source_extract/main.bbl

\begin{thebibliography}{59}
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\newblock Efficient topology optimization in matlab using 88 lines of code.
\newblock \emph{Structural and Multidisciplinary Optimization}, 43\penalty0 (1):\penalty0 1--16, 2011.

\bibitem[Aroyo(2016)]{aroyoInternationalTablesCrystallography2016}
Aroyo, M.~I. (ed.).
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\bibitem[Artin(2011)]{artinAlgebra2011}
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\newblock \emph{Algebra}.
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\bibitem[Atiyah \& Macdonald(1994)Atiyah and Macdonald]{atiyahIntroductionCommutativeAlgebra1994}
Atiyah, M. and Macdonald, I.~G.
\newblock \emph{Introduction {{To Commutative Algebra}}}.
\newblock Westview Press, 1994.

\bibitem[Bendsoe \& Sigmund(2013)Bendsoe and Sigmund]{bendsoe2013topology}
Bendsoe, M.~P. and Sigmund, O.
\newblock \emph{Topology optimization: theory, methods, and applications}.
\newblock Springer Science \& Business Media, 2013.

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

\bibitem[Bourbaki(2002)]{bourbakiLieGroupsLie2002}
Bourbaki, N.
\newblock \emph{Lie {{Groups}} and {{Lie Algebras}}: {{Chapters}} 4-6}.
\newblock Elements of Mathematics. Springer, 2002.

\bibitem[Bruns \& Herzog(1998)Bruns and Herzog]{brunsCohenMacaulayRings1998}
Bruns, W. and Herzog, J.
\newblock \emph{Cohen-{{Macaulay}} Rings}.
\newblock Number~39 in Cambridge Studies in Advanced Mathematics. Cambridge Univ. Press, rev. ed. edition, 1998.

\bibitem[Chen et~al.(2023)Chen, Zhang, Han, Chen, Shi, Xu, and Xu]{chen2023vlp}
Chen, F.-L., Zhang, D.-Z., Han, M.-L., Chen, X.-Y., Shi, J., Xu, S., and Xu, B.
\newblock Vlp: A survey on vision-language pre-training.
\newblock \emph{Machine Intelligence Research}, 20\penalty0 (1):\penalty0 38--56, 2023.

\bibitem[Chen et~al.(2026)Chen, Lin, Chen, Zhang, Huang, Li, and Sun]{chen2026optimization}
Chen, H., Lin, N., Chen, L., Zhang, R., Huang, W., Li, C., and Sun, H.
\newblock Optimization and generation in aerodynamics inverse design.
\newblock \emph{arXiv preprint arXiv:2602.03582}, 2026.

\bibitem[Chen et~al.(2025)Chen, Wang, Zhou, Gao, Su, Zhu, and Li]{chen2025microdreamer}
Chen, L., Wang, Z., Zhou, Z., Gao, T., Su, H., Zhu, J., and Li, C.
\newblock Microdreamer: Efficient 3d generation in ~20 seconds by score-based iterative reconstruction.
\newblock \emph{IEEE Transactions on Pattern Analysis and Machine Intelligence}, 2025.

\bibitem[Chung et~al.(2023)Chung, Kim, McCann, Klasky, and Ye]{chung2023diffusion}
Chung, H., Kim, J., McCann, M.~T., Klasky, M.~L., and Ye, J.~C.
\newblock Diffusion posterior sampling for general noisy inverse problems.
\newblock In \emph{11th International Conference on Learning Representations, ICLR 2023}, 2023.

\bibitem[Derksen \& Kemper(2002)Derksen and Kemper]{derksenComputationalInvariantTheory2002}
Derksen, H. and Kemper, G.
\newblock \emph{Computational Invariant Theory}.
\newblock Number Volume 130 in Encyclopaedia of {{Mathematical Sciences}}. Springer, 2002.

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

\bibitem[Farkas(1986)]{farkasReflectionGroupsMultiplicative1986}
Farkas, D.~R.
\newblock Reflection {{Groups}} and {{Multiplicative Invariants}}.
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\bibitem[{Google DeepMind}(2025)]{googledeepmind2025gemini}
{Google DeepMind}.
\newblock {Gemini}, 2025.
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Guan, K., Wang, X., Lai, Z., Cheng, X., Zhang, P., Liu, X., Song, R., and Cao, M.
\newblock Taming text-to-sounding video generation via advanced modality condition and interaction.
\newblock \emph{arXiv preprint arXiv:2510.03117}, 2025.

\bibitem[Guillemin \& Pollack(1974)Guillemin and Pollack]{guilleminDifferentialTopology1974}
Guillemin, V. and Pollack, A.
\newblock \emph{Differential Topology}.
\newblock Prentice Hall, 1974.

\bibitem[Hiller(1986)]{hillerCrystallographyCohomologyGroups1986}
Hiller, H.
\newblock Crystallography and {{Cohomology}} of {{Groups}}.
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\bibitem[Ho et~al.(2020)Ho, Jain, and Abbeel]{ho2020denoising}
Ho, J., Jain, A., and Abbeel, P.
\newblock Denoising diffusion probabilistic models.
\newblock \emph{Advances in neural information processing systems}, 33:\penalty0 6840--6851, 2020.

\bibitem[Hu et~al.(2022)Hu, Shen, Wallis, Allen-Zhu, Li, Wang, Wang, Chen, et~al.]{hu2022lora}
Hu, E.~J., Shen, Y., Wallis, P., Allen-Zhu, Z., Li, Y., Wang, S., Wang, L., Chen, W., et~al.
\newblock Lora: Low-rank adaptation of large language models.
\newblock \emph{ICLR}, 1\penalty0 (2):\penalty0 3, 2022.

\bibitem[Humphreys(1992)]{humphreysReflectionGroupsCoxeter1992}
Humphreys, J.~E.
\newblock \emph{Reflection Groups and {{Coxeter}} Groups}.
\newblock Number~29 in Cambridge Studies in Advanced Mathematics. Cambridge Univ. Press, 1 ed. edition, 1992.

\bibitem[Jiao et~al.(2023)Jiao, Huang, Lin, Han, Chen, Lu, and Liu]{jiao2023crystal}
Jiao, R., Huang, W., Lin, P., Han, J., Chen, P., Lu, Y., and Liu, Y.
\newblock Crystal structure prediction by joint equivariant diffusion.
\newblock \emph{Advances in Neural Information Processing Systems}, 36:\penalty0 17464--17497, 2023.

\bibitem[Jiao et~al.(2024)Jiao, Huang, Liu, Zhao, and Liu]{jiaospace}
Jiao, R., Huang, W., Liu, Y., Zhao, D., and Liu, Y.
\newblock Space group constrained crystal generation.
\newblock In \emph{The Twelfth International Conference on Learning Representations}, 2024.

\bibitem[Kim(2001)]{kimRingInvariantReal2001}
Kim, J.
\newblock The ring of invariant real functions on the {{Brillouin}} zone.
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Levy, D., Panigrahi, S.~S., Kaba, S.-O., Zhu, Q., Lee, K. L.~K., Galkin, M., Miret, S., and Ravanbakhsh, S.
\newblock Symmcd: symmetry-preserving crystal generation with diffusion models.
\newblock \emph{arXiv preprint arXiv:2502.03638}, 2025.

\bibitem[Li et~al.(2016)Li, Chen, Liu, and Tong]{liStructuralTopologyOptimization2016}
Li, Q., Chen, W., Liu, S., and Tong, L.
\newblock Structural topology optimization considering connectivity constraint.
\newblock \emph{Structural and Multidisciplinary Optimization}, 54\penalty0 (4):\penalty0 971--984, 2016.

\bibitem[Li et~al.(2026)Li, Li, Liu, Cen, Wu, Xu, Rong, Zhao, and Huang]{li2026unipath}
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\newblock One path to model them all: Learnable-time flow matching for crystal structure and energy prediction.
\newblock In \emph{32nd SIGKDD Conference on Knowledge Discovery and Data Mining - AI for Sciences Track}, 2026.

\bibitem[Lipman et~al.(2023)Lipman, Chen, Ben-Hamu, Nickel, and Le]{lipmanflow}
Lipman, Y., Chen, R.~T., Ben-Hamu, H., Nickel, M., and Le, M.
\newblock Flow matching for generative modeling.
\newblock In \emph{The Eleventh International Conference on Learning Representations}, 2023.

\bibitem[Lorenz(2005)]{lorenzMultiplicativeInvariantTheory2005}
Lorenz, M.
\newblock \emph{Multiplicative Invariant Theory}.
\newblock Number 135 in Encyclopaedia of Mathematical Sciences. Springer, 2005.

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Lu, C., Zhou, Y., Bao, F., Chen, J., Li, C., and Zhu, J.
\newblock Dpm-solver++: Fast solver for guided sampling of diffusion probabilistic models.
\newblock \emph{Machine Intelligence Research}, 22\penalty0 (4):\penalty0 730--751, 2025.

\bibitem[Mao et~al.(2020)Mao, He, and Zhao]{mao2020designing}
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\newblock Designing complex architectured materials with generative adversarial networks.
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\newblock Structure and appearance optimization for controllable shape design.
\newblock \emph{ACM Transactions on Graphics (TOG)}, 34\penalty0 (6):\penalty0 1--11, 2015.

\bibitem[Mildenhall et~al.(2021)Mildenhall, Srinivasan, Tancik, Barron, Ramamoorthi, and Ng]{mildenhall2021nerf}
Mildenhall, B., Srinivasan, P.~P., Tancik, M., Barron, J.~T., Ramamoorthi, R., and Ng, R.
\newblock Nerf: Representing scenes as neural radiance fields for view synthesis.
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\newblock Instant neural graphics primitives with a multiresolution hash encoding.
\newblock \emph{ACM Transactions on Graphics (TOG)}, 2022.

\bibitem[Mumford(2004)]{mumfordRedBookVarieties2004}
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\newblock \emph{The Red Book of Varieties and Schemes}.
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\bibitem[{OpenAI}(2024)]{openai2024sora}
{OpenAI}.
\newblock {Video Generation Models as World Simulators}, February 2024.
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\bibitem[{OpenAI}(2025)]{openai2025gpt52}
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\bibitem[Plesken(1991)]{pleskenApplicationsRepresentationTheory1991}
Plesken, W.
\newblock Some applications of representation theory.
\newblock In Michler, G.~O. and Ringel, C.~M. (eds.), \emph{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 Articles}} on {{Topics}} of {{Representation Theory}}}, pp.\  477--496. Birkh\"auser, 1991.

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

\bibitem[Poole et~al.(2023)Poole, Jain, Barron, and Mildenhall]{poole2023dreamfusion}
Poole, B., Jain, A., Barron, J.~T., and Mildenhall, B.
\newblock Dreamfusion: Text-to-3d using 2d diffusion.
\newblock In \emph{The Eleventh International Conference on Learning Representations}, 2023.

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\newblock Learning transferable visual models from natural language supervision.
\newblock In \emph{International conference on machine learning}, pp.\  8748--8763. PmLR, 2021.

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

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\newblock Laion-5b: An open large-scale dataset for training next generation image-text models.
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\newblock International Union of Crystallography, 2 edition, 2010.

\bibitem[Shubnikov \& Koptsik(1974)Shubnikov and Koptsik]{shubnikovSymmetryScienceArt1974}
Shubnikov, A.~V. and Koptsik, V.~A.
\newblock \emph{{Symmetry in science and art}}.
\newblock Plenum Press, 1974.

\bibitem[Stanley(1979)]{stanleyInvariantsFiniteGroups1979}
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\newblock Invariants of finite groups and their applications to combinatorics.
\newblock \emph{Bulletin (New Series) of the American Mathematical Society}, 1\penalty0 (3):\penalty0 475--511, 1979.

\bibitem[Sturmfels(2008)]{sturmfels2008algorithms}
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\newblock \emph{Algorithms in invariant theory}.
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\bibitem[Tahara(1971)]{taharaFiniteSubgroupsGL1971}
Tahara, K.-I.
\newblock On the {{Finite Subgroups}} of {{GL}} (3, {{Z}}).
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\newblock Tripoptimizer: Generative three-dimensional shape optimization and drag prediction using triplane variational autoencoder networks.
\newblock \emph{Physics of Fluids}, 37\penalty0 (12), 2025.

\bibitem[von Platen et~al.(2022)von Platen, Patil, Lozhkov, Cuenca, Lambert, Rasul, Davaadorj, Nair, Paul, Berman, Xu, Liu, and Wolf]{von-platen-etal-2022-diffusers}
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\newblock Diffusers: State-of-the-art diffusion models.
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\newblock Harmonycut: Supporting creative chinese paper-cutting design with form and connotation harmony.
\newblock In \emph{Proceedings of the 2025 CHI Conference on Human Factors in Computing Systems}, pp.\  1--22, 2025.

\bibitem[Wondratschek \& M{\"u}ller(2011)Wondratschek and M{\"u}ller]{wondratschekInternationalTablesCrystallography2011}
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\newblock Siamese foundation models for crystal structure prediction.
\newblock \emph{Nature Communications}, 2026{\natexlab{a}}.
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\bibitem[Wu et~al.(2026{\natexlab{b}})Wu, Jiao, Li, Li, Li, Jin, and Huang]{wu2026dmflow}
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\newblock Dmflow: Disordered materials generation by flow matching.
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\bibitem[Xia \& Breitkopf(2015)Xia and Breitkopf]{xia2015design}
Xia, L. and Breitkopf, P.
\newblock Design of materials using topology optimization and energy-based homogenization approach in matlab.
\newblock \emph{Structural and multidisciplinary optimization}, 52\penalty0 (6):\penalty0 1229--1241, 2015.

\bibitem[Ye et~al.(2024)Ye, Lin, Han, Xu, Liu, Liang, Ma, Zou, and Ermon]{ye2024tfg}
Ye, H., Lin, H., Han, J., Xu, M., Liu, S., Liang, Y., Ma, J., Zou, J., and Ermon, S.
\newblock Tfg: Unified training-free guidance for diffusion models.
\newblock \emph{Advances in Neural Information Processing Systems}, 37:\penalty0 22370--22417, 2024.

\bibitem[Zhang et~al.(2023)Zhang, Rao, and Agrawala]{zhang2023adding}
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\newblock Adding conditional control to text-to-image diffusion models.
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\bibitem[Zhong et~al.(2023)Zhong, Punpongsanon, Iwai, and Sato]{zhongTopologyOptimizationTextguided2023}
Zhong, S., Punpongsanon, P., Iwai, D., and Sato, K.
\newblock Topology optimization with text-guided stylization.
\newblock \emph{Structural and Multidisciplinary Optimization}, 66\penalty0 (12):\penalty0 256, 2023.

\end{thebibliography}