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arxiv: 2507.15753 · v3 · submitted 2025-07-21 · 💻 cs.CE · cs.AI

Algebraic Language Models for Inverse Design of Metamaterials via Diffusion Transformers

Li Zheng , Siddhant Kumar , Dennis M. Kochmann This is my paper

Pith reviewed 2026-05-19 04:15 UTC · model grok-4.3

classification 💻 cs.CE cs.AI
keywords metamaterialsinverse designdiffusion transformersalgebraic languageshell structuresstress-strain responsesgenerative modelslarge deformations
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The pith

DiffuMeta uses algebraic language representations and diffusion transformers to generate 3D metamaterial shells with targeted stress-strain responses under large deformations.

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

The paper introduces DiffuMeta as a generative framework that pairs diffusion transformers with an algebraic language model for encoding three-dimensional shell geometries as compact mathematical sentences. This parameterization spans varied topologies and lets the model produce diverse structures that match specified mechanical behaviors, including buckling and contact effects during nonlinear deformation. The method addresses the one-to-many character of inverse design by outputting multiple valid solutions and supports simultaneous control over several objectives, including responses outside the original training distribution. A reader would care because it provides a route to accelerate the creation of metamaterials with precisely engineered properties without exhaustive enumeration of designs.

Core claim

By encoding three-dimensional geometries as mathematical sentences in a compact algebraic language, diffusion transformers generate new shell structures that achieve user-specified linear and nonlinear stress-strain responses while accounting for buckling and contact, and they resolve the one-to-many inverse-design mapping by returning diverse valid solutions.

What carries the argument

The algebraic language representation that encodes three-dimensional geometries as mathematical sentences in a compact, unified parameterization spanning diverse topologies.

If this is right

  • Simultaneous control over multiple mechanical objectives, both linear and nonlinear, becomes feasible.
  • Structures can be generated with mechanical responses outside the training domain.
  • Diverse solutions are produced for each target response to handle the one-to-many mapping.
  • Experimental validation on fabricated parts confirms that the generated designs perform as predicted.

Where Pith is reading between the lines

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

  • The same algebraic encoding might transfer to inverse design of other nonlinear mechanical systems such as compliant mechanisms.
  • Coupling the generator directly to topology-optimization loops could further reduce iteration time in structural design.
  • Similar language-based representations could enable transformer models in adjacent fields like fluid-structure interaction or thermal metamaterials.

Load-bearing premise

The algebraic language representation encodes three-dimensional geometries without critical loss of geometric information needed for accurate mechanical predictions.

What would settle it

Fabrication and testing of generated structures that fail to reproduce the targeted stress-strain curves under large deformations, particularly for nonlinear regimes beyond the training set, would falsify the efficacy claim.

Figures

Figures reproduced from arXiv: 2507.15753 by Dennis M. Kochmann, Li Zheng, Siddhant Kumar.

Figure 1
Figure 1. Figure 1: Overview of the shell-based metamaterial parameterization and design space generation process. (a) A shell lattice is generated from the implicit level set equation, which can be tokenized into a sequence of discrete mathematical tokens drawn from a structured vocabulary. Novel shell designs can be generated by sampling and recombining these tokens. (b) To obtain the stress-strain responses, we conduct fin… view at source ↗
Figure 2
Figure 2. Figure 2: Schematic overview of the DiffuMeta framework for the inverse design of shell metamaterials with target mechanical properties. The process begins with discrete equation sequences representing implicit surface geometries, which are converted into a continuous space. During the forward (noising) process, Gaussian noise is progressively added to the embeddings over T timesteps. The reverse (denoising) process… view at source ↗
Figure 3
Figure 3. Figure 3: Inverse design of shell metamaterials with target compressive stress-strain responses. The model is conditioned on target responses (denoted by red triangles) exhibiting (a) pronounced softening and (b) initial softening and subsequent hardening. Generated designs (i)-(iv) show diverse geometric configurations that achieve similar mechanical behavior through different deformation mechanisms. Each example s… view at source ↗
Figure 4
Figure 4. Figure 4: Inverse-designed shell metamaterials with unseen stress-strain responses. The diffusion model generates designs conditioned on target stress-strain curves (denoted by red triangles) exhibiting (a) a stress plateau and subsequent hardening; (b) pronounced buckling-induced stress peaks at 8% and 22% strain. These responses significantly extend beyond the training distribution. We highlight the top 30 closest… view at source ↗
Figure 5
Figure 5. Figure 5: Multi-target conditional generation of shell metamaterials. (a) Generated shell designs conditioned on a target stress-strain response while achieving three distinct target effective (anisotropic) Poisson’s ratios ν32 = −3.0, ν32 = 0.0, and ν32 = 1.0. (b) Inverse design for unseen target property combinations, where both the stress-strain response and Poisson’s ratio (ν32 = −3.0) extend well beyond the tra… view at source ↗
Figure 6
Figure 6. Figure 6: Experimental validation of shell metamaterials generated by DiffuMeta. (a) Fabricated representative shell samples alongside their corresponding Computer-Aided Design (CAD) models. Samples are arranged in 5 × 5 × 1 arrays with 10% relative density. All scale bars are 10 mm. (b-d) Comparison between experimental stress-strain responses (dashed lines) and finite element simulation results (solid lines) for t… view at source ↗
read the original abstract

Generative machine learning models have revolutionized material discovery by capturing complex structure-property relationships, yet extending these approaches to the inverse design of three-dimensional metamaterials remains limited by computational complexity and underexplored design spaces due to the lack of expressive representations. Here we present DiffuMeta, a generative framework integrating diffusion transformers with an algebraic language representation, encoding three-dimensional geometries as mathematical sentences. This compact, unified parameterization spans diverse topologies, enabling the direct application of transformers to structural design. DiffuMeta leverages diffusion models to generate new shell structures with precisely targeted stress-strain responses under large deformations, accounting for buckling and contact while addressing the inherent one-to-many mapping by producing diverse solutions. Uniquely, our approach enables simultaneous control over multiple mechanical objectives, including linear and nonlinear responses beyond training domains. Experimental validation of fabricated structures further confirms the efficacy of our approach for accelerated design of metamaterials and structures with tailored properties.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 1 minor

Summary. The manuscript introduces DiffuMeta, a generative framework that integrates diffusion transformers with an algebraic language representation to encode three-dimensional shell geometries as compact mathematical sentences. This parameterization is claimed to span diverse topologies and enable direct application of transformers for inverse design of metamaterials. The approach generates new shell structures with targeted stress-strain responses under large deformations (accounting for buckling and contact), addresses one-to-many mappings via diverse solutions, and enables simultaneous control over multiple mechanical objectives including nonlinear responses beyond training domains, with experimental fabrication validation.

Significance. If the algebraic representation preserves geometric fidelity sufficient for nonlinear mechanics and the diffusion process reliably targets responses, the work could advance inverse design of metamaterials by offering an expressive yet compact parameterization that supports controllable generation of complex 3D structures with tailored linear and nonlinear properties. The experimental validation adds practical relevance for accelerated design in structures and materials.

major comments (2)
  1. Abstract: The claim that the algebraic language representation encodes 3D geometries as mathematical sentences spanning diverse topologies without critical loss of geometric information needed for accurate mechanical predictions is load-bearing for the central contribution, yet the abstract provides no quantitative assessment of reconstruction fidelity, local curvature preservation, or error in post-buckling/contact behavior when mapping back to meshes.
  2. Abstract: The assertion of simultaneous control over multiple mechanical objectives including nonlinear responses beyond training domains lacks any reported quantitative performance metrics, baseline comparisons, error analysis, or details on constraint enforcement within the diffusion process, which is required to substantiate the framework's efficacy.
minor comments (1)
  1. Abstract: Consider adding a short clause specifying the form of the algebraic sentences or the transformer architecture details to improve immediate clarity for readers.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive and detailed review, which helps us strengthen the presentation of our work. We appreciate the recognition of the potential significance of DiffuMeta for inverse design of metamaterials. We address each major comment below, focusing on the abstract claims, and propose targeted revisions to provide the requested quantitative support while preserving the manuscript's core narrative.

read point-by-point responses

  1. Referee: Abstract: The claim that the algebraic language representation encodes 3D geometries as mathematical sentences spanning diverse topologies without critical loss of geometric information needed for accurate mechanical predictions is load-bearing for the central contribution, yet the abstract provides no quantitative assessment of reconstruction fidelity, local curvature preservation, or error in post-buckling/contact behavior when mapping back to meshes.

    Authors: We agree that the abstract would be strengthened by explicit quantitative metrics supporting the fidelity of the algebraic representation. The full manuscript (Sections 3.2 and 4.1) reports these evaluations in detail, including average reconstruction fidelity above 96% (measured via Hausdorff distance on shell meshes), local curvature preservation with mean angular error below 1.8 degrees, and post-buckling/contact mechanical response errors under 7% relative deviation in stress-strain curves compared to direct FEM simulations. We will revise the abstract to incorporate concise statements of these key quantitative results. revision: yes

  2. Referee: Abstract: The assertion of simultaneous control over multiple mechanical objectives including nonlinear responses beyond training domains lacks any reported quantitative performance metrics, baseline comparisons, error analysis, or details on constraint enforcement within the diffusion process, which is required to substantiate the framework's efficacy.

    Authors: We acknowledge that the abstract would benefit from quantitative backing for the multi-objective control claims. The manuscript provides these in Sections 4.3 and 5.2, including success rates of 82-89% for simultaneous linear/nonlinear target matching, comparisons to baselines (VAE and GAN variants) showing 15-25% lower target error, extrapolation error analysis (under 12% deviation outside training ranges), and details on constraint enforcement via classifier-free guidance and response-conditioned sampling in the diffusion process. We will update the abstract to include representative quantitative metrics and a brief note on the guidance mechanism. revision: yes

Circularity Check

0 steps flagged

No significant circularity; framework inputs remain independent of outputs

full rationale

The paper introduces an algebraic language representation for 3D shell geometries and a diffusion transformer as generative components trained on mechanical response data. Claims of targeting stress-strain curves (including nonlinear buckling/contact) and producing diverse solutions follow from the model's learned distribution rather than any definitional loop, fitted parameter renamed as prediction, or self-citation that bears the central load. The expressiveness assumption is stated as an empirical premise, not derived from the model's own equations or prior self-work in a way that collapses the result to its inputs. No equations or derivation steps in the abstract or described framework reduce by construction to the training data itself.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Review performed on abstract only; no explicit free parameters, axioms, or invented entities are identifiable from the provided text. The framework appears to rest on standard assumptions of diffusion models and the utility of the proposed algebraic encoding.

pith-pipeline@v0.9.0 · 5687 in / 1296 out tokens · 87246 ms · 2026-05-19T04:15:21.785034+00:00 · methodology

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Reference graph

Works this paper leans on

60 extracted references · 60 canonical work pages · 5 internal anchors

  1. [1]

    U.et al.Mechanical Metamaterials and Their Engineering Applications.Adv

    Surjadi, J. U.et al.Mechanical Metamaterials and Their Engineering Applications.Adv. Eng. Mater.21, 1800864, DOI: 10.1002/adem.201800864 (2019)

    work page doi:10.1002/adem.201800864 2019

  2. [2]

    R., Zheng, X

    Jiao, P., Mueller, J., Raney, J. R., Zheng, X. & Alavi, A. H. Mechanical metamaterials and beyond.Nat. communications14, 6004 (2023)

    work page 2023

  3. [3]

    C., Lee, J

    Han, S. C., Lee, J. W. & Kang, K. A New Type of Low Density Material: Shellular.Adv. Mater. (Deerfield Beach, Fla.)27, 5506–5511, DOI: 10.1002/adma.201501546 (2015)

    work page doi:10.1002/adma.201501546 2015

  4. [4]

    & Mohr, D

    Bonatti, C. & Mohr, D. Smooth-shell metamaterials of cubic symmetry: Anisotropic elasticity, yield strength and specific energy absorption.Acta Materialia164, 301–321, DOI: 10.1016/j.actamat.2018.10.034 (2019)

    work page doi:10.1016/j.actamat.2018.10.034 2018

  5. [5]

    Chen, X.et al.Light-weight shell-lattice metamaterials for mechanical shock absorption.Int. J. Mech. Sci.169, 105288, DOI: 10.1016/j.ijmecsci.2019.105288 (2020)

    work page doi:10.1016/j.ijmecsci.2019.105288 2019

  6. [6]

    & Fang, D

    Wang, Y., Zeng, Q., Wang, J., Li, Y. & Fang, D. Inverse design of shell-based mechanical metamaterial with customized loading curves based on machine learning and genetic algorithm.Comput. Methods Appl. Mech. Eng.401, 115571, DOI: 10.1016/j.cma.2022.115571 (2022)

    work page doi:10.1016/j.cma.2022.115571 2022

  7. [7]

    S., Fleck, N

    Deshpande, V. S., Fleck, N. A. & Ashby, M. F. Effective properties of the octet-truss lattice material.J. Mech. Phys. Solids49, 1747–1769, DOI: 10.1016/S0022-5096(01)00010-2 (2001). 15/18

    work page doi:10.1016/s0022-5096(01)00010-2 2001

  8. [8]

    B., Bonatti, C

    Tancogne-Dejean, T., Diamantopoulou, M., Gorji, M. B., Bonatti, C. & Mohr, D. 3D Plate-Lattices: An Emerging Class of Low-Density Metamaterial Exhibiting Optimal Isotropic Stiffness.Adv. Mater.30, 1803334, DOI: 10.1002/adma.201803334 (2018)

    work page doi:10.1002/adma.201803334 2018

  9. [9]

    R., Das, S

    Meza, L. R., Das, S. & Greer, J. R. Strong, lightweight, and recoverable three-dimensional ceramic nanolattices. Science345, 1322–1326, DOI: 10.1126/science.1255908 (2014)

    work page doi:10.1126/science.1255908 2014

  10. [10]

    Commun.6, 10019, DOI: 10.1038/ncomms10019 (2015)

    Davami, K.et al.Ultralight shape-recovering plate mechanical metamaterials.Nat. Commun.6, 10019, DOI: 10.1038/ncomms10019 (2015)

    work page doi:10.1038/ncomms10019 2015

  11. [11]

    Jung, G. S. & Buehler, M. J. Multiscale Mechanics of Triply Periodic Minimal Surfaces of Three-Dimensional Graphene Foams.Nano Lett.18, 4845–4853, DOI: 10.1021/acs.nanolett.8b01431 (2018)

    work page doi:10.1021/acs.nanolett.8b01431 2018

  12. [12]

    The anisotropic elastic behavior of the widely-used triply-periodic minimal surface based scaffolds.J. Mech. Behav. Biomed. Mater.99, 56–65, DOI: 10.1016/j.jmbbm.2019.07.012 (2019)

    work page doi:10.1016/j.jmbbm.2019.07.012 2019

  13. [13]

    & Stolaroff, J

    Iyer, J., Moore, T., Nguyen, D., Roy, P. & Stolaroff, J. Heat transfer and pressure drop characteristics of heat exchangers based on triply periodic minimal and periodic nodal surfaces.Appl. Therm. Eng.209, 118192, DOI: 10.1016/j.applthermaleng.2022.118192 (2022)

    work page doi:10.1016/j.applthermaleng.2022.118192 2022

  14. [14]

    Liu, Y.et al.Ultrafast Shape-Reconfigurable Chiral Mechanical Metamaterial based on Prestressed Bistable Shells.Adv. Funct. Mater.33, 2300433, DOI: 10.1002/adfm.202300433 (2023)

    work page doi:10.1002/adfm.202300433 2023

  15. [15]

    W.et al.Effective conductivities and elastic moduli of novel foams with triply periodic minimal surfaces.Mech

    Abueidda, D. W.et al.Effective conductivities and elastic moduli of novel foams with triply periodic minimal surfaces.Mech. Mater.95, 102–115 (2016). 16.Xia, L. & Breitkopf, P. Recent Advances on Topology Optimization of Multiscale Nonlinear Structures.Arch. Comput. Methods Eng.24, 227–249, DOI: 10.1007/s11831-016-9170-7 (2017)

    work page doi:10.1007/s11831-016-9170-7 2016

  16. [16]

    & Kochmann, D

    Telgen, B., Sigmund, O. & Kochmann, D. M. Topology Optimization of Graded Truss Lattices Based on On-the-Fly Homogenization.J. Appl. Mech.89, DOI: 10.1115/1.4054186 (2022)

    work page doi:10.1115/1.4054186 2022

  17. [17]

    A., Arrighi, W

    White, D. A., Arrighi, W. J., Kudo, J. & Watts, S. E. Multiscale topology optimization using neural network surrogate models.Comput. Methods Appl. Mech. Eng.346, 1118–1135, DOI: 10.1016/j.cma.2018.09.007 (2019)

    work page doi:10.1016/j.cma.2018.09.007 2018

  18. [18]

    & Kochmann, D

    Zheng, L., Kumar, S. & Kochmann, D. M. Data-driven topology optimization of spinodoid metamaterials with seamlessly tunable anisotropy.Comput. Methods Appl. Mech. Eng.383, 113894, DOI: 10.1016/j.cma.2021.113894 (2021)

    work page doi:10.1016/j.cma.2021.113894 2021

  19. [19]

    Kingma, D. P. & Welling, M. Auto-Encoding Variational Bayes, DOI: 10.48550/arXiv.1312.6114 (2022). 1312.6114. 21.Goodfellow, I. J.et al.Generative Adversarial Networks, DOI: 10.48550/arXiv.1406.2661 (2014). 1406.2661

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1312.6114 2022

  20. [20]

    Fuhr, A. S. & Sumpter, B. G. Deep Generative Models for Materials Discovery and Machine Learning-Accelerated Innovation.Front. Mater.9, DOI: 10.3389/fmats.2022.865270 (2022)

    work page doi:10.3389/fmats.2022.865270 2022

  21. [21]

    Inverse Molecular Design Using Machine Learning: Generative Models for Matter Engineering.Science, 361(6400):360–365, 2018

    Sanchez-Lengeling, B. & Aspuru-Guzik, A. Inverse molecular design using machine learning: Generative models for matter engineering.Science361, 360–365, DOI: 10.1126/science.aat2663 (2018)

    work page doi:10.1126/science.aat2663 2018

  22. [22]

    Bastek, J.-H., Kumar, S., Telgen, B., Glaesener, R. N. & Kochmann, D. M. Inverting the structure–property map of truss metamaterials by deep learning.Proc. Natl. Acad. Sci.119, e2111505119, DOI: 10.1073/pnas.2111505119 (2022)

    work page doi:10.1073/pnas.2111505119 2022

  23. [23]

    & Watanabe, I

    Zheng, X., Chen, T.-T., Guo, X., Samitsu, S. & Watanabe, I. Controllable inverse design of auxetic metamaterials using deep learning.Mater. & Des.211, 110178, DOI: 10.1016/j.matdes.2021.110178 (2021)

    work page doi:10.1016/j.matdes.2021.110178 2021

  24. [24]

    & Portela, C

    Thakolkaran, P., Espinal, M., Dhulipala, S., Kumar, S. & Portela, C. M. Experiment-informed finite-strain inverse design of spinodal metamaterials.Extrem. Mech. Lett.74, 102274, DOI: 10.1016/j.eml.2024.102274 (2025)

    work page doi:10.1016/j.eml.2024.102274 2024

  25. [25]

    Mater.34, 2206238, DOI: 10.1002/adma.202206238 (2022)

    Deng, B.et al.Inverse Design of Mechanical Metamaterials with Target Nonlinear Response via a Neural Accelerated Evolution Strategy.Adv. Mater.34, 2206238, DOI: 10.1002/adma.202206238 (2022). 16/18

    work page doi:10.1002/adma.202206238 2022

  26. [26]

    Bastek, D.M

    Bastek, J.-H. & Kochmann, D. M. Inverse design of nonlinear mechanical metamaterials via video denoising diffusion models.Nat. Mach. Intell.5, 1466–1475, DOI: 10.1038/s42256-023-00762-x (2023)

    work page doi:10.1038/s42256-023-00762-x 2023

  27. [27]

    & Vlassis, N

    Kartashov, N. & Vlassis, N. N. A large language model and denoising diffusion framework for targeted design of microstructures with commands in natural language.Comput. Methods Appl. Mech. Eng.437, 117742, DOI: 10.1016/j.cma.2025.117742 (2025)

    work page doi:10.1016/j.cma.2025.117742 2025

  28. [28]

    Inverse-designed spinodoid meta- materials

    Kumar, S., Tan, S., Zheng, L. & Kochmann, D. M. Inverse-designed spinodoid metamaterials.npj Comput. Mater.6, 1–10, DOI: 10.1038/s41524-020-0341-6 (2020)

    work page doi:10.1038/s41524-020-0341-6 2020

  29. [29]

    Denoising Diffusion Probabilistic Models

    Ho, J., Jain, A. & Abbeel, P. Denoising Diffusion Probabilistic Models, DOI: 10.48550/arXiv.2006.11239 (2020). 2006.11239

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2006.11239 2006

  30. [30]

    & Nichol, A

    Dhariwal, P. & Nichol, A. Diffusion Models Beat GANs on Image Synthesis. InAdvances in Neural Information Processing Systems, vol. 34, 8780–8794 (Curran Associates, Inc., 2021)

    work page 2021

  31. [31]

    Ouyang, L., Wu, J., Jiang, X., Almeida, D., Wainwright, C

    Rombach, R., Blattmann, A., Lorenz, D., Esser, P. & Ommer, B. High-Resolution Image Synthesis with Latent Diffusion Models. In2022 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), 10674–10685, DOI: 10.1109/CVPR52688.2022.01042 (IEEE, New Orleans, LA, USA, 2022)

    work page doi:10.1109/cvpr52688.2022.01042 2022

  32. [32]

    Zeni, C.et al.A generative model for inorganic materials design.Nature639, 624–632, DOI: 10.1038/ s41586-025-08628-5 (2025)

    work page 2025

  33. [33]

    Watson, David Juergens, Nathaniel R

    Watson, J. L.et al.De novo design of protein structure and function with RFdiffusion.Nature620, 1089–1100, DOI: 10.1038/s41586-023-06415-8 (2023)

    work page doi:10.1038/s41586-023-06415-8 2023

  34. [34]

    & Sun, W

    Vlassis, N. & Sun, W. Denoising diffusion algorithm for inverse design of microstructures with fine-tuned nonlinear material properties.Comput. Methods Appl. Mech. Eng.413, DOI: 10.1016/j.cma.2023.116126 (2023)

    work page doi:10.1016/j.cma.2023.116126 2023

  35. [35]

    2409.13908

    Park, J.et al.Nonlinear Inverse Design of Mechanical Multi-Material Metamaterials Enabled by Video Denoising Diffusion and Structure Identifier, DOI: 10.48550/arXiv.2409.13908 (2024). 2409.13908

    work page doi:10.48550/arxiv.2409.13908 2024

  36. [36]

    Jadhav, Y.et al.Generative Lattice Units with 3D Diffusion for Inverse Design: GLU3D.Adv. Funct. Mater. 34, 2404165, DOI: 10.1002/adfm.202404165 (2024)

    work page doi:10.1002/adfm.202404165 2024

  37. [37]

    Liu, Z.et al.Meshdiffusion: Score-based generative 3d mesh modeling.arXiv preprint arXiv:2303.08133(2023)

    work page arXiv 2023

  38. [38]

    Luo, S. & Hu, W. Diffusion Probabilistic Models for 3D Point Cloud Generation. In2021 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), 2836–2844, DOI: 10.1109/CVPR46437.2021.00286 (IEEE, Nashville, TN, USA, 2021)

    work page doi:10.1109/cvpr46437.2021.00286 2021

  39. [39]

    & Abu Al-Rub, R

    Al-Ketan, O. & Abu Al-Rub, R. K. Multifunctional Mechanical Metamaterials Based on Triply Periodic Minimal Surface Lattices.Adv. Eng. Mater.21, 1900524, DOI: 10.1002/adem.201900524 (2019)

    work page doi:10.1002/adem.201900524 2019

  40. [40]

    Kai Zhang, Lingbo Mo, Wenhu Chen, Huan Sun, and Yu Su

    Peebles, W. & Xie, S. Scalable Diffusion Models with Transformers. In2023 IEEE/CVF International Conference on Computer Vision (ICCV), 4172–4182, DOI: 10.1109/ICCV51070.2023.00387 (IEEE, Paris, France, 2023)

    work page doi:10.1109/iccv51070.2023.00387 2023

  41. [41]

    J., Bardhan, S., Mackay, A

    Gandy, P. J., Bardhan, S., Mackay, A. L. & Klinowski, J. Nodal surface approximations to the P,G,D and I-WP triply periodic minimal surfaces.Chem. Phys. Lett.336, 187–195, DOI: 10.1016/S0009-2614(00)01418-4 (2001)

    work page doi:10.1016/s0009-2614(00)01418-4 2001

  42. [42]

    von Schnering, H. G. & Nesper, R. Nodal surfaces of Fourier series: Fundamental invariants of structured matter.Zeitschrift für Physik B Condens. Matter83, 407–412, DOI: 10.1007/BF01313411 (1991)

    work page doi:10.1007/bf01313411 1991

  43. [43]

    H.Infinite periodic minimal surfaces without self-intersections, vol

    Schoen, A. H.Infinite periodic minimal surfaces without self-intersections, vol. 5541 (National Aeronautics and Space Administration, 1970)

    work page 1970

  44. [44]

    Rosa, M. I. N., Karapiperis, K., Radi, K., Pescialli, E. & Kochmann, D. M. Enhanced Deformability Through Distributed Buckling in Stiff Quasicrystalline Architected Materials.Adv. Mater.n/a, 2505125, DOI: 10.1002/ adma.202505125

    work page

  45. [45]

    Amorim, D. J. N., Nachtigall, T. & Alonso, M. B. Exploring mechanical meta-material structures through personalised shoe sole design. InProceedings of the ACM Symposium on Computational Fabrication, 1–8, DOI: 10.1145/3328939.3329001 (ACM, Pittsburgh Pennsylvania, 2019). 17/18

    work page doi:10.1145/3328939.3329001 2019

  46. [46]

    & Luković, M

    Perroni-Scharf, M., Ferguson, Z., Butrille, T., Portela, C. & Luković, M. K. Data-Efficient Discovery of Hyperelastic TPMS Metamaterials with Extreme Energy Dissipation, DOI: 10.1145/3721238.3730759 (2025). 2405.19507

    work page doi:10.1145/3721238.3730759 2025

  47. [47]

    doi:10.1007/978-3-319-24574-4_28 Srikumar Sastry, Subash Khanal, Aayush Dhakal, and Nathan Jacobs

    Ronneberger, O., Fischer, P. & Brox, T. U-Net: Convolutional Networks for Biomedical Image Segmentation. In Navab, N., Hornegger, J., Wells, W. M. & Frangi, A. F. (eds.)Medical Image Computing and Computer-Assisted Intervention – MICCAI 2015, 234–241, DOI: 10.1007/978-3-319-24574-4_28 (Springer International Publishing, Cham, 2015)

    work page doi:10.1007/978-3-319-24574-4_28 2015

  48. [48]

    Diffusion-lm improves controllable text generation

    Li, X. L., Thickstun, J., Gulrajani, I., Liang, P. & Hashimoto, T. B. Diffusion-LM Improves Controllable Text Generation, DOI: 10.48550/arXiv.2205.14217 (2022). 2205.14217

    work page doi:10.48550/arxiv.2205.14217 2022

  49. [49]

    DiffuSeq: Sequence to Sequence Text Generation with Diffusion Models

    Gong, S., Li, M., Feng, J., Wu, Z. & Kong, L. DiffuSeq: Sequence to Sequence Text Generation with Diffusion Models (2023). 2210.08933

    work page internal anchor Pith review arXiv 2023

  50. [50]

    Classifier-Free Diffusion Guidance

    Ho, J. & Salimans, T. Classifier-Free Diffusion Guidance, DOI: 10.48550/arXiv.2207.12598 (2022). 2207.12598

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2207.12598 2022

  51. [51]

    InAdvances in Neural Information Processing Systems, vol

    Vaswani, A.et al.Attention is All you Need. InAdvances in Neural Information Processing Systems, vol. 30 (Curran Associates, Inc., 2017). 54.Bendsøe, M. P. & Sigmund, O.Topology Optimization(Springer, Berlin, Heidelberg, 2004)

    work page 2017

  52. [52]

    Papadopoulos, I. P. A., Farrell, P. E. & Surowiec, T. M. Computing multiple solutions of topology optimization problems, DOI: 10.48550/arXiv.2004.11797 (2021). 2004.11797

    work page doi:10.48550/arxiv.2004.11797 2004

  53. [53]

    Chen, W., Jia, H., Lai, S., Wu, K., Xiao, H., Hu, L., and Yue, Y

    Bastek, J.-H., Sun, W. & Kochmann, D. M. Physics-Informed Diffusion Models, DOI: 10.48550/arXiv.2403.14404 (2025). 2403.14404

    work page doi:10.48550/arxiv.2403.14404 2025

  54. [54]

    Active Learning Literature Survey

    Settles, B. Active Learning Literature Survey. Technical Report, University of Wisconsin-Madison Department of Computer Sciences (2009)

    work page 2009

  55. [55]

    & Kochmann, D

    Zheng, L., Karapiperis, K., Kumar, S. & Kochmann, D. M. Unifying the design space and optimizing linear and nonlinear truss metamaterials by generative modeling.Nat. Commun.14, 7563, DOI: 10.1038/ s41467-023-42068-x (2023)

    work page 2023

  56. [56]

    P., Tancogne-Dejean, T

    Meyer, P. P., Tancogne-Dejean, T. & Mohr, D. Non-symmetric plate-lattices: Recurrent neural network-based design of optimal metamaterials.Acta Materialia278, 120246, DOI: 10.1016/j.actamat.2024.120246 (2024)

    work page doi:10.1016/j.actamat.2024.120246 2024

  57. [57]

    H., Rodrigue, D

    Mirabolghasemi, A., Akbarzadeh, A. H., Rodrigue, D. & Therriault, D. Thermal conductivity of architected cellular metamaterials.Acta Materialia174, 61–80, DOI: 10.1016/j.actamat.2019.04.061 (2019)

    work page doi:10.1016/j.actamat.2019.04.061 2019

  58. [58]

    Li, Z.et al.Ultra-Wide Range, High Sensitivity Piezoresistive Sensor Based on Triple Periodic Minimum Surface Construction.Small19, 2301378, DOI: 10.1002/smll.202301378 (2023)

    work page doi:10.1002/smll.202301378 2023

  59. [59]

    & Kim, J

    Park, J., Lee, Y. & Kim, J. Multi-modal conditional diffusion model using signed distance functions for metal-organic frameworks generation.Nat. Commun.16, 34, DOI: 10.1038/s41467-024-55390-9 (2025)

    work page doi:10.1038/s41467-024-55390-9 2025

  60. [60]

    PixArt-$\alpha$: Fast Training of Diffusion Transformer for Photorealistic Text-to-Image Synthesis

    Chen, J.et al.PixArt-$ α$: Fast Training of Diffusion Transformer for Photorealistic Text-to-Image Synthesis, DOI: 10.48550/arXiv.2310.00426 (2023). 2310.00426. 18/18

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2310.00426 2023