models for atomistic machine learningProceedings of the 42nd International Conference on Machine Learning(Proceedings of Machine Learning Research vol 267) ed Singh A, Fazel M,...

  32. [40]

    Mater.11178 URLhttps://doi.org/10.1038/s41524-025-01650-1

    Loew A, Sun D, Wang H C, Botti S and Marques M A L 2025npj Comput. Mater.11178 URLhttps://doi.org/10.1038/s41524-025-01650-1

  33. [41]

    Ngo K and Ravanbakhsh S Scaling Laws and Symmetry, Evidence from Neural Force Fields URLhttps://arxiv.org/abs/2510.09768v1

  34. [42]

    Brehmer J, Behrends S, de Haan P and Cohen T Does equivariance matter at scale? (Preprint2410.23179)

  35. [43]

    Qu E and Krishnapriyan A 2024Advances in Neural Information Processing Systems37 139030–139053

  36. [44]

    Mazitov A, Bigi F, Kellner M, Pegolo P, Tisi D, Fraux G, Pozdnyakov S, Loche P and Ceriotti M ArXiv:2503.14118 URLhttp://arxiv.org/abs/2503.14118

  37. [46]

    Xie Y and Smidt T 2025arXiv preprint arXiv:2506.02269

  38. [47]

    Accessed 2025-11-10 URLhttps://github.com/NVIDIA/cuEquivariance

    NVIDIA Corporation 2025 cuequivariance cUDA kernels and APIs for equivariant neural networks. Accessed 2025-11-10 URLhttps://github.com/NVIDIA/cuEquivariance

  39. [49]

    Paszke A, Gross S, Massa F, Lerer A, Bradbury J, Chanan G, Killeen T, Lin Z, Gimelshein N, Antiga L, Desmaison A, K¨ opf A, Yang E, DeVito Z, Raison M, Tejani A, Chilamkurthy S, Steiner B, Fang L, Bai J and Chintala S 2019 Pytorch: An imperative style, high-performance deep le...

  40. [50]

    TorchScript is deprecated; usetorch.export

    PyTorch Contributors 2025TorchScript (torch.jit) — PyTorch Documentationlast updated 2025-07-16. TorchScript is deprecated; usetorch.export. Accessed 2025-11-10 URL https://docs.pytorch.org/docs/stable/jit.html

  41. [51]

    Han K, Deng B, Farimani A B and Ceder G 2025 ArXiv:2506.02023 [cs] URL https://arxiv.org/abs/2506.02023

  42. [52]

    Loshchilov I and Hutter F 2019 ArXiv:1711.05101 URL https://arxiv.org/abs/1711.05101

  43. [53]

    Jordan K 2024 Muon: An optimizer for the hidden layers in neural networks accessed 2025-11-10 URLhttps://kellerjordan.github.io/posts/muon/

  44. [54]

    Chen C and Ong S P 2022Nat. Comput. Sci.2718–728 URL https://doi.org/10.1038/s43588-022-00349-3 23 ML4AtomsSix Open Questions for MLIPsAuthoret al

  45. [55]

    Merchant A, Batzner S, Schoenholz S S, Aykol M, Cheon G and Cubuk E D 2023Nature 62480–85 URLhttps://www.nature.com/articles/s41586-023-06735-9

  46. [56]

    Mater.11 1–11 URLhttps://doi.org/10.1038/s41524-025-01727-x

    Radova M, Stark W G, Allen C S, Maurer R J and Bart´ ok A P 2025npj Comput. Mater.11 1–11 URLhttps://doi.org/10.1038/s41524-025-01727-x

  47. [57]

    Messerly M, Matin S, Allen A E A, Nebgen B, Barros K, Smith J S, Lubbers N and Messerly R 2025arXiv preprint arXiv:2505.01590ArXiv:2505.01590 URL https://arxiv.org/abs/2505.01590

  48. [58]

    Koker T, Kotak M and Smidt T 2025 ArXiv:2508.16067 URL https://arxiv.org/abs/2508.16067

  49. [59]

    Cheng B, Engel E A, Behler J, Dellago C and Ceriotti M 2019Proc. Natl. Acad. Sci. U.S.A. 1161110–1115 URLhttps://www.pnas.org/doi/abs/10.1073/pnas.1815117116

  50. [60]

    Commun.114895 URL https://doi.org/10.1038/s41467-020-18556-9

    von Lilienfeld O A and Burke K 2020Nat. Commun.114895 URL https://doi.org/10.1038/s41467-020-18556-9

  51. [61]

    Yang H, Hu C, Zhou Y, Liu X, Shi Y, Li J, Li G, Chen Z, Chen S, Zeni C, Horton M, Pinsler R, Fowler A, Z¨ ugner D, Xie T, Smith J, Sun L, Wang Q, Kong L, Liu C, Hao H and Lu Z 2024 ArXiv:2405.04967 URLhttps://arxiv.org/abs/2405.04967

  52. [62]

    Today Phys.48101560 URL https://doi.org/10.1016/j.mtphys.2024.101560

    Schmidt J, Cerqueira T F, Romero A H, Loew A, J¨ ager F, Wang H C, Botti S and Marques M A 2024Mater. Today Phys.48101560 URL https://doi.org/10.1016/j.mtphys.2024.101560

  53. [63]

    Barroso-Luque L, Shuaibi M, Fu X, Wood B M, Dzamba M, Gao M, Rizvi A, Zitnick C L and Ulissi Z W 2024arXiv preprint arXiv:2410.12771ArXiv:2410.12771 URL https://doi.org/10.48550/arXiv.2410.12771

  54. [64]

    Kaplan A D, Liu R, Qi J, Ko T W, Deng B, Riebesell J, Ceder G, Persson K A and Ong S P 2025 ArXiv:2503.04070 URLhttps://doi.org/10.48550/arXiv.2503.04070

  55. [65]

    Qi J, Ko T W, Wood B C, Pham T A and Ong S P 2024npj Computational Materials1043 URLhttps://www.nature.com/articles/s41524-024-01227-4

  56. [66]

    Mazitov A, Chorna S, Fraux G, Bercx M, Pizzi G, De S and Ceriotti M 2025Scientific data 121857

  57. [67]

    Mater.7185 URL https://doi.org/10.1038/s41524-021-00650-1

    Choudhary K and DeCost B 2021npj Comput. Mater.7185 URL https://doi.org/10.1038/s41524-021-00650-1

  58. [68]

    Gibson J B, Janicki T D, Hire A C, Bishop C, Lane J M D and Hennig R G 2024 ArXiv:2409.07610 URLhttps://doi.org/10.48550/arXiv.2409.07610

  59. [69]

    Nakkiran P, Kaplun G, Bansal Y, Yang T, Barak B and Sutskever I 2019 ArXiv:1912.02292 URLhttps://doi.org/10.48550/arXiv.1912.02292

  60. [70]

    Kaplan J, McCandlish S, Henighan T, Brown T B, Chess B, Child R, Gray S, Radford A, Wu J and Amodei D 2020arXiv preprint arXiv:2001.08361ArXiv:2001.08361 URL https://arxiv.org/abs/2001.08361

  61. [71]

    Frey N C, Soklaski R, Axelrod S, Samsi S, G´ omez-Bombarelli R, Coley C W and Gadepally V 2023Nat. Mach. Intell.51297–1305 URLhttps://doi.org/10.1038/s42256-023-00740-3

  62. [72]

    Kuryla D, Berger F, Cs´ anyi G and Michaelides A 2025arXiv preprint arXiv:2510.19774 ArXiv:2510.19774 URLhttps://doi.org/10.48550/arXiv.2510.19774

  63. [73]

    Perdew J P, Burke K and Ernzerhof M 1996Phys. Rev. Lett.77(18) 3865–3868 URL https://link.aps.org/doi/10.1103/PhysRevLett.77.3865

  64. [74]

    Meggiolaro D and De Angelis F 2018ACS Energy Letters32206–2222 ISSN 2380-8195 URL http://dx.doi.org/10.1021/acsenergylett.8b01212

  65. [75]

    Taheri A, Da Silva C and Amon C H 2018Journal of Applied Physics123ISSN 1089-7550 URLhttp://dx.doi.org/10.1063/1.5027619 24 ML4AtomsSix Open Questions for MLIPsAuthoret al

  66. [76]

    Stephens P J, Devlin F J, Chabalowski C F and Frisch M J 1994The Journal of Physical Chemistry9811623–11627 (Preprinthttps://doi.org/10.1021/j100096a001) URL https://doi.org/10.1021/j100096a001

  67. [77]

    Krukau A V, Vydrov O A, Izmaylov A F and Scuseria G E 2006The Journal of Chemical Physics125224106 ISSN 0021-9606 (Preprinthttps://pubs.aip.org/aip/jcp/ article-pdf/doi/10.1063/1.2404663/13263224/224106_1_online.pdf) URL https://doi.org/10.1063/1.2404663

  68. [78]

    Deng B, Choi Y, Zhong P, Riebesell J, Anand S, Li Z, Jun K, Persson K A and Ceder G 2025 npj Computational Materials119 URL https://www.nature.com/articles/s41524-024-01500-6

  69. [79]

    Fu X, Wood B M, Barroso-Luque L, Levine D S, Gao M, Dzamba M and Zitnick C L 2025 arXiv preprint arXiv:2502.12147ArXiv:2502.12147 URL https://doi.org/10.48550/arXiv.2502.12147

  70. [80]

    Mater.11URL https://doi.org/10.1038/s41524-025-01550-4

    Ko T W and Ong S P 2025npj Comput. Mater.11URL https://doi.org/10.1038/s41524-025-01550-4

  71. [81]

    Learn.: Sci

    Oerder R, Schmieden G and Hamaekers J 2025Mach. Learn.: Sci. Technol.6045004 URL https://doi.org/10.1088/2632-2153/ae0d41

  72. [82]

    Schmitz N F, Ploumhans B and Herbst M F126 ISSN 2057-3960

  73. [83]

    Kohn W 1996Physical Review Letters763168–3171 ISSN 1079-7114 URL http://dx.doi.org/10.1103/PhysRevLett.76.3168

  74. [84]

    Prodan E and Kohn W 2005Proceedings of the National Academy of Sciences102 11635–11638 ISSN 1091-6490 URLhttp://dx.doi.org/10.1073/pnas.0505436102

  75. [85]

    Grisafi A and Ceriotti M 2019J. Chem. Phys.151URL https://doi.org/10.1063/1.5128375

  76. [86]

    Yue S, Muniz M C, Calegari Andrade M F, Zhang L, Car R and Panagiotopoulos A Z 2021 J. Chem. Phys.154URLhttps://doi.org/10.1063/5.0031215

  77. [87]

    Ko T W, Finkler J A, Goedecker S and Behler J 2023J. Chem. Theory Comput.19 3567–3579 URLhttps://doi.org/10.1021/acs.jctc.2c01146

  78. [88]

    Ko T W, Finkler J A, Goedecker S and Behler J 2021Acc. Chem. Res.54808–817 URL https://doi.org/10.1021/acs.accounts.0c00689

  79. [89]

    Energy Mater.412562–12569 URLhttps://doi.org/10.1021/acsaem.1c02363

    Staacke C G, Heenen H H, Scheurer C, Cs´ anyi G, Reuter K and Margraf J T 2021ACS Appl. Energy Mater.412562–12569 URLhttps://doi.org/10.1021/acsaem.1c02363

  80. [90]

    Cheng B 2025npj Computational Materials1180

  81. [91]

    Niblett S P, Galib M and Limmer D T 2021The Journal of Chemical Physics155ISSN 1089-7690 URLhttp://dx.doi.org/10.1063/5.0067565

  82. [92]

    Janeˇ cek J 2006J. Phys. Chem. B1106264–6269 URL https://doi.org/10.1021/jp056344z

  83. [93]

    Rumiantsev E, Langer M F, Sodjargal T E, Ceriotti M and Loche P 2025 Learning long-range representations with equivariant messages URLhttps://arxiv.org/abs/2507.19382

  84. [94]

    Commun.12URLhttp://dx.doi.org/10.1038/s41467-021-27504-0

    Unke O T, Chmiela S, Gastegger M, Sch¨ utt K T, Sauceda H E and M¨ uller K R 2021Nat. Commun.12URLhttp://dx.doi.org/10.1038/s41467-021-27504-0

  85. [95]

    Pozdnyakov S and Ceriotti M3679469–79501 URL https://proceedings.neurips.cc/paper_files/paper/2023/hash/ fb4a7e3522363907b26a86cc5be627ac-Abstract-Conference.html

  86. [96]

    Unke O T, Chmiela S, Sauceda H E, Gastegger M, Poltavsky I, Sch¨ utt K T, Tkatchenko A and M¨ uller K R12110142–10186 ISSN 0009-2665 25 ML4AtomsSix Open Questions for MLIPsAuthoret al

  87. [97]

    Frank J T, Unke O T and M¨ uller K R 2023 ArXiv:2205.14276 URL https://arxiv.org/abs/2205.14276

  88. [98]

    Batatia I, Schaaf L L, Chen H, Cs´ anyi G, Ortner C and Faber F A 2024 ArXiv:2310.10434 URLhttps://arxiv.org/abs/2310.10434

  89. [99]

    Freysoldt C, Neugebauer J and Van de Walle C G 2011physica status solidi (b)248 1067–1076 URLhttps://doi.org/10.1002/pssb.201046289

  90. [100]

    French R H, Parsegian V A, Podgornik R, Rajter R F, Jagota A, Luo J, Asthagiri D, Chaudhury M K, Chiang Y m, Granick Set al.2010Rev. Mod. Phys.821887–1944 URL https://doi.org/10.1103/RevModPhys.82.1887

  91. [101]

    Resta R 1994Rev. Mod. Phys.66899–915 URL https://doi.org/10.1103/RevModPhys.66.899

  92. [102]

    Gonze X and Lee C 1997Phys. Rev. B5510355–10368 URL https://doi.org/10.1103/PhysRevB.55.10355

  93. [103]

    Ho L, Clarke W, Micolich A, Danneau R, Klochan O, Simmons M, Hamilton A, Pepper M and Ritchie D 2008Phys. Rev. B77201402 URL https://doi.org/10.1103/PhysRevB.77.201402

  94. [104]

    Defenu N, Donner T, Macr` ı T, Pagano G and Ruffo S 2023Rev. Mod. Phys.95035002 URL https://doi.org/10.1103/RevModPhys.95.035002

  95. [105]

    Keldysh L V 1979JETP Letters29658–661 URL https://doi.org/10.1142/9789811279461_0024

  96. [106]

    Cardy J 1996Scaling and Renormalization in Statistical Physics(Cambridge University Press)

  97. [107]

    Bamberger J, Gutteridge B, le Roux S, Bronstein M M and Dong X 2025 ArXiv:2506.05971 URLhttps://arxiv.org/abs/2506.05971

  98. [108]

    Di Giovanni F, Giusti L, Barbero F, Luise G, Lio’ P and Bronstein M 2023 ArXiv:2302.02941 URLhttps://arxiv.org/abs/2302.02941

  99. [109]

    Balcilar M, H´ eroux P, Ga¨ uz` ere B, Vasseur P, Adam S and Honeine P 2021 ArXiv:2106.04319 URLhttps://arxiv.org/abs/2106.04319

  100. [110]

    Cai C, Hy T S, Yu R and Wang Y 2023 ArXiv:2301.11956 [cs] URL http://arxiv.org/abs/2301.11956

  101. [111]

    Katharopoulos A, Vyas A, Pappas N and Fleuret F 2020 Transformers are RNNs: Fast autoregressive transformers with linear attentionProceedings of the 37th International Conference on Machine Learning(Proceedings of Machine Learning Researchvol 119) ed III H D and Singh A (PMLR)...

  102. [112]

    Gu A, Goel K and R´ e C 2021 Efficiently modeling long sequences with structured state spaces URLhttps://arxiv.org/abs/2111.00396

  103. [113]

    Gu A and Dao T 2023 Mamba: Linear-time sequence modeling with selective state spaces URLhttps://arxiv.org/abs/2312.00752

  104. [114]

    Frank J T, Chmiela S, M¨ uller K R and Unke O T Euclidean Fast Attention: Machine Learning Global Atomic Representations at Linear Cost (Preprint2412.08541)

  105. [115]

    Xu K, Hu W, Leskovec J and Jegelka S 2019 ArXiv:1810.00826 URL https://arxiv.org/abs/1810.00826

  106. [116]

    Kreiman T and Krishnapriyan A S 2026Digital Discovery5415–439

  107. [117]

    Li Q, Han Z and Wu X M 2018 ArXiv:1801.07606 URL https://arxiv.org/abs/1801.07606

  108. [118]

    Oono K and Suzuki T 2021 ArXiv:1905.10947 URLhttps://arxiv.org/abs/1905.10947 26 ML4AtomsSix Open Questions for MLIPsAuthoret al

  109. [119]

    Alon U and Yahav E 2021 ArXiv:2006.05205 URLhttps://arxiv.org/abs/2006.05205

  110. [120]

    Topping J, Giovanni F D, Chamberlain B P, Dong X and Bronstein M M 2022 ArXiv:2111.14522 URLhttps://arxiv.org/abs/2111.14522

  111. [121]

    Arnaiz-Rodriguez A and Errica F 2025 ArXiv:2505.15547 URL https://arxiv.org/abs/2505.15547

  112. [122]

    Giovanni F D, Rusch T K, Bronstein M M, Deac A, Lackenby M, Mishra S and Veliˇ ckovi´ c P 2024 ArXiv:2306.03589 URLhttps://arxiv.org/abs/2306.03589

  113. [123]

    Blayney H, ´Alvaro Arroyo, Dong X and Bronstein M M 2025 ArXiv:2510.08450 URL https://arxiv.org/abs/2510.08450

  114. [124]

    ´Alvaro Arroyo, Gravina A, Gutteridge B, Barbero F, Gallicchio C, Dong X, Bronstein M and Vandergheynst P 2025 ArXiv:2502.10818 URLhttps://arxiv.org/abs/2502.10818

  115. [125]

    Gutteridge B, Dong X, Bronstein M and Di Giovanni F 2023 ArXiv:2305.08018 URL https://arxiv.org/abs/2305.08018

  116. [126]

    Karhadkar K, Banerjee P K and Mont´ ufar G 2023 ArXiv:2210.11790 URL https://arxiv.org/abs/2210.11790

  117. [127]

    Gilmer J, Schoenholz S S, Riley P F, Vinyals O and Dahl G E 2017 ArXiv:1704.01212 URL https://arxiv.org/abs/1704.01212

  118. [128]

    Scarselli F, Gori M, Tsoi A C, Hagenbuchner M and Monfardini G 2009IEEE Transactions on Neural Networks2061–80 URLhttp://dx.doi.org/10.1109/TNN.2008.2005605

  119. [129]

    Southern J, Giovanni F D, Bronstein M and Lutzeyer J F 2025 ArXiv:2405.13526 [cs] URL http://arxiv.org/abs/2405.13526

  120. [130]

    Hwang E, Thost V, Dasgupta S S and Ma T 2022 An analysis of virtual nodes in graph neural networks for link prediction (extended abstract)The First Learning on Graphs ConferenceURLhttps://openreview.net/forum?id=dI6KBKNRp7

  121. [131]

    Sestak F, Schneckenreiter L, Brandstetter J, Hochreiter S, Mayr A and Klambauer G 2024 ArXiv:2404.07194 [cs] URLhttp://arxiv.org/abs/2404.07194

  122. [132]

    Liu X, Cheng J, Song Y and Jiang X 2022 ArXiv:2206.08561 [cs] URL http://arxiv.org/abs/2206.08561

  123. [133]

    Li X, Zhou Z, Yao J, Rong Y, Zhang L and Han B 2024 ArXiv:2311.01276 [cs] URL http://arxiv.org/abs/2311.01276

  124. [134]

    Kiani B T, Fesser L and Weber M 2024 ArXiv:2410.05499 URL https://arxiv.org/abs/2410.05499

  125. [135]

    Li Y, Wang Y, Huang L, Yang H, Wei X, Zhang J, Wang T, Wang Z, Shao B and Liu T Y 2023ICLR 2024URLarXivpreprintarXiv:2304.13542

  126. [136]

    Mathys J and Errica F 2025 ArXiv:2509.01381 URLhttps://arxiv.org/abs/2509.01381

  127. [137]

    Fey M, Yuen J G and Weichert F 2020 ArXiv:2006.12179 URL https://arxiv.org/abs/2006.12179

  128. [138]

    Sun Y, Lu Y, Li Y Y, Jing Z, Leung C K and Hu P 2025Communications Chemistry8URL http://dx.doi.org/10.1038/s42004-025-01683-z

  129. [139]

    Han S, Fu H, Wu Y, Zhao G, Song Z, Huang F, Zhang Z, Liu S and Zhang W 2023Briefings in bioinformaticsURLhttps://api.semanticscholar.org/CorpusID:260969566

  130. [140]

    Ji Y, Liang J and Xu Z 2025Phys. Rev. Lett.135URL http://dx.doi.org/10.1103/ssp9-7s81

  131. [141]

    Sci.1–1 URL http://dx.doi.org/10.1109/TPAMI.2021.3054830 27 ML4AtomsSix Open Questions for MLIPsAuthoret al

    Cea T, Pantale’on P A, Walet N R and Guinea F 2021Nano Mater. Sci.1–1 URL http://dx.doi.org/10.1109/TPAMI.2021.3054830 27 ML4AtomsSix Open Questions for MLIPsAuthoret al

  132. [142]

    Wu F, Radev D and Li S Z 2021 ArXiv:2110.01191 URL https://arxiv.org/abs/2110.01191

  133. [143]

    Liao Y L and Smidt T 2023 ArXiv:2206.11990 URLhttps://arxiv.org/abs/2206.11990

  134. [144]

    Liao Y L, Wood B, Das A and Smidt T 2024 ArXiv:2306.12059 URL https://arxiv.org/abs/2306.12059

  135. [145]

    Frank J T, Unke O T, M¨ uller K R and Chmiela S156539 ISSN 2041-1723

  136. [146]

    Kabylda A, Frank J T, Su´ arez-Dou S, Khabibrakhmanov A, Medrano Sandonas L, Unke O T, Chmiela S, M¨ uller K R and Tkatchenko A 2025J. Am. Chem. Soc.14733723–33734 URLhttp://dx.doi.org/10.1021/jacs.5c09558

  137. [147]

    Behler J and Cs´ anyi G 2021Eur. Phys. J. B94URL http://dx.doi.org/10.1140/epjb/s10051-021-00156-1

  138. [148]

    Kulichenko M, Nebgen B, Lubbers N, Smith J S, Barros K, Allen A E A, Habib A, Shinkle E, Fedik N, Li Y W, Messerly R A and Tretiak S 2024Chemical Reviews12413681–13714 URLhttps://pubs.acs.org/doi/10.1021/acs.chemrev.4c00572

  139. [149]

    Bart´ ok A P, Payne M C, Kondor R and Cs´ anyi G 2010Physical Review Letters104ISSN 1079-7114 URLhttp://dx.doi.org/10.1103/PhysRevLett.104.136403

  140. [150]

    Artrith N, Morawietz T and Behler J 2011Physical Review B83ISSN 1550-235X URL http://dx.doi.org/10.1103/PhysRevB.83.153101

  141. [151]

    Toukmaji A Y and Board J A 1996Computer Physics Communications9573–92 ISSN 0010-4655 URLhttp://dx.doi.org/10.1016/0010-4655(96)00016-1

  142. [152]

    King D S, Kim D, Zhong P and Cheng B 2025Nature Communications168763

  143. [153]

    Zhong P, Kim D, King D S and Cheng B 2025npj Computational Materials11ISSN 2057-3960 URLhttp://dx.doi.org/10.1038/s41524-025-01911-z

  144. [154]

    Kim D, Wang X, Zhong P, King D S, Inizan T J and Cheng B 2025 ArXiv:2507.14302 URL https://arxiv.org/abs/2507.14302

  145. [155]

    Loche P, Huguenin-Dumittan K K, Honarmand M, Xu Q, Rumiantsev E, How W B, Langer M F and Ceriotti M 2025J. Chem. Phys.162URL http://dx.doi.org/10.1063/5.0251713

  146. [156]

    Ramasubramanian H, Vazquez-Mayagoitia A, Sivaraman G and Thakur A C 2025 Reciprocal space attention for learning long-range interactions URL https://arxiv.org/abs/2510.13055

  147. [157]

    Fuchs P, Sanocki M and Zavadlav J 2025npj Comput Mater11287 URL https://www.nature.com/articles/s41524-025-01790-4

  148. [158]

    Khajehpasha E R, Finkler J A, K¨ uhne T D and Ghasemi S A 2022Phys. Rev. B105URL http://dx.doi.org/10.1103/PhysRevB.105.144106

  149. [159]

    Xie X, Persson K A and Small D W 2020J. Chem. Theory Comput.164256–4270 URL http://dx.doi.org/10.1021/acs.jctc.0c00217

  150. [160]

    Ko T W, Liu R, Mishra A R, Yu Z, Qi J and Ong S P 2025 ArXiv:2511.07249 URL https://arxiv.org/abs/2511.07249

  151. [161]

    Thomas J, Baldwin W J, Cs´ anyi G and Ortner C 2024 ArXiv:2406.10915 URL https://arxiv.org/abs/2406.10915

  152. [162]

    Kim D and Cheng B 2026The Journal of Chemical Physics164ISSN 1089-7690 URL http://dx.doi.org/10.1063/5.0316886

  153. [163]

    Grasselli F, Rossi K, de Gironcoli S and Grisafi A Long-range electrostatics in atomistic machine learning: A physical perspective (Preprint2602.11071) 28 ML4AtomsSix Open Questions for MLIPsAuthoret al

  154. [164]

    Batatia I, Benner P, Yuan C, Elena A M, Kov’acs D P, Riebesell J, Advincula X R, Asta M, Baldwin W J, Bernstein N, Bhowmik A, Blau S M, Cuarare V, Darby J P, De S, Pia F D, Deringer V L, Elijovsius R, El-Machachi Z, Fako E, Ferrari A C, Genreith-Schriever A R, George J, Goodal...

  155. [165]

    Anstine D M and Isayev O 2023The Journal of Physical Chemistry A1272417 – 2431 URL https://pubs.acs.org/doi/10.1021/acs.jpca.2c06778

  156. [166]

    Liu X, Zeng K, Luo Z, Wang Y, Zhao T and Xu Z Fine-Tuning Universal Machine-Learned Interatomic Potentials: A Tutorial on Methods and Applications arXiv:2506.21935 URL http://arxiv.org/abs/2506.21935

  157. [167]

    Bart´ ok A P, Kermode J, Bernstein N and Cs´ anyi G 2018Phys. Rev. X8041048 URL https://link.aps.org/doi/10.1103/PhysRevX.8.041048

  158. [168]

    Botu V and Ramprasad R 2015Phys. Rev. B92094306 URL https://link.aps.org/doi/10.1103/PhysRevB.92.094306

  159. [169]

    Mater.9174 URL https://www.nature.com/articles/s41524-023-01123-3

    Liu Y, He X and Mo Y 2023npj Comput. Mater.9174 URL https://www.nature.com/articles/s41524-023-01123-3

  160. [170]

    Han B and Cheng Y ArXiv:2506.01860 URLhttp://arxiv.org/abs/2506.01860

  161. [171]

    P´ ota B, Ahlawat P, Cs´ anyi G and Simoncelli M ArXiv:2408.00755 URL http://arxiv.org/abs/2408.00755

  162. [172]

    Jakob K, Reuter K and Margraf J T 2025Adv. Intell. Discov.URL https://doi.org/10.1002/aidi.202500031

  163. [173]

    Berger E, Bagheri M and Komsa H P ArXiv:2504.06993 URL http://arxiv.org/abs/2504.06993

  164. [174]

    Prakash P, Gibson J B, Li Z, Gianluca G D, Esquivel J, Fuemmeler E, Geisler B, Kim J S, Roitberg A, Tadmor E B, Liu M, Martiniani S, Stewart G R, Hamlin J J, Hirschfeld P J and Hennig R G ArXiv:2509.25186 URLhttp://arxiv.org/abs/2509.25186

  165. [175]

    Lu Y, Andersen H, Wu R, Ganose A M, Wen B, Pujari A, Wang T, Borowiec J, Parkin I P, De Volder M and Boruah B D202308869 URL https://onlinelibrary.wiley.com/doi/abs/10.1002/smll.202308869

  166. [176]

    Rong Z, Kitchaev D A, Canepa P, Huang W and Ceder G 2016The Journal of chemical physics145 7074112

  167. [177]

    Hirschfeld J A and Lustfeld H 2011Phys. Rev. B84224308 URL https://doi.org/10.1103/PhysRevB.84.224308

  168. [178]

    Yajima T, Hinuma Y, Hori S, Iwasaki R, Kanno R, Ohhara T, Nakao A, Munakata K and Hiroi Z 2021J. Mater. Chem. A911278–11284 URL https://doi.org/10.1039/D1TA00552A

  169. [179]

    Maxson T, Soyemi A, Zhang X, Chen B W J and Szilv´ asi T 2025J. Chem. Inf. Model.65 8097–8112 URLhttps://doi.org/10.1021/acs.jcim.5c01262

  170. [180]

    Eckhoff M and Reiher M 2025J. Chem. Theory Comput.219641–9656 URL https://doi.org/10.1021/acs.jctc.5c01127

  171. [181]

    Zeng J, Zhang D, Lu D, Mo P, Li Z, Chen Y, Rynik M, Huang L, Li Z, Shi S, Wang Y, Ye H, Tuo P, Yang J, Ding Y, Li Y, Tisi D, Zeng Q, Bao H, Xia Y, Huang J, Muraoka K, Wang Y, Chang J, Yuan F, Bore S L, Cai C, Lin Y, Wang B, Xu J, Zhu J X, Luo C, Zhang Y, Goodall R E A, Liang W...

  172. [182]

    Zeni C, Pinsler R, Z¨ ugner D, Fowler A, Horton M, Fu X, Shysheya S, Crabb´ e J, Sun L, Smith J, Nguyen B, Schulz H, Lewis S, Huang C W, Lu Z, Zhou Y, Yang H, Hao H, Li J, Tomioka R and Xie T 2024 ArXiv:2312.03687 [cond-mat] URLhttp://arxiv.org/abs/2312.03687

  173. [183]

    Kov´ acs D P, Moore J H, Browning N J, Batatia I, Horton J T, Pu Y, Kapil V, Witt W C, Magd˘ au I B, Cole D J and Cs´ anyi G 2025J. Am. Chem. Soc.14717598–17611 URL https://pubs.acs.org/doi/10.1021/jacs.4c07099

  174. [184]

    Esders M, Schnake T, Lederer J, Kabylda A, Montavon G, Tkatchenko A and M¨ uller K R 2025J. Chem. Theory Comput.21714–729 URL https://pubs.acs.org/doi/10.1021/acs.jctc.4c01424

  175. [185]

    Han Z, Kivelson S A and Volkov P A 2023Phys. Rev. Lett.132 22226001 URL https://doi.org/10.1103/PhysRevLett.132.226001

  176. [186]

    Stepanov P, Das I, Lu X, Fahimniya A, Watanabe K, Taniguchi T, Koppens F H L, Lischner J, Levitov L S and Efetov D K 2020Nature583375–378 URL https://doi.org/10.1038/s41586-020-2459-6

  177. [187]

    Commun.1510852 URL https://doi.org/10.1038/s41467-024-55138-5

    Choi H, Kim J, Park J, Lee J, Heo W, Kwon J, Lee S H, Ahmed F, Watanabe K, Taniguchi T, Sun Z, Jo M H and Choi H 2024Nat. Commun.1510852 URL https://doi.org/10.1038/s41467-024-55138-5

  178. [188]

    Commun.136164 URLhttps://doi.org/10.1038/s41467-022-33811-x

    Muir J B, Levinsen J, Earl S K, Conway M A, Cole J H, Wurdack M, Mishra R, Ing D J, Estrecho E, Lu Y, Efimkin D K, Tollerud J O, Ostrovskaya E A, Parish M M and Davis J A 2022Nat. Commun.136164 URLhttps://doi.org/10.1038/s41467-022-33811-x

  179. [189]

    Biswas S, Champagne A, Haber J B, Pokawanvit S, Wong J, Akbari H, Krylyuk S, Watanabe K, Taniguchi T, Davydov A V, Balushi Z Y A, Qiu D Y, da Jornada F H, Neaton J B and Atwater H A 2023ACS Nano177685–7694 URL https://doi.org/10.1021/acsnano.3c00145

  180. [190]

    hao Chan Y, Haber J B, Naik M H, Neaton J B, Qiu D Y, da Jornada F H and Louie S G 2023Nano Lett.233971–3977 URLhttps://doi.org/10.1021/acs.nanolett.3c00732

  181. [191]

    van Efferen C, Patzold L, Tounsi T Y, Schobert A, Winter M, in ’t Veld Y, Georger M, Safeer A, Kramer C, Fischer J, Berges J, Michely T, Mozara R, Wehling T and Jolie W 2025 Phys. Rev. X15031030 URLhttps://doi.org/10.1103/l8lg-ny6m

  182. [192]

    Barrier J, Peng L, Xu S, Fal’ko V I, Watanabe K, Tanigushi T, Geim A K, Adam S and Berdyugin A I 2024arXivArXiv:2412.01577 URLhttps://arxiv.org/abs/2412.01577

  183. [193]

    Kobchikova P P, Bakirov B A, Ryltsev R E, Xiao H and Khodov I A 2025Biophys. Rev. URLhttps://doi.org/10.1007/s12551-025-01351-5

  184. [194]

    Simul.47786 – 803 URL https://doi.org/10.1080/08927022.2020.1828583

    Joshi S Y and Deshmukh S A 2020Mol. Simul.47786 – 803 URL https://doi.org/10.1080/08927022.2020.1828583

  185. [195]

    Methods18382 – 388 URLhttps://doi.org/10.1038/s41592-021-01098-3

    Souza P C T, Alessandri R, Barnoud J, Thallmair S, Faustino I, Grunewald F, Patmanidis I, Abdizadeh H, Bruininks B M H, Wassenaar T A, Kroon P C, Melcr J, Nieto V, Corradi V, Khan H M, Domanski J J, Javanainen M, Martinez-Seara H, Reuter N, Best R B, Vattulainen I, Monticelli ...

  186. [196]

    Chaisson E H, Heberle F A and Doktorova M 2023Membranes13URL https://doi.org/10.3390/membranes13070629

  187. [197]

    Rev.14111 – 143 URL https://doi.org/10.1007/s12551-021-00913-7

    Carey A B, Ashenden A and Koper I 2022Biophys. Rev.14111 – 143 URL https://doi.org/10.1007/s12551-021-00913-7

  188. [198]

    Lee J, Patel D S, Staahle J, Park S J, Kern N R, Kim S, Lee J, Cheng X, Valvano M A, Holst O, Knirel Y A, Qi Y, Jo S, Klauda J B, Widmalm G and Im W 2018J. Chem. Theory Comput.15775–786 URLhttps://doi.org/10.1021/acs.jctc.8b01066 30 ML4AtomsSix Open Questions for MLIPsAuthoret al

  189. [199]

    Pattern Anal

    Xie J, Ma Z, Lei a J, Zhang G, Xue J H, Tan Z H and Guo J 2021IEEE Trans. Pattern Anal. Mach. Intell.1–1 arXiv:2010.05244 [cs] URLhttp://arxiv.org/abs/2010.05244

  190. [200]

    Mater.6124 URL https://www.nature.com/articles/s41524-020-00390-8

    Wen M and Tadmor E B 2020npj Comput. Mater.6124 URL https://www.nature.com/articles/s41524-020-00390-8

  191. [201]

    Kurniawan Y, Wen M, Tadmor E B and Transtrum M K 2025 ArXiv:2508.06456 [cond-mat] URLhttp://arxiv.org/abs/2508.06456

  192. [202]

    Sci.107913–7922 URL https://xlink.rsc.org/?DOI=C9SC02298H

    Janet J P, Duan C, Yang T, Nandy A and Kulik H J 2019Chem. Sci.107913–7922 URL https://xlink.rsc.org/?DOI=C9SC02298H

  193. [203]

    Podryabinkin E V, Tikhonov E V, Shapeev A V and Oganov A R 2019Phys. Rev. B99 064114 URLhttps://link.aps.org/doi/10.1103/PhysRevB.99.064114

  194. [204]

    Jacobsen T, Jørgensen M and Hammer B 2018Phys. Rev. Lett.120026102 URL https://doi.org/10.1103/PhysRevLett.120.026102

  195. [205]

    Rodriguez A, Smith J S and Mendoza-Cortes J L216698–6710 arXiv:2503.07839 URL http://arxiv.org/abs/2503.07839

  196. [206]

    Birschitzky V C, Leoni L, Reticcioli M and Franchini C134216301 URL https://link.aps.org/doi/10.1103/PhysRevLett.134.216301

  197. [207]

    Pl´ e T, Lagard` ere L and Piquemal J P1412554–12569 URL https://xlink.rsc.org/?DOI=D3SC02581K

  198. [208]

    Chun H, Hong M, Noh S H and Han B214030–4039 URL https://pubs.acs.org/doi/10.1021/acs.jctc.5c00090

  199. [209]

    Hu Y, Sheng Y, Huang J, Xu X, Yang Y, Zhang M, Wu Y, Ye C, Yang J and Zhang W122 e2503439122 URLhttps://pnas.org/doi/10.1073/pnas.2503439122

  200. [210]

    R¨ ocken S and Zavadlav J1069 URL https://www.nature.com/articles/s41524-024-01251-4

  201. [211]

    Lundberg S M, Erion G, Chen H, DeGrave A, Prutkin J M, Nair B, Katz R, Himmelfarb J, Bansal N and Lee S I 2020Nat. Mach. Intell.256–67 URL https://www.nature.com/articles/s42256-019-0138-9

  202. [212]

    Oviedo F, Ferres J L, Buonassisi T and Butler K T 2022Accounts of Materials Research3 597–607 URLhttps://pubs.acs.org/doi/10.1021/accountsmr.1c00244

  203. [213]

    Ribeiro M T, Singh S and Guestrin C 2016 ArXiv:1602.04938 [cs] URL http://arxiv.org/abs/1602.04938

  204. [214]

    Shapley L S 1953 A Value for n-Person GamesContributions to the Theory of Games (AM-28), Volume IIed Kuhn H W and Tucker A W (Princeton University Press) pp 307–318 ISBN 978-1-4008-8197-0 URL https://www.degruyter.com/document/doi/10.1515/9781400881970-018/html

  205. [215]

    Lundberg S M and Lee S I 2017 A unified approach to interpreting model predictions Proceedings of the 31st International Conference on Neural Information Processing Systems NIPS’17 (Curran Associates Inc.) p 4768–4777 URLhttps://proceedings.neurips.cc/ paper_files/paper/2017/f...

  206. [216]

    Morita K, Davies D W, Butler K T and Walsh A 2020J. Chem. Phys.153024503 URL https://pubs.aip.org/jcp/article/153/2/024503/1061513/ Modeling-the-dielectric-constants-of-crystals

  207. [217]

    Ver´ ıssimo R F, Matias P H F, Barbosa M R, Neto F O S, Neto B A D and De Oliveira H C B 2025J. Chem. Inf. Model.657874–7886 URL https://pubs.acs.org/doi/10.1021/acs.jcim.4c02414

  208. [218]

    Dangayach R, Jeong N, Demirel E, Uzal N, Fung V and Chen Y 2025Environmental Science & Technology59993–1012 URLhttps://pubs.acs.org/doi/10.1021/acs.est.4c08298 31 ML4AtomsSix Open Questions for MLIPsAuthoret al

  209. [219]

    Akkas S and Azad A 2024 GNNShap: Scalable and Accurate GNN Explanation using Shapley ValuesProceedings of the ACM Web Conference 2024(Singapore Singapore: ACM) pp 827–838 ISBN 979-8-4007-0171-9 URL https://dl.acm.org/doi/10.1145/3589334.3645599

  210. [220]

    Muschalik M, Fumagalli F, Frazzetto P, Strotherm J, Hermes L, Sperduti A, H¨ ullermeier E and Hammer B 2025 ArXiv:2501.16944 [cs] URLhttp://arxiv.org/abs/2501.16944

  211. [221]

    Rowe P, Csanyi G, Alfe D and Michaelides A 2018Phys. Rev. B97054303 URL https://doi.org/10.1103/PhysRevB.97.054303

  212. [222]

    Thiemann F L, Rowe P, Muller E A and Michaelides A 2020J. Phys. Chem. C124 22278–22290 URLhttps://doi.org/10.1021/acs.jpcc.0c05831

  213. [223]

    Mater.101–11 URL https://doi.org/10.1038/s41524-024-01357-9

    Siddiqui A and Hine N 2024npj Comput. Mater.101–11 URL https://doi.org/10.1038/s41524-024-01357-9

  214. [224]

    Huang B, Clark G, Navarro-Moratalla E, Klein D R, Cheng R, Seyler K L, Zhong D, Schmidgall E R, McGuire M A, Cobden D H, Yao W, Xiao D, Jarillo-Herrero P and Xu X 2017Nature546270–273 URLhttps://doi.org/10.1038/nature22391

  215. [225]

    Nanotechnol.10 765–769 URLhttps://doi.org/10.1038/nnano.2015.143

    Xi X, Zhao L, Wang Z, Berger H, Forro L, Shan J and Mak K F 2015Nat. Nanotechnol.10 765–769 URLhttps://doi.org/10.1038/nnano.2015.143

  216. [226]

    Duvjir G, Choi B K, Jang I, Ulstrup S, Kang S, Ly T T, Kim S, Choi Y H, Jozwiak C, Bostwick A, Rotenberg E, Park J G, Sankar R, Kim K S, Kim J and Chang Y J 2018Nano Lett.185432–5438 URLhttps://doi.org/10.1021/acs.nanolett.8b01764

  217. [227]

    24(39) 12088–12094 URLhttps://doi.org/10.1021/acs.nanolett.4c02750

    Cheung C T S, Goodwin Z A H, Han Y, Lu J, Mostofi A A and Lischner J 2024Nano Lett. 24(39) 12088–12094 URLhttps://doi.org/10.1021/acs.nanolett.4c02750

  218. [228]

    Rivano N, Libbi F, Tan C W, Cheung C, Lado J, Mostofi A, Kim P, Lischner J, Fumega A O, Kozinsky B and Goodwin Z A H 2025arXivArXiv:2504.13675 URL http://arxiv.org/abs/2504.13675

  219. [229]

    Commun.586902–6905 URL https://doi.org/10.1039/D2CC02519A

    Marmolejo-Tejada J M and Mosquera M A 2022Chem. Commun.586902–6905 URL https://doi.org/10.1039/D2CC02519A

  220. [230]

    Wang W, Zhou G D, Lin W H, Feng Z, Wang Y, Liang M, Zhang Z, Wu M, Liu L, Watanabe K, Taniguchi T, Yang W, Zhang G, Liu K, Gao J, Liu Y, Xie X, Song Z and Lu X 2024Phys. Rev. Lett.132246501 URLhttps://doi.org/10.1103/PhysRevLett.132.246501

  221. [231]

    Phys.1842–47 URL https://doi.org/10.1038/s41567-021-01418-6

    Polshyn H, Zhang Y, Kumar M A, Soejima T, Ledwith P J, Watanabe K, Taniguchi T, Vishwanath A, Zaletel M P and Young A F 2021Nat. Phys.1842–47 URL https://doi.org/10.1038/s41567-021-01418-6

  222. [232]

    Phys.: Condens

    Goodwin Z A H and Fal’ko V I 2022J. Phys.: Condens. Matter34494001 URL https://doi.org/10.1088/1361-648X/ac99ca

  223. [233]

    Zhang L and Luo D 2025arXivArXiv:2509.09275 URL https://arxiv.org/abs/2509.09275

  224. [234]

    Wang T, He X, Li M, Li Y, Bi R, Wang Y, Cheng C, Shen X, Meng J, Zhang H, Liu H, Wang Z, Li S, Shao B and Liu T Y 2024Nature6351019–1027 URL https://doi.org/10.1038/s41586-024-08127-z

  225. [235]

    Batatia I, Benner P, Chiang Y, Elena A M, Kov´ acs D P, Riebesell J, Advincula X R, Asta M, Avaylon M, Baldwin W J, Berger F, Bernstein N, Bhowmik A, Blau S M, C˘ arare V, Darby J P, De S, Della Pia F, Deringer V L, Elijoˇ sius R, El-Machachi Z, Falcioni F, Fako E, Ferrari A C...

  226. [236]

    Bertani M and Pedone A 2025J. Phys. Chem. C12912697–12709 URL https://pubs.acs.org/doi/10.1021/acs.jpcc.5c01857

  227. [237]

    Kov´ acs D P, Oord C v d, Kucera J, Allen A E, Cole D J, Ortner C and Cs´ anyi G 2021J. Chem. Theory Comput.17(12) 7696–7711 URL https://doi.org/10.1021/acs.jctc.1c00647

  228. [238]

    Witt W C, van der Oord C, Gelˇ zinyt˙ e E, J¨ arvinen T, Ross A, Darby J P, Ho C H, Baldwin W J, Sachs M, Kermode Jet al.2023J. Chem. Phys.159164101 URL https://doi.org/10.1063/5.0158783

  229. [239]

    Fu X, Musaelian A, Johansson A, Jaakkola T and Kozinsky B 2023 ArXiv:2310.13756 [physics] URLhttp://arxiv.org/abs/2310.13756

  230. [240]

    Tan C W, Descoteaux M L, Kotak M, Nascimento G d M, Kavanagh S R, Zichi L, Wang M, Saluja A, Hu Y R, Smidt Tet al.2025arXiv preprint arXiv:2504.16068ArXiv:2504.16068 URLhttps://arxiv.org/abs/2504.16068

  231. [241]

    Aykent S and Xia T 2025 GotenNet: Rethinking efficient 3D equivariant graph neural networksThe Thirteenth International Conference on Learning RepresentationsURL https://openreview.net/forum?id=5wxCQDtbMo

  232. [242]

    Qu E, Wood B M, Krishnapriyan A S and Ulissi Z W A recipe for scalable attention-based MLIPs: Unlocking long-range accuracy with all-to-all node attention (Preprint2603.06567)

  233. [243]

    Dao T, Fu D Y, Ermon S, Rudra A and R´ e C FlashAttention: Fast and Memory-Efficient Exact Attention with IO-Awareness (Preprint2205.14135)

  234. [244]

    Langer M F, Pozdnyakov S N and Ceriotti M504LT01 ISSN 2632-2153

  235. [245]

    Jacobs R, Morgan D, Attarian S, Meng J, Shen C, Wu Z, Xie C Y, Yang J H, Artrith N, Blaiszik B, Ceder G, Choudhary K, Csanyi G, Cubuk E D, Deng B, Drautz R, Fu X, Godwin J, Honavar V, Isayev O, Johansson A, Kozinsky B, Martiniani S, Ong S P, Poltavsky I, Schmidt K, Takamoto S,...

  236. [246]

    Hinton G, Vinyals O and Dean J 2015 ArXiv:1503.02531 [stat] URL http://arxiv.org/abs/1503.02531

  237. [247]

    Amin I, Raja S and Krishnapriyan A S 2025 Towards fast, specialized machine learning force fields: Distilling foundation models via energy hessiansThe Thirteenth International Conference on Learning RepresentationsURL https://openreview.net/forum?id=1durmugh3I

  238. [248]

    Gardner J L A, Toit D F T d, Mahmoud C B, Beaulieu Z F, Juraskova V, Pa¸ sca L B, Rosset L A M, Duarte F, Martelli F, Pickard C J and Deringer V L 2025 ArXiv:2506.10956 URL http://arxiv.org/abs/2506.10956

  239. [249]

    Taniguchi T 2025Faraday Discussions256139–155 publisher: Royal Society of Chemistry URLhttps://pubs.rsc.org/en/content/articlelanding/2025/fd/d4fd00090k

  240. [250]

    Matin S, Allen A E A, Shinkle E, Pachalieva A, Craven G T, Nebgen B, Smith J S, Messerly R, Li Y W, Tretiak S, Barros K and Lubbers N 2025 ArXiv:2502.05379 [physics] URL http://arxiv.org/abs/2502.05379

  241. [251]

    Leimeroth N, Erhard L C, Albe K and Rohrer J 2025 ArXiv:2505.02503 URL http://arxiv.org/abs/2505.02503

  242. [252]

    Thompson A P, Aktulga H M, Berger R, Bolintineanu D S, Brown W M, Crozier P S, In ’T Veld P J, Kohlmeyer A, Moore S G, Nguyen T D, Shan R, Stevens M J, Tranchida J, Trott C and Plimpton S J 2022Computer Physics Communications271108171 URL https://linkinghub.elsevier.com/retrie...

  243. [253]

    Johansson A, Weinberg E, Trott C R, McCarthy M J and Moore S G 2025 ArXiv:2508.13523 URLhttp://arxiv.org/abs/2508.13523 33 ML4AtomsSix Open Questions for MLIPsAuthoret al

  244. [254]

    Bharadwaj V, Glover A, Buluc A and Demmel J 2025 ArXiv:2501.13986 URL http://arxiv.org/abs/2501.13986

  245. [255]

    Learn.: Sci

    Stark W G, Van Der Oord C, Batatia I, Zhang Y, Jiang B, Cs´ anyi G and Maurer R J 2024 Mach. Learn.: Sci. Technol.5030501 URL https://iopscience.iop.org/article/10.1088/2632-2153/ad5f11

  246. [256]

    Musaelian A, Johansson A, Batzner S and Kozinsky B 2023 Scaling the Leading Accuracy of Deep Equivariant Models to Biomolecular Simulations of Realistic SizeProceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis(Denver CO U...

  247. [257]

    Firoz J, Pellegrini F, Geiger M, Hsu D, Bilbrey J A, Chou H Y, Stadler M, Hoehnerbach M, Wang T, Lin D, Kucukbenli E, Sprueill H W, Batatia I, Xantheas S S, Lee M, Mundy C, Csanyi G, Smith J S, Sadayappan P and Choudhury S Optimizing Data Distribution and Kernel Performance fo...

  248. [258]

    Wood B M, Dzamba M, Fu X, Gao M, Shuaibi M, Barroso-Luque L, Abdelmaqsoud K, Gharakhanyan V, Kitchin J R, Levine D S, Michel K, Sriram A, Cohen T, Das A, Rizvi A, Sahoo S J, Ulissi Z W and Zitnick C L UMA: A Family of Universal Models for Atoms (Preprint2506.23971)

  249. [259]

    Liu Y, Zhang D, Peng A, E W, Zhang L and Wang H Scaling Machine Learning Interatomic Potentials with Mixtures of Experts (Preprint2603.07977)

  250. [260]

    Alampara N, Schilling-Wilhelmi M and Jablonka K M 2025Comput. Mater. Sci.259URL https://doi.org/10.1016/j.commatsci.2025.114041

  251. [261]

    Econ.1114041

    Goodhart C 1975Monet. Econ.1114041

  252. [262]

    Yankelovich D 1971ales Management, the Marketing MagazineURLhttps: //archive.org/details/sim_sales-management_1971-11-15_107_11/page/26/mode/2up

  253. [263]

    Deng L 2012IEEE Signal Processing Magazine29141–142

  254. [264]

    Chiang Y, Kreiman T, Weaver E, Amin I, Kuner M, Zhang C, Kaplan A, Chrzan D, Blau S M, Krishnapriyan A Set al.2025 Mlip arena: advancing fairness and transparency in machine learning interatomic potentials through an open and accessible benchmark platform AI for Accelerated Ma...

  255. [265]

    Kasoar E, Hart J, Batatia I, Elena A and Cs´ anyi G ml-peg URL https://github.com/ddmms/ml-peg

  256. [266]

    Kaplan A D, Liu R, Qi J, Ko T W, Deng B, Riebesell J, Ceder G, Persson K A and Ong S P 2025arXiv preprint arXiv:2503.04070

  257. [267]

    FAIR Chemistry Leaderboard - a Hugging Face Space by facebook — huggingface.co https://huggingface.co/spaces/facebook/fairchem_leaderboard[Accessed 27-04-2026]

  258. [268]

    Mater.10144 URL https://doi.org/10.1038/s41524-024-01316-4

    Omee S S, Fu N, Dong R, Hu M and Hu J 2024npj Comput. Mater.10144 URL https://doi.org/10.1038/s41524-024-01316-4

  259. [269]

    Lottick K, Susai S, Friedler S A and Wilson J P 2019 ArXiv:1911.08354 URL https://arxiv.org/abs/1911.08354

  260. [270]

    Horiz.URLhttps://doi.org/10.1039/D5MH01404B 34

    Walker M and Butler K 2025Mater. Horiz.URLhttps://doi.org/10.1039/D5MH01404B 34

Pith tools

Reviewed June 27, 2026 · model on record in the stance chip above.