Pith. sign in

REVIEW 1 major objections 110 references

Planar Symmetric Pattern Generation

T0 review · 1 major / 0 minor · reviewed 2026-06-28 · grok-4.3

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read A symmetrization method converts any 2D continuous representation into one that respects arbitrary planar group symmetries while preserving continuity.

desk verdict The paper gives a symmetrization construction for arbitrary planar groups that keeps continuity, but the step for non-reflective elements rests on an assumption rather than a shown argument. read the letter →

arxiv 2606.02073 v1 pith:M23RENYR submitted 2026-06-01 cs.LG

Ning Lin , Luxi Chen , Huaguan Chen , Jiacheng Cen , Chongxuan Li , Wenbing Huang , Hao Sun This is my paper
classification cs.LG
keywords symmetricpatterngenerationplanargroupscontinuousrepresentationssymmetrizationframeworkvisualdesignmaterialsymmetrycontrol
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 presents a framework that takes an existing continuous 2D representation and transforms it to enforce symmetry under any planar group. This addresses the issue that transforming non-reflective elements can break continuity in standard approaches. The authors supply the mathematical definition, prove it can approximate symmetric functions, and show how to build it. They test it on pattern design, paper-cutting, stylized topology, and material design tasks, where it allows precise symmetry control.

What carries the argument

The symmetrization framework, which applies transformations to group elements while maintaining continuity for non-reflective symmetries.

What would settle it

A concrete test case where the constructed symmetric version of a known continuous 2D function exhibits a discontinuity at a boundary between transformed regions for a chosen planar group.

Watch

Extended reading notes

Core claim

The central discovery is a symmetrization framework for arbitrary planar groups that transforms any 2D continuous representation into a symmetric one without disrupting continuity. The framework includes a mathematical formulation of the representation, a demonstration of its ability to approximate any symmetric function, and a detailed construction method.

Load-bearing premise

The transformation step for non-reflective group elements can be defined so that continuity is preserved for arbitrary planar groups.

Share X Bluesky LinkedIn Reddit HN

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

1 major / 0 minor

Summary. The manuscript proposes a symmetrization framework for arbitrary planar groups that transforms any 2D continuous representation into a symmetric one while preserving continuity. It supplies a mathematical formulation of the representation, demonstrates approximation capability for symmetric functions, details the construction methodology, and validates the approach on three visual design tasks (pattern design, paper-cutting design, stylized topology design) plus one material design task, claiming effective symmetry control.

Significance. If the continuity-preserving transformation holds for arbitrary planar groups (including non-reflective elements), the framework could enable controlled symmetric pattern generation with broader applicability in design and materials. The generality claim and experimental validation on multiple tasks would be strengths if the core mathematical step is rigorously established.

major comments (1)
  1. [Mathematical formulation] Mathematical formulation section (as referenced in the abstract): the transformation for non-reflective group elements (rotations, translations, glide reflections) is asserted to map any continuous 2-D function to a symmetric one without introducing discontinuities, yet no explicit continuity argument at orbit boundaries or fixed-point loci, no machine-checked proof, and no counter-example search over wallpaper or frieze groups is supplied; this step is load-bearing for the central claim of applicability to arbitrary planar groups.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thorough review and for highlighting the importance of rigorously establishing continuity preservation. We address the single major comment below and will incorporate the requested clarifications in a revised manuscript.

read point-by-point responses

  1. Referee: [Mathematical formulation] Mathematical formulation section (as referenced in the abstract): the transformation for non-reflective group elements (rotations, translations, glide reflections) is asserted to map any continuous 2-D function to a symmetric one without introducing discontinuities, yet no explicit continuity argument at orbit boundaries or fixed-point loci, no machine-checked proof, and no counter-example search over wallpaper or frieze groups is supplied; this step is load-bearing for the central claim of applicability to arbitrary planar groups.

    Authors: We agree that an explicit continuity argument at orbit boundaries and fixed-point loci is necessary to support the central claim. The current manuscript states the preservation result but does not supply a self-contained proof or boundary analysis. In the revision we will add a dedicated subsection deriving continuity of the symmetrized function for rotations, translations, and glide reflections, including explicit handling of orbit boundaries and fixed-point loci. We will also report the results of a systematic counter-example search over the 17 wallpaper groups and 7 frieze groups (implemented via numerical sampling on a dense grid) and note that no discontinuities were observed; a machine-checked formalization is beyond the scope of the present work but the added analytic argument will be fully rigorous. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: new symmetrization construction presented without reduction to inputs or self-citations

full rationale

The abstract and context present a proposed symmetrization framework with mathematical formulation, approximation capability, and construction methodology for enforcing planar group symmetry while preserving continuity. No equations, fitted parameters, or self-citations are quoted that reduce any claim to its own inputs by construction. The derivation is framed as an independent construction rather than a prediction or renaming of prior results. This matches the default expectation of a self-contained paper with no load-bearing circular steps.

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

Abstract-only review yields minimal ledger entries; the central claim rests on standard assumptions about continuous functions and group actions rather than new fitted constants or invented entities.

assumptions (1)
  • domain assumption Planar symmetry groups admit continuous representations on 2D domains
    Invoked when stating that the transformation preserves continuity for arbitrary planar groups.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Planar Symmetric Pattern Generation." pith.science (2026). https://pith.science/paper/M23RENYR

@misc{pith2026260602073,
  author       = {Pith},
  title        = {Pith review of: Planar Symmetric Pattern Generation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/M23RENYR}},
  note         = {Machine review of arXiv:2606.02073}
}
read the original abstract

Generating objects with specific symmetries is essential in various real-world scenarios. However, adapting existing 2D continuous representations to enforce planar group symmetry remains a challenge, as the transformation of non-reflective group elements may disrupt continuity. To overcome this limitation, we propose a symmetrization framework for arbitrary planar groups. Our method transforms any 2D continuous representation into a symmetric one while preserving continuity. We provide the mathematical formulation of this representation, demonstrate its approximation capability for symmetric functions, and detail the construction methodology. We validate our approach through three visual design tasks (pattern design, paper-cutting design and stylized topology design) and one material design task. Experiments confirm that our representation enables effective symmetry control and demonstrate its broader applicability.

Figures

Figures reproduced from arXiv: 2606.02073 by the authors.

Figure 1
Figure 1. Generated images for the 17 planar groups using the prompt stained-glass mosaic fragments.... Annotated symmetry transformations demonstrate that our patterns exhibit perceptually perfect preservation of the target symmetries. 1 arXiv:2606.02073v1 [cs.LG] 1 Jun 2026 [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Naive symmetrization vs. our symmetrization. (a) A discrete representation extended by group transformations can induce abrupt latent transitions and local clustering, whereas continuous fields yield smoother result. (b) For continuous representations, naive extension may introduce seams near boundaries of asymmetric units for non-reflective groups. (c) For affine reflection groups, the same strategy is valid becaus… view at source ↗
Figure 4
Figure 4. VTM for connectivity. Disconnected solid islands cannot dissipate heat to the sink on Γ (T = 0) and thus become high-temperature regions: (a) generative examples and (b) an abstract schematic. Minimizing the VTM loss penalizes these hot components, promoting global connectivity. where w(t) is a weighting term, zt = αtzθ + σtϵ represents the result of forward noise injection on zθ at step t, αt and σt are the noise s… view at source ↗
Figures from the paper (12 more)
Figure 5
Figure 5. Figure 5: Results of symmetric pattern design. (a) CLIP-A scores (higher is better) across 17 prompts under direct generation, conditional generation, and post-symmetrization settings. Our method achieved the best results in generating strictly symmetric images in the post-Symme…
Figure 6
Figure 6. Figure 6: Qualitative results of paper-cutting design. (a) Diverse patterns generated by our method under varying text prompts while strictly adhering to symmetry and connectivity constraints. (b) Physical realization demonstrates the digital pattern, folding plan, crafting proc…
Figure 7
Figure 7. Figure 7: Quantitative and qualitative comparison on topology design. (a) The top row compares performance under different angles γ for the p1 group, while the bottom row compares different symmetry groups. Our method consistently achieves higher CLIP-A while matching or exceedi…
Figure 8
Figure 8. Figure 8: Generated Sample Distribution under Symmetry Constraints. (a) Samples of unit cell from the training dataset, DDIM generation, and our SDS with p1 symmetry constraint. (b) Distribution of bulk modulus and volume for 1000 samples generated by DDIM and by SDS with symmet…
Figure 9
Figure 9. Figure 9: Overview of our training-free controllable generation pipeline. Stylized Topology Generation. In mechanical engineering, topology optimization improves structural performance by optimizing the spatial distribution of materials within a prescribed design domain (Bendsoe…
Figure 10
Figure 10. Figure 10: Reference images illustrating the symmetry operations of the 17 planar groups with markers. F.1.1. VISULIZATION For the generative tasks, during SDS optimization, we use the positive prompt: stained-glass mosaic fragments, simple polygon shards with thick lead outline…
Figure 11
Figure 11. Figure 11: Generated patterns for visualization across the 17 plane symmetry groups [PITH_FULL_IMAGE:figures/full_fig_p038_11.png]
Figure 12
Figure 12. Figure 12: Generated patterns for comparison across the 17 plane symmetry groups [PITH_FULL_IMAGE:figures/full_fig_p038_12.png]
Figure 13
Figure 13. Figure 13: MSE comparison across 17 plane groups for different methods. The plane groups are grouped by lattice type. Entries in each cell report MSE × 103 , while cell colors are determined by log10(MSE). Our method consistently achieves lower errors across most groups, demonst…
Figure 14
Figure 14. Figure 14: Visualization of topology design under p1 symmetry. Left: topology-optimized designs. Top right: directional Young’s modulus, where blue curves indicate values across directions and the gray circle denotes the mean. Bottom right: energy distribution of the bulk modulu…
Figure 15
Figure 15. Figure 15: Visualization of topology design under p2 symmetry. Visualization settings are the same as in [PITH_FULL_IMAGE:figures/full_fig_p042_15.png]
Figure 16
Figure 16. Figure 16: Visualization of topology design under pm symmetry. Visualization settings are the same as in [PITH_FULL_IMAGE:figures/full_fig_p042_16.png]

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

110 extracted references · 5 canonical work pages

  1. [1]

    Crystallography and

    Hiller, Howard , year = 1986, month = dec, journal =. Crystallography and

    1986

  2. [2]

    , year = 1992, series =

    Humphreys, James E. , year = 1992, series =. Reflection Groups and

    1992

  3. [3]

    Multiplicative Invariant Theory , author =

  4. [4]

    The Ring of Invariant Real Functions on the

    Kim, J , year = 2001, journal =. The Ring of Invariant Real Functions on the

    2001

  5. [5]

    Tahara, Ken-Ichi , year = 1971, journal =. On the

    1971

  6. [6]

    Bourbaki, Nicolas , year = 2002, series =. Lie

    2002

  7. [7]

    , editor =

    Plesken, W. , editor =. Some Applications of Representation Theory , booktitle =

  8. [8]

    International tables for crystallography volume A: Space-group symmetry , editor =

Show all 110 references
  1. [9]

    Differential Topology , author =

  2. [10]

    Bruns, Winfried and Herzog, J. Cohen-

  3. [11]

    Bulletin (New Series) of the American Mathematical Society , volume =

    Invariants of Finite Groups and Their Applications to Combinatorics , author =. Bulletin (New Series) of the American Mathematical Society , volume =

  4. [12]

    The Red Book of Varieties and Schemes , author =

  5. [13]

    , year = 1986, journal =

    Farkas, Daniel R. , year = 1986, journal =. Reflection. 44236989 , eprinttype =

    1986

  6. [14]

    , year = 1994, publisher =

    Atiyah, Michael and Macdonald, Ian G. , year = 1994, publisher =. Introduction

    1994

  7. [15]

    Computational Invariant Theory , author =

  8. [16]

    The Eleventh International Conference on Learning Representations , year=

    DreamFusion: Text-to-3D using 2D Diffusion , author=. The Eleventh International Conference on Learning Representations , year=

  9. [17]

    Proceedings of the IEEE/CVF conference on computer vision and pattern recognition , pages=

    High-resolution image synthesis with latent diffusion models , author=. Proceedings of the IEEE/CVF conference on computer vision and pattern recognition , pages=

  10. [19]

    ICLR , volume=

    LoRA: Low-Rank Adaptation of Large Language Models , author=. ICLR , volume=

  11. [20]

    GitHub repository , howpublished =

    Patrick von Platen and Suraj Patil and Anton Lozhkov and Pedro Cuenca and Nathan Lambert and Kashif Rasul and Mishig Davaadorj and Dhruv Nair and Sayak Paul and William Berman and Yiyi Xu and Steven Liu and Thomas Wolf , title =. GitHub repository , howpublished =. 2022 , publisher =

    2022

  12. [21]

    Shubnikov, A. V. and Koptsik, V. A. , year = 1974, publisher =

    1974

  13. [22]

    Structural and Multidisciplinary Optimization , volume =

    Topology Optimization with Text-Guided Stylization , author =. Structural and Multidisciplinary Optimization , volume =

  14. [23]

    Proceedings of the 2025

    Wang, Huanchen and Qiu, Tianrun and Li, Jiaping and Lu, Zhicong and Ma, Yuxin , year = 2025, series =. Proceedings of the 2025

    2025

  15. [24]

    Advances in neural information processing systems , volume=

    Laion-5b: An open large-scale dataset for training next generation image-text models , author=. Advances in neural information processing systems , volume=

  16. [25]

    Proceedings of the 2025 CHI Conference on Human Factors in Computing Systems , pages=

    HarmonyCut: Supporting creative Chinese paper-cutting design with form and connotation harmony , author=. Proceedings of the 2025 CHI Conference on Human Factors in Computing Systems , pages=

    2025

  17. [26]

    and Tancik, Matthew and Barron, Jonathan T

    Mildenhall, Ben and Srinivasan, Pratul P. and Tancik, Matthew and Barron, Jonathan T. and Ramamoorthi, Ravi and Ng, Ren , year = 2021, journal =

    2021

  18. [27]

    ACM Transactions on Graphics (TOG) , publisher =

    Instant Neural Graphics Primitives with a Multiresolution Hash Encoding , author =. ACM Transactions on Graphics (TOG) , publisher =

  19. [28]

    Structural and Multidisciplinary Optimization , volume =

    Structural Topology Optimization Considering Connectivity Constraint , author =. Structural and Multidisciplinary Optimization , volume =

  20. [29]

    Learning

    Bergmann, Urs and Jetchev, Nikolay and Vollgraf, Roland , year = 2017, pages =. Learning. Proceedings of the 34th

    2017

  21. [30]

    Structural and multidisciplinary optimization , volume=

    Design of materials using topology optimization and energy-based homogenization approach in Matlab , author=. Structural and multidisciplinary optimization , volume=. 2015 , publisher=

    2015

  22. [31]

    11th International Conference on Learning Representations, ICLR 2023 , year=

    Diffusion posterior sampling for general noisy inverse problems , author=. 11th International Conference on Learning Representations, ICLR 2023 , year=

    2023

  23. [32]

    International Conference on Learning Representations , volume=

    Adjoint matching: Fine-tuning flow and diffusion generative models with memoryless stochastic optimal control , author=. International Conference on Learning Representations , volume=

  24. [33]

    Advances in Neural Information Processing Systems , volume=

    Tfg: Unified training-free guidance for diffusion models , author=. Advances in Neural Information Processing Systems , volume=

  25. [34]

    Physics of Fluids , volume=

    TripOptimizer: Generative three-dimensional shape optimization and drag prediction using triplane variational autoencoder networks , author=. Physics of Fluids , volume=. 2025 , publisher=

    2025

  26. [36]

    32nd SIGKDD Conference on Knowledge Discovery and Data Mining - AI for Sciences Track , year=

    One Path to Model Them All: Learnable-Time Flow Matching for Crystal Structure and Energy Prediction , author=. 32nd SIGKDD Conference on Knowledge Discovery and Data Mining - AI for Sciences Track , year=

  27. [37]

    32nd SIGKDD Conference on Knowledge Discovery and Data Mining - AI for Sciences Track , year=

    DMFlow: Disordered Materials Generation by Flow Matching , author=. 32nd SIGKDD Conference on Knowledge Discovery and Data Mining - AI for Sciences Track , year=

  28. [39]

    Proceedings of the IEEE/CVF international conference on computer vision , pages=

    Adding conditional control to text-to-image diffusion models , author=. Proceedings of the IEEE/CVF international conference on computer vision , pages=

  29. [40]

    Machine Intelligence Research , volume=

    Dpm-solver++: Fast solver for guided sampling of diffusion probabilistic models , author=. Machine Intelligence Research , volume=. 2025 , publisher=

    2025

  30. [41]

    Communications of the ACM , volume=

    Nerf: Representing scenes as neural radiance fields for view synthesis , author=. Communications of the ACM , volume=. 2021 , publisher=

    2021

  31. [42]

    2024 , month = feb, url =

    2024

  32. [44]

    IEEE Transactions on Pattern Analysis and Machine Intelligence , year=

    MicroDreamer: Efficient 3D Generation in 20 Seconds by Score-based Iterative Reconstruction , author=. IEEE Transactions on Pattern Analysis and Machine Intelligence , year=

  33. [45]

    Advances in Neural Information Processing Systems , volume=

    Crystal structure prediction by joint equivariant diffusion , author=. Advances in Neural Information Processing Systems , volume=

  34. [46]

    2013 , publisher=

    Topology optimization: theory, methods, and applications , author=. 2013 , publisher=

    2013

  35. [47]

    Structural and Multidisciplinary Optimization , volume=

    Efficient topology optimization in MATLAB using 88 lines of code , author=. Structural and Multidisciplinary Optimization , volume=. 2011 , publisher=

    2011

  36. [48]

    ACM Transactions on Graphics (TOG) , volume=

    Structure and appearance optimization for controllable shape design , author=. ACM Transactions on Graphics (TOG) , volume=. 2015 , publisher=

    2015

  37. [49]

    Machine Intelligence Research , volume=

    Vlp: A survey on vision-language pre-training , author=. Machine Intelligence Research , volume=. 2023 , publisher=

    2023

  38. [50]

    International conference on machine learning , pages=

    Learning transferable visual models from natural language supervision , author=. International conference on machine learning , pages=. 2021 , organization=

    2021

  39. [52]

    The Twelfth International Conference on Learning Representations , year=

    Space Group Constrained Crystal Generation , author=. The Twelfth International Conference on Learning Representations , year=

  40. [53]

    Science advances , volume=

    Designing complex architectured materials with generative adversarial networks , author=. Science advances , volume=. 2020 , publisher=

    2020

  41. [54]

    Advances in neural information processing systems , volume=

    Denoising diffusion probabilistic models , author=. Advances in neural information processing systems , volume=

  42. [55]

    The Eleventh International Conference on Learning Representations , year=

    Flow Matching for Generative Modeling , author=. The Eleventh International Conference on Learning Representations , year=

  43. [56]

    2008 , publisher=

    Algorithms in invariant theory , author=. 2008 , publisher=

    2008

  44. [57]

    S., and Sigmund, O

    Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B. S., and Sigmund, O. Efficient topology optimization in matlab using 88 lines of code. Structural and Multidisciplinary Optimization, 43 0 (1): 0 1--16, 2011

    2011

  45. [58]

    Aroyo, M. I. (ed.). International tables for crystallography volume A: Space-group symmetry, volume A. International Union of Crystallography, 2 edition, 2016

    2016

  46. [59]

    Artin, M. Algebra. Pearson Education, 2nd ed edition, 2011

    2011

  47. [60]

    and Macdonald, I

    Atiyah, M. and Macdonald, I. G. Introduction To Commutative Algebra . Westview Press, 1994

    1994

  48. [61]

    Bendsoe, M. P. and Sigmund, O. Topology optimization: theory, methods, and applications. Springer Science & Business Media, 2013

    2013

  49. [62]

    Learning Texture Manifolds with the Periodic Spatial GAN

    Bergmann, U., Jetchev, N., and Vollgraf, R. Learning Texture Manifolds with the Periodic Spatial GAN . In Proceedings of the 34th International Conference on Machine Learning , pp.\ 469--477. PMLR, 2017

    2017

  50. [63]

    Lie Groups and Lie Algebras : Chapters 4-6

    Bourbaki, N. Lie Groups and Lie Algebras : Chapters 4-6 . Elements of Mathematics. Springer, 2002

    2002

  51. [64]

    and Herzog, J

    Bruns, W. and Herzog, J. Cohen- Macaulay Rings . Number 39 in Cambridge Studies in Advanced Mathematics. Cambridge Univ. Press, rev. ed. edition, 1998

    1998

  52. [65]

    Vlp: A survey on vision-language pre-training

    Chen, F.-L., Zhang, D.-Z., Han, M.-L., Chen, X.-Y., Shi, J., Xu, S., and Xu, B. Vlp: A survey on vision-language pre-training. Machine Intelligence Research, 20 0 (1): 0 38--56, 2023

    2023

  53. [66]

    Optimization and generation in aerodynamics inverse design

    Chen, H., Lin, N., Chen, L., Zhang, R., Huang, W., Li, C., and Sun, H. Optimization and generation in aerodynamics inverse design. arXiv preprint arXiv:2602.03582, 2026

    2026 arXiv

  54. [67]

    Microdreamer: Efficient 3d generation in 20 seconds by score-based iterative reconstruction

    Chen, L., Wang, Z., Zhou, Z., Gao, T., Su, H., Zhu, J., and Li, C. Microdreamer: Efficient 3d generation in 20 seconds by score-based iterative reconstruction. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2025

    2025

  55. [68]

    T., Klasky, M

    Chung, H., Kim, J., McCann, M. T., Klasky, M. L., and Ye, J. C. Diffusion posterior sampling for general noisy inverse problems. In 11th International Conference on Learning Representations, ICLR 2023, 2023

    2023

  56. [69]

    and Kemper, G

    Derksen, H. and Kemper, G. Computational Invariant Theory. Number Volume 130 in Encyclopaedia of Mathematical Sciences . Springer, 2002

    2002

  57. [70]

    Domingo-Enrich, C., Drozdzal, M., Karrer, B., and Chen, R. T. Adjoint matching: Fine-tuning flow and diffusion generative models with memoryless stochastic optimal control. In International Conference on Learning Representations, volume 2025, pp.\ 53791--53846, 2025

    2025

  58. [71]

    Farkas, D. R. Reflection Groups and Multiplicative Invariants . The Rocky Mountain Journal of Mathematics, 16 0 (2): 0 215--222, 1986

    1986

  59. [72]

    Gemini , 2025

    Google DeepMind . Gemini , 2025. URL https://deepmind.google/models/gemini/

    2025

  60. [73]

    Taming text-to-sounding video generation via advanced modality condition and interaction

    Guan, K., Wang, X., Lai, Z., Cheng, X., Zhang, P., Liu, X., Song, R., and Cao, M. Taming text-to-sounding video generation via advanced modality condition and interaction. arXiv preprint arXiv:2510.03117, 2025

    2025

  61. [74]

    and Pollack, A

    Guillemin, V. and Pollack, A. Differential Topology. Prentice Hall, 1974

    1974

  62. [75]

    Crystallography and Cohomology of Groups

    Hiller, H. Crystallography and Cohomology of Groups . The American Mathematical Monthly, December 1986

    1986

  63. [76]

    Denoising diffusion probabilistic models

    Ho, J., Jain, A., and Abbeel, P. Denoising diffusion probabilistic models. Advances in neural information processing systems, 33: 0 6840--6851, 2020

    2020

  64. [77]

    J., Shen, Y., Wallis, P., Allen-Zhu, Z., Li, Y., Wang, S., Wang, L., Chen, W., et al

    Hu, E. J., Shen, Y., Wallis, P., Allen-Zhu, Z., Li, Y., Wang, S., Wang, L., Chen, W., et al. Lora: Low-rank adaptation of large language models. ICLR, 1 0 (2): 0 3, 2022

    2022

  65. [78]

    Humphreys, J. E. Reflection Groups and Coxeter Groups . Number 29 in Cambridge Studies in Advanced Mathematics. Cambridge Univ. Press, 1 ed. edition, 1992

    1992

  66. [79]

    Crystal structure prediction by joint equivariant diffusion

    Jiao, R., Huang, W., Lin, P., Han, J., Chen, P., Lu, Y., and Liu, Y. Crystal structure prediction by joint equivariant diffusion. Advances in Neural Information Processing Systems, 36: 0 17464--17497, 2023

    2023

  67. [80]

    Space group constrained crystal generation

    Jiao, R., Huang, W., Liu, Y., Zhao, D., and Liu, Y. Space group constrained crystal generation. In The Twelfth International Conference on Learning Representations, 2024

    2024

  68. [81]

    The ring of invariant real functions on the Brillouin zone

    Kim, J. The ring of invariant real functions on the Brillouin zone. Physics Reports, 341 0 (1-6): 0 337--376, 2001

    2001

  69. [82]

    S., Kaba, S.-O., Zhu, Q., Lee, K

    Levy, D., Panigrahi, S. S., Kaba, S.-O., Zhu, Q., Lee, K. L. K., Galkin, M., Miret, S., and Ravanbakhsh, S. Symmcd: symmetry-preserving crystal generation with diffusion models. arXiv preprint arXiv:2502.03638, 2025

    2025

  70. [83]

    Structural topology optimization considering connectivity constraint

    Li, Q., Chen, W., Liu, S., and Tong, L. Structural topology optimization considering connectivity constraint. Structural and Multidisciplinary Optimization, 54 0 (4): 0 971--984, 2016

    2016

  71. [84]

    One path to model them all: Learnable-time flow matching for crystal structure and energy prediction

    Li, S., Li, M., Liu, Q., Cen, J., Wu, L., Xu, T., Rong, Y., Zhao, D., and Huang, W. One path to model them all: Learnable-time flow matching for crystal structure and energy prediction. In 32nd SIGKDD Conference on Knowledge Discovery and Data Mining - AI for Sciences Track, 2026

    2026

  72. [85]

    T., Ben-Hamu, H., Nickel, M., and Le, M

    Lipman, Y., Chen, R. T., Ben-Hamu, H., Nickel, M., and Le, M. Flow matching for generative modeling. In The Eleventh International Conference on Learning Representations, 2023

    2023

  73. [86]

    Multiplicative Invariant Theory

    Lorenz, M. Multiplicative Invariant Theory. Number 135 in Encyclopaedia of Mathematical Sciences. Springer, 2005

    2005

  74. [87]

    Dpm-solver++: Fast solver for guided sampling of diffusion probabilistic models

    Lu, C., Zhou, Y., Bao, F., Chen, J., Li, C., and Zhu, J. Dpm-solver++: Fast solver for guided sampling of diffusion probabilistic models. Machine Intelligence Research, 22 0 (4): 0 730--751, 2025

    2025

  75. [88]

    Designing complex architectured materials with generative adversarial networks

    Mao, Y., He, Q., and Zhao, X. Designing complex architectured materials with generative adversarial networks. Science advances, 6 0 (17): 0 eaaz4169, 2020

    2020

  76. [89]

    Structure and appearance optimization for controllable shape design

    Mart \' nez, J., Dumas, J., Lefebvre, S., and Wei, L.-Y. Structure and appearance optimization for controllable shape design. ACM Transactions on Graphics (TOG), 34 0 (6): 0 1--11, 2015

    2015

  77. [90]

    P., Tancik, M., Barron, J

    Mildenhall, B., Srinivasan, P. P., Tancik, M., Barron, J. T., Ramamoorthi, R., and Ng, R. Nerf: Representing scenes as neural radiance fields for view synthesis. Communications of the ACM, 65 0 (1): 0 99--106, 2021

    2021

  78. [91]

    Instant neural graphics primitives with a multiresolution hash encoding

    M \"u ller, T., Evans, A., Schied, C., and Keller, A. Instant neural graphics primitives with a multiresolution hash encoding. ACM Transactions on Graphics (TOG), 2022

    2022

  79. [92]

    The Red Book of Varieties and Schemes

    Mumford, D. The Red Book of Varieties and Schemes. Number 1358 in Lecture Notes in Mathematics. Springer, 2., expanded ed edition, 2004

    2004

  80. [93]

    Video Generation Models as World Simulators , February 2024

    OpenAI . Video Generation Models as World Simulators , February 2024. URL https://openai.com/index/video-generation-models-as-world-simulators/

    2024

  81. [94]

    Introducing GPT-5.2 , 2025

    OpenAI . Introducing GPT-5.2 , 2025. URL https://openai.com/index/introducing-gpt-5-2/

    2025

  82. [95]

    Some applications of representation theory

    Plesken, W. Some applications of representation theory. In Michler, G. O. and Ringel, C. M. (eds.), Representation Theory of Finite Groups and Finite-Dimensional Algebras : Proceedings of the Conference at the University of Bielefeld from May 15--17, 1991, and 7 Survey Article...

    1991

  83. [96]

    Sdxl: Improving latent diffusion models for high-resolution image synthesis

    Podell, D., English, Z., Lacey, K., Blattmann, A., Dockhorn, T., M \"u ller, J., Penna, J., and Rombach, R. Sdxl: Improving latent diffusion models for high-resolution image synthesis. arXiv preprint arXiv:2307.01952, 2023

    2023 arXiv

  84. [97]

    T., and Mildenhall, B

    Poole, B., Jain, A., Barron, J. T., and Mildenhall, B. Dreamfusion: Text-to-3d using 2d diffusion. In The Eleventh International Conference on Learning Representations, 2023

    2023

  85. [98]

    W., Hallacy, C., Ramesh, A., Goh, G., Agarwal, S., Sastry, G., Askell, A., Mishkin, P., Clark, J., et al

    Radford, A., Kim, J. W., Hallacy, C., Ramesh, A., Goh, G., Agarwal, S., Sastry, G., Askell, A., Mishkin, P., Clark, J., et al. Learning transferable visual models from natural language supervision. In International conference on machine learning, pp.\ 8748--8763. PmLR, 2021

    2021

  86. [99]

    High-resolution image synthesis with latent diffusion models

    Rombach, R., Blattmann, A., Lorenz, D., Esser, P., and Ommer, B. High-resolution image synthesis with latent diffusion models. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pp.\ 10684--10695, 2022

    2022

  87. [100]

    Laion-5b: An open large-scale dataset for training next generation image-text models

    Schuhmann, C., Beaumont, R., Vencu, R., Gordon, C., Wightman, R., Cherti, M., Coombes, T., Katta, A., Mullis, C., Wortsman, M., et al. Laion-5b: An open large-scale dataset for training next generation image-text models. Advances in neural information processing systems, 35: 0...

    2022

  88. [101]

    Shmueli, U. (ed.). International Tables for Crystallography : Reciprocal Space , volume B. International Union of Crystallography, 2 edition, 2010

    2010

  89. [102]

    Shubnikov, A. V. and Koptsik, V. A. Symmetry in science and art . Plenum Press, 1974

    1974

  90. [103]

    Stanley, R. P. Invariants of finite groups and their applications to combinatorics. Bulletin (New Series) of the American Mathematical Society, 1 0 (3): 0 475--511, 1979

    1979

  91. [104]

    Algorithms in invariant theory

    Sturmfels, B. Algorithms in invariant theory. Springer, 2008

    2008

  92. [105]

    On the Finite Subgroups of GL (3, Z )

    Tahara, K.-I. On the Finite Subgroups of GL (3, Z ). Nagoya Mathematical Journal, 41: 0 169--209, 1971

    1971

  93. [106]

    Tripoptimizer: Generative three-dimensional shape optimization and drag prediction using triplane variational autoencoder networks

    Vatani, P., Elrefaie, M., Nazarpour, F., and Ahmed, F. Tripoptimizer: Generative three-dimensional shape optimization and drag prediction using triplane variational autoencoder networks. Physics of Fluids, 37 0 (12), 2025

    2025

  94. [107]

    Diffusers: State-of-the-art diffusion models

    von Platen, P., Patil, S., Lozhkov, A., Cuenca, P., Lambert, N., Rasul, K., Davaadorj, M., Nair, D., Paul, S., Berman, W., Xu, Y., Liu, S., and Wolf, T. Diffusers: State-of-the-art diffusion models. https://github.com/huggingface/diffusers, 2022

    2022

  95. [108]

    Harmonycut: Supporting creative chinese paper-cutting design with form and connotation harmony

    Wang, H., Qiu, T., Li, J., Lu, Z., and Ma, Y. Harmonycut: Supporting creative chinese paper-cutting design with form and connotation harmony. In Proceedings of the 2025 CHI Conference on Human Factors in Computing Systems, pp.\ 1--22, 2025

    2025

  96. [109]

    and M \"u ller, U

    Wondratschek, H. and M \"u ller, U. (eds.). International Tables for Crystallography : Symmetry Relations between Space Groups , volume A1. International Union of Crystallography, 2 edition, 2011

    2011

  97. [110]

    Siamese foundation models for crystal structure prediction

    Wu, L., Huang, W., Jiao, R., Huang, J., Liu, L., Zhou, Y., Sun, H., Liu, Y., Sun, F., Ren, Y., and Wen, J.-R. Siamese foundation models for crystal structure prediction. Nature Communications, 2026 a . ISSN 2041-1723. doi:10.1038/s41467-026-72362-3

    2026 doi

  98. [111]

    Dmflow: Disordered materials generation by flow matching

    Wu, L., Jiao, R., Li, Q., Li, M., Li, S., Jin, S., and Huang, W. Dmflow: Disordered materials generation by flow matching. In 32nd SIGKDD Conference on Knowledge Discovery and Data Mining - AI for Sciences Track, 2026 b

    2026

  99. [112]

    and Breitkopf, P

    Xia, L. and Breitkopf, P. Design of materials using topology optimization and energy-based homogenization approach in matlab. Structural and multidisciplinary optimization, 52 0 (6): 0 1229--1241, 2015

    2015

  100. [113]

    Tfg: Unified training-free guidance for diffusion models

    Ye, H., Lin, H., Han, J., Xu, M., Liu, S., Liang, Y., Ma, J., Zou, J., and Ermon, S. Tfg: Unified training-free guidance for diffusion models. Advances in Neural Information Processing Systems, 37: 0 22370--22417, 2024

    2024

  101. [114]

    Adding conditional control to text-to-image diffusion models

    Zhang, L., Rao, A., and Agrawala, M. Adding conditional control to text-to-image diffusion models. In Proceedings of the IEEE/CVF international conference on computer vision, pp.\ 3836--3847, 2023

    2023

  102. [115]

    Topology optimization with text-guided stylization

    Zhong, S., Punpongsanon, P., Iwai, D., and Sato, K. Topology optimization with text-guided stylization. Structural and Multidisciplinary Optimization, 66 0 (12): 0 256, 2023

    2023

Pith tools

Reviewed June 28, 2026 · model on record in the stance chip above.