Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
166 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
42 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Energy-conserving equivariant GNN for elasticity of lattice architected metamaterials (2401.16914v2)

Published 30 Jan 2024 in cs.LG and cond-mat.mtrl-sci

Abstract: Lattices are architected metamaterials whose properties strongly depend on their geometrical design. The analogy between lattices and graphs enables the use of graph neural networks (GNNs) as a faster surrogate model compared to traditional methods such as finite element modelling. In this work, we generate a big dataset of structure-property relationships for strut-based lattices. The dataset is made available to the community which can fuel the development of methods anchored in physical principles for the fitting of fourth-order tensors. In addition, we present a higher-order GNN model trained on this dataset. The key features of the model are (i) SE(3) equivariance, and (ii) consistency with the thermodynamic law of conservation of energy. We compare the model to non-equivariant models based on a number of error metrics and demonstrate its benefits in terms of predictive performance and reduced training requirements. Finally, we demonstrate an example application of the model to an architected material design task. The methods which we developed are applicable to fourth-order tensors beyond elasticity such as piezo-optical tensor etc.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (34)
  1. Cormorant: Covariant molecular neural networks. In H. Wallach, H. Larochelle, A. Beygelzimer, F. AlcheBuc, E. Fox, and R. Garnett (eds.), Advances in Neural Information Processing Systems, volume 32. Curran Associates, Inc., 2019. URL https://proceedings.neurips.cc/paper/2019/file/03573b32b2746e6e8ca98b9123f2249b-Paper.pdf.
  2. Spectral decomposition of a 4th-order covariance tensor: Applications to diffusion tensor MRI. Signal Processing, 87(2):220–236, 2007. ISSN 0165-1684. doi: https://doi.org/10.1016/j.sigpro.2006.02.050. URL https://www.sciencedirect.com/science/article/pii/S0165168406001678.
  3. Inverting the structure–property map of truss metamaterials by deep learning. Proceedings of the National Academy of Sciences, 119(1):e2111505119, January 2022. ISSN 0027-8424, 1091-6490. doi: 10.1073/pnas.2111505119. URL https://pnas.org/doi/full/10.1073/pnas.2111505119.
  4. Mace: Higher order equivariant message passing neural networks for fast and accurate force fields. In S. Koyejo, S. Mohamed, A. Agarwal, D. Belgrave, K. Cho, and A. Oh (eds.), Advances in Neural Information Processing Systems, volume 35, pp.  11423–11436. Curran Associates, Inc., 2022. URL https://proceedings.neurips.cc/paper_files/paper/2022/file/4a36c3c51af11ed9f34615b81edb5bbc-Paper-Conference.pdf.
  5. E(3)-equivariant graph neural networks for data-efficient and accurate interatomic potentials. Nature Communications, 13(1):2453, May 2022. ISSN 2041-1723. doi: 10.1038/s41467-022-29939-5. URL https://doi.org/10.1038/s41467-022-29939-5.
  6. Geometric and physical quantities improve e(3) equivariant message passing. In International Conference on Learning Representations, 2022. URL https://openreview.net/forum?id=_xwr8gOBeV1.
  7. Micro-architectured materials: past, present and future. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 466(2121):2495–2516, June 2010. doi: 10.1098/rspa.2010.0215. URL https://doi.org/10.1098/rspa.2010.0215.
  8. e3nn/e3nn: 2022-12-12, December 2022. URL https://doi.org/10.5281/zenodo.7430260.
  9. Neural message passing for quantum chemistry. In Proceedings of the 34th International Conference on Machine Learning - Volume 70, ICML’17, pp.  1263–1272. JMLR.org, 2017.
  10. Peter Helnwein. Some remarks on the compressed matrix representation of symmetric second-order and fourth-order tensors. Computer Methods in Applied Mechanics and Engineering, 190(22):2753–2770, 2001. ISSN 0045-7825. doi: https://doi.org/10.1016/S0045-7825(00)00263-2. URL https://www.sciencedirect.com/science/article/pii/S0045782500002632.
  11. Predicting deformation mechanisms in architected metamaterials using gnn, 2022. URL https://doi.org/10.48550/arXiv.2202.09427.
  12. Neural network layers for prediction of positive definite elastic stiffness tensors, 2022. URL https://doi.org/10.48550/arXiv.2203.13938.
  13. On the expressive power of geometric graph neural networks, 2023.
  14. Prediction and control of fracture paths in disordered architected materials using graph neural networks. Communications Engineering, 2(1):32, June 2023. ISSN 2731-3395. doi: 10.1038/s44172-023-00085-0. URL https://doi.org/10.1038/s44172-023-00085-0.
  15. Physics-informed machine learning. Nature Reviews Physics, 3(6):422–440, June 2021. ISSN 2522-5820. doi: 10.1038/s42254-021-00314-5. URL https://doi.org/10.1038/s42254-021-00314-5.
  16. Clebsch–gordan nets: a fully fourier space spherical convolutional neural network. In S. Bengio, H. Wallach, H. Larochelle, K. Grauman, N. Cesa-Bianchi, and R. Garnett (eds.), Advances in Neural Information Processing Systems, volume 31. Curran Associates, Inc., 2018. URL https://proceedings.neurips.cc/paper_files/paper/2018/file/a3fc981af450752046be179185ebc8b5-Paper.pdf.
  17. Inverse-designed spinodoid metamaterials. npj Computational Materials, 6(1):73, June 2020. ISSN 2057-3960. doi: 10.1038/s41524-020-0341-6. URL https://doi.org/10.1038/s41524-020-0341-6.
  18. Equiformer: Equivariant graph attention transformer for 3d atomistic graphs. In The Eleventh International Conference on Learning Representations, 2023. URL https://openreview.net/forum?id=KwmPfARgOTD.
  19. Exploring the property space of periodic cellular structures based on crystal networks. Proceedings of the National Academy of Sciences, 118(7), February 2021. doi: 10.1073/pnas.2003504118. URL https://doi.org/10.1073/pnas.2003504118.
  20. Predicting stress, strain and deformation fields in materials and structures with graph neural networks. Scientific Reports, 12(1):21834, December 2022. ISSN 2045-2322. doi: 10.1038/s41598-022-26424-3. URL https://doi.org/10.1038/s41598-022-26424-3.
  21. Graph-based metamaterials: Deep learning of structure-property relations. Materials & Design, 223:111175, 2022. ISSN 0264-1275. doi: https://doi.org/10.1016/j.matdes.2022.111175. URL https://www.sciencedirect.com/science/article/pii/S0264127522007973.
  22. Tomáš Mánik. A natural vector/matrix notation applied in an efficient and robust return-mapping algorithm for advanced yield functions. European Journal of Mechanics - A/Solids, 90:104357, 2021. ISSN 0997-7538. doi: https://doi.org/10.1016/j.euromechsol.2021.104357. URL https://www.sciencedirect.com/science/article/pii/S0997753821001236.
  23. The reticular chemistry structure resource (RCSR) database of, and symbols for, crystal nets. Accounts of Chemical Research, 41(12):1782–1789, October 2008. doi: 10.1021/ar800124u. URL https://doi.org/10.1021/ar800124u.
  24. Three-dimensional euclidean nets from two-dimensional hyperbolic tilings: kaleidoscopic examples. Acta Crystallographica Section A Foundations of Crystallography, 65(2):81–108, January 2009. doi: 10.1107/s0108767308040592. URL https://doi.org/10.1107/s0108767308040592.
  25. Using graph neural networks to approximate mechanical response on 3d lattice structures. AAG2020, 2020. URL https://thinkshell.fr/wp-content/uploads/2019/10/AAG2020_24_Ross.pdf.
  26. E(n) equivariant graph neural networks. In Marina Meila and Tong Zhang (eds.), Proceedings of the 38th International Conference on Machine Learning, volume 139 of Proceedings of Machine Learning Research, pp.  9323–9332. PMLR, 18–24 Jul 2021. URL https://proceedings.mlr.press/v139/satorras21a.html.
  27. Tensor field networks: Rotation- and translation-equivariant neural networks for 3d point clouds, 2018.
  28. William Thomson. Xxi. elements of a mathematical theory of elasticity. Philosophical Transactions of the Royal Society of London, (146):481–498, 1856.
  29. Inverse-designed growth-based cellular metamaterials. Mechanics of Materials, 182:104668, 2023. ISSN 0167-6636. doi: https://doi.org/10.1016/j.mechmat.2023.104668. URL https://www.sciencedirect.com/science/article/pii/S016766362300114X.
  30. 3d steerable cnns: Learning rotationally equivariant features in volumetric data. In S. Bengio, H. Wallach, H. Larochelle, K. Grauman, N. Cesa-Bianchi, and R. Garnett (eds.), Advances in Neural Information Processing Systems, volume 31. Curran Associates, Inc., 2018. URL https://proceedings.neurips.cc/paper_files/paper/2018/file/488e4104520c6aab692863cc1dba45af-Paper.pdf.
  31. Crystal graph convolutional neural networks for an accurate and interpretable prediction of material properties. Phys. Rev. Lett., 120:145301, Apr 2018. doi: 10.1103/PhysRevLett.120.145301. URL https://link.aps.org/doi/10.1103/PhysRevLett.120.145301.
  32. Learning constitutive relations using symmetric positive definite neural networks. Journal of Computational Physics, 428:110072, 2021. ISSN 0021-9991. doi: https://doi.org/10.1016/j.jcp.2020.110072. URL https://www.sciencedirect.com/science/article/pii/S0021999120308469.
  33. Learning the nonlinear dynamics of mechanical metamaterials with graph networks. International Journal of Mechanical Sciences, 238:107835, 2023. ISSN 0020-7403. doi: https://doi.org/10.1016/j.ijmecsci.2022.107835. URL https://www.sciencedirect.com/science/article/pii/S0020740322007147.
  34. Unifying the design space of truss metamaterials by generative modeling, 2023.
Citations (3)
View on Semantic Scholar

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now