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Designing metamaterials with programmable nonlinear responses and geometric constraints in graph space

Nature Machine Intelligence volume 7, pages 1023–1036 (2025)Cite this article

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Abstract

Advances in data-driven design and additive manufacturing have substantially accelerated the development of truss metamaterials—three-dimensional truss networks—offering exceptional mechanical properties at a fraction of the weight of conventional solids. While existing design approaches can generate metamaterials with target linear properties, such as elasticity, they struggle to capture complex nonlinear behaviours and to incorporate geometric and manufacturing constraints—including defects—crucial for engineering applications. Here we present GraphMetaMat, an autoregressive graph-based framework capable of designing three-dimensional truss metamaterials with programmable nonlinear responses, originating from hard-to-capture physics such as buckling, frictional contact and wave propagation, along with arbitrary geometric constraints and defect tolerance. Integrating graph neural networks, physics biases, imitation learning, reinforcement learning and tree search, we show that GraphMetaMat can target stress–strain curves across four orders of magnitude and vibration transmission responses with varying attenuation gaps, unattainable by previous methods. We further demonstrate the use of GraphMetaMat for the inverse design of novel material topologies with tailorable high-energy absorption and vibration damping that outperform existing polymeric foams and phononic crystals, potentially suitable for protective equipment and electric vehicles. This work sets the stage for the automatic design of manufacturable, defect-tolerant materials with on-demand functionalities.

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Fig. 1: Metamaterial-to-graph space and generation of diverse graph-response pairs.
Fig. 2: Overview of GraphMetaMat.
Fig. 3: Inverse design of representative user-defined nonlinear responses (unknown curve space).
Fig. 4: Defect-aware metamaterials design.
Fig. 5: Metamaterial generation for impact protection and vibration attenuation.

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Data availability

All training and test datasets, benchmark models and data are available via figshare at https://doi.org/10.6084/m9.figshare.28773806 (ref. 72).

Code availability

The code developed in this study is available via GitHub at https://github.com/marcomau06/GraphMetaMat and via Zenodo at https://doi.org/10.5281/ZENODO.15498444 (ref. 73).

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Acknowledgements

We thank O. Chen for her assistance in characterizing the commercial chest protector foams, and H. Lu for his help in setting up the vibration experiments. We also used OpenAI’s ChatGPT (GPT-4o) to assist in proofreading portions of the abstract and main text, with subsequent human revision and editing. We acknowledge support from the following funding sources: NSF DMREF grant 2119643 (M.M., D.X., D.Y., D.H., A.S., M.B., W.W. and X.R.Z.); NSF grant 1829071 (D.X. and W.W.); NSF grant 2106859 (W.W.); NSF grant 2312501 (D.X. and Y.S.); ONR grant N00014-23-1-2797 (X.R.Z.); DARPA grant HR00112490370 (Y.S.); and NSF DMREF grant 2119545 (Y.-T.W., M.O. and Y.J.).

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Author notes

  1. These authors contributed equally: Marco Maurizi, Derek Xu, Yu-Tong Wang.

Authors and Affiliations

  1. Department of Materials Science and Engineering, University of California Berkeley, Berkeley, CA, USA

    Marco Maurizi, Desheng Yao, David Hahn, Anish Satpati & Xiaoyu Rayne Zheng

  2. Department of Computer Science, University of California Los Angeles, Los Angeles, CA, USA

    Derek Xu, Wei Wang & Yizhou Sun

  3. Graduate Program in Acoustics, Penn State University, University Park, PA, USA

    Yu-Tong Wang, Mourad Oudich & Yun Jing

  4. Institut Jean Lamour, Université de Lorraine, CNRS, Nancy, France

    Mourad Oudich

  5. Department of Civil and Environmental Engineering, University of California Los Angeles, Los Angeles, CA, USA

    Mathieu Bauchy & Xiaoyu Rayne Zheng

  6. Energy Technology Area, Lawrence Berkeley National Laboratory, Berkeley, CA, USA

    Xiaoyu Rayne Zheng

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Contributions

X.R.Z., M.M., W.W., Y.S., M.B. and Y.J. contributed to the conceptualization of the study. The methodology was developed by M.M., D.X., Y.-T.W., D.Y., X.R.Z., W.W., Y.S., M.B. and Y.J. M.M., D.X., Y.-T.W., D.Y., D.H., M.O. and A.S. carried out the investigation. M.M., Y.-T.W. and D.Y. led the visualization efforts. Funding was acquired by X.R.Z., W.W., Y.S., M.B. and Y.J. Project administration was carried out by X.R.Z., M.M., W.W., Y.S., M.B. and Y.J., with supervision provided by X.R.Z., W.W., Y.S., M.B. and Y.J. M.M. and D.X. wrote the original draft of the manuscript, and M.M., X.R.Z., D.X., Y.-T.W., D.Y., M.O., W.W., Y.S. and Y.J. contributed to the review and editing process.

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Correspondence to Marco Maurizi or Xiaoyu Rayne Zheng.

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Nature Machine Intelligence thanks Xiaoyang Zheng, Jan-Hendrik Bastek and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Extended data

Extended Data Fig. 1 Schematic of the policy network (top) and forward model (bottom).

The physics bias for stress predictions is encoded in the ‘Magnitude Decoder’ of the forward model, predicting the coefficients C and n of the power-scaling law, \({y}_{max}={C}_{\rho }^{-n}\), with the relative density \(\bar{\rho }\).

Extended Data Fig. 2 Ablation study to quantify the impact of search and policy training.

Distribution of the relative error (NMAE) for stress-strain inverse design on the (a) 90:5:5 split test set, (b) user-defined target curves, and (c) energy absorbing-oriented target curves. Distribution of the accuracy for wave transmission inverse design on the (d) 90:5:5 split test set, (e) user-defined target sequences, and (f) noise control-oriented target responses. The relative error and accuracy are computed between the predicted responses of the generated metamaterials and the target responses. These results demonstrate how all components, from IL to RL and MCTS, contribute to the performance of our model.

Extended Data Fig. 3 Impact of printability constraint.

(a) Distribution of the relative error (NMAE) between the target and predicted test stress-strain responses, with and without printability constraint. (b) Distribution of the degree of support, Rsupp of the generated graphs on the test stress-strain target responses, with and without printability constraint. (c) Two examples of generated structures with and without printability constraint along with the corresponding FE-reconstructed stress-strain curves. These results reveal that, for certain target responses, printable structures can outperform non-printable truss networks. Overall, GraphMetaMat demonstrates the new capability of encoding manufacturing constraints, and offers flexibility in balancing performance and manufacturability constraints, as highlighted in (a) and (b).

Extended Data Fig. 4 Inverse design of representative unseen nonlinear responses (known curve space).

(a) Target stress-strain curves from the test set. NMAE values are computed between the target and FE-reconstructed curves. Stress is normalized by the constitutive material’s Young’s modulus Es. (b) Target transmission curves from the test set. Accuracy values are computed between the target and FE-reconstructed curves using a transmission threshold of Tth = −10 dB. In both (a) and (b), predictions are shown as mean curves, with shaded regions denoting the estimated uncertainty from the forward model, defined as the range (minimum to maximum) across five snapshots using the snapshot ensemble technique59 (n = 5).

Extended Data Fig. 5 Benchmark summary between GraphMetaMat and state-of-the-art inverse-design methods.

VAE-based method refers to ref. 51. Tandem-network method refers to ref. 50. Denoising diffusion method refers to ref. 28. GAN-based method refers to ref. 45.

Extended Data Fig. 6 Example of defect-aware inverse design.

(a) Target and predicted defect-free stress-strain curves of five generated designs. (b) Defect sensitivity of the five designs, defined as \(1-\bar{J}\), where \(\bar{J}\) represents the average curve similarity between pristine and flawed design responses across different defect parameters, such as the fraction of broken struts. Error bars represent the estimated uncertainty from the forward model, defined as the range (minimum to maximum) across five snapshots using the snapshot ensemble technique59 (n = 5). (c) Design 2 identified as the metamaterial with optimal defect tolerance.

Extended Data Table 1 Performance comparison between GraphMetaMat and prior generative methods
Full size table

Supplementary information

Supplementary Information (download PDF )

Supplementary Text, Figs. 1–57, Tables 1–11 and References 1–48.

Supplementary Video 1 (download MP4 )

Graph generation for a target nonlinear stress–strain response with linear strain hardening during RL training. The target curve is from the validation dataset.

Supplementary Video 2 (download MP4 )

Graph generation for a target nonlinear stress–strain response with nonlinear strain hardening during RL training. The target curve is from the validation dataset.

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Maurizi, M., Xu, D., Wang, YT. et al. Designing metamaterials with programmable nonlinear responses and geometric constraints in graph space. Nat Mach Intell 7, 1023–1036 (2025). https://doi.org/10.1038/s42256-025-01067-x

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