Intelligent Computation and Analysis of Mechanical Behaviour in Piezoelectric Metamaterials Based on Physics-Informed Neural Networks

Danyang Qiu; Yaoxin Huang; Xinru  Li; Ningping  Zhan

doi:10.62177/jaet.v3i1.979

Authors

Danyang Qiu Ningbo University
Yaoxin Huang Ningbo University
Xinru Li Ningbo University
Ningping Zhan Ningbo University

DOI:

https://doi.org/10.62177/jaet.v3i1.979

Keywords:

Piezoelectric Metamaterials, Physical Information Neural Network, Multi-Scale Modeling, Multiphysics Coupling, Topology Optimization

Abstract

Piezoelectric metamaterials, serving as critical functional media in high-end equipment, face significant design challenges due to the mesh bottlenecks of traditional finite element methods and the interpretability shortcomings of purely data-driven models. Physical Information Neural Networks (PINNs) establish a robust scientific machine learning paradigm by embedding physical equations, offering an innovative solution to these predicaments. This paper systematically reviews recent advancements of PINNs in piezoelectric metamaterial analysis and design: drawing upon multiscale modelling theory, it elucidates PINNs' mesh-free advantages in handling high-dimensional parameters and their exceptional capability in solving small-sample inverse problems; subsequently, it explores their application paradigms in constructing high-fidelity forward surrogate models and accelerating efficient topology optimisation. Finally, this paper summarises key computational challenges in multi-physics coupling scenarios and outlines potential pathways towards achieving high-fidelity intelligent design, aiming to bridge the existing gap between theoretical modelling and engineering practice in piezoelectric metamaterials.

Downloads

Download data is not yet available.

References

Bischof, R., & Kraus, M. A. (2025). Multi-objective loss balancing for physics-informed deep learning. Computer Methods in Applied Mechanics and Engineering, 439, 117914. https://doi.org/10.1016/j.cma.2025.117914

Burbano, A., Zorin, D., & Jarosz, W. (Eds.). (2024). Data-efficient discovery of hyperelastic TPMS metamaterials with extreme energy dissipation. In Special Interest Group on Computer Graphics and Interactive Techniques Conference Conference Papers 24. ACM.

Chen, K., Dong, X., Gao, P., Chen, Q., Peng, Z., & Meng, G. (2025). Physics-informed neural networks for topological metamaterial design and mechanical applications. International Journal of Mechanical Sciences, 301, 110489. https://doi.org/10.1016/j.ijmecsci.2025.110489

Chen, S., Tong, X., Huo, Y., Liu, S., Yin, Y., Tan, M., Cai, K., & Ji, W. (2024). Piezoelectric biomaterials inspired by nature for applications in biomedicine and nanotechnology. Advanced Materials, 36, 2406192. https://doi.org/10.1002/adma.202406192

Chen, Y., Lu, L., Karniadakis, G. E., & Dal Negro, L. (2020). Physics-informed neural networks for inverse problems in nano-optics and metamaterials. Optics Express, 28, 11618. https://doi.org/10.1364/OE.384875

Danawe, H., & Tol, S. (2023). Electro-momentum coupling tailored in piezoelectric metamaterials with resonant shunts. APL Materials, 11, 091118. https://doi.org/10.1063/5.0165267

Didilis, K., Selicani, G. V., Tinti, V. B., Mobin, M., Brouczek, D., Staal, L., Marani, D., Insinga, A. R., Haugen, A. B., & Esposito, V. (2026). Topology-driven electromechanical actuation in 3D-printed TPMS piezoelectric ceramics. Acta Materialia, 303, 121724. https://doi.org/10.1016/j.actamat.2025.121724

Fan, G., Guo, N., Ahmed, M., & Ghazouani, N. (2025). Physics-informed neural network-enhanced simulation to estimate piezoelectric energy harvesting in cantilevered metamaterial concrete systems under wind excitation. Mechanics of Advanced Materials and Structures, 1–17. https://doi.org/10.1080/15376494.2025.2476205

Fang, D., & Tan, J. (2022). Immersed boundary-physics informed machine learning approach for fluid–solid coupling. Ocean Engineering, 263, 112360. https://doi.org/10.1016/j.oceaneng.2022.112360

Fang, Z., & Zhan, J. (2020). Deep physical informed neural networks for metamaterial design. IEEE Access, 8, 24506–24513. https://doi.org/10.1109/ACCESS.2019.2963375

Faraci, D., Comi, C., & Marigo, J.-J. (2022). Band gaps in metamaterial plates: Asymptotic homogenization and bloch-floquet approaches. Journal of Elasticity, 148, 55–79. https://doi.org/10.1007/s10659-022-09879-3

Gao, M., He, Z., Liu, J., Lü, C., & Wang, G. (2025). A dynamic homogenization method for elastic wave band gap and initial-boundary value problem analysis of piezoelectric composites with elastic and viscoelastic periodic layers. Journal of the Mechanics and Physics of Solids, 197, 106048. https://doi.org/10.1016/j.jmps.2025.106048

Gao, Z., Lei, Y., Li, Z., Yang, J., Yu, B., Yuan, X., Hou, Z., Hong, J., & Dong, S. (2025). Artificial piezoelectric metamaterials. Progress in Materials Science, 151, 101434. https://doi.org/10.1016/j.pmatsci.2025.101434

Guo, J., Zhu, H., Yang, Y., & Guo, C. (2025). Advances in physics-informed neural networks for solving complex partial differential equations and their engineering applications: A systematic review. Engineering Applications of Artificial Intelligence, 161, 112044. https://doi.org/10.1016/j.engappai.2025.112044

Haghighat, E., Raissi, M., Moure, A., Gomez, H., & Juanes, R. (2021). A physics-informed deep learning framework for inversion and surrogate modeling in solid mechanics. Computer Methods in Applied Mechanics and Engineering, 379, 113741. https://doi.org/10.1016/j.cma.2021.113741

Harandi, A., Moeineddin, A., Kaliske, M., Reese, S., & Rezaei, S. (2024). Mixed formulation of physics-informed neural networks for thermo-mechanically coupled systems and heterogeneous domains. Numerical Methods in Engineering, 125, e7388. https://doi.org/10.1002/nme.7388

He, Z., Liu, J., & Chen, Q. (2023). Higher-order asymptotic homogenization for piezoelectric composites. International Journal of Solids and Structures, 264, 112092. https://doi.org/10.1016/j.ijsolstr.2022.112092

Institute of Electrical and Electronics Engineers. (1988). IEEE standard on piezoelectricity. https://doi.org/10.1109/IEEESTD.1988.79638

Jagtap, A. D., Kharazmi, E., & Karniadakis, G. E. (2020). Conservative physics-informed neural networks on discrete domains for conservation laws: Applications to forward and inverse problems. Computer Methods in Applied Mechanics and Engineering, 365, 113028. https://doi.org/10.1016/j.cma.2020.113028

Jeong, H., Batuwatta-Gamage, C., Bai, J., Xie, Y. M., Rathnayaka, C., Zhou, Y., & Gu, Y. (2023). A complete physics-informed neural network-based framework for structural topology optimization. Computer Methods in Applied Mechanics and Engineering, 417, 116401. https://doi.org/10.1016/j.cma.2023.116401

Ji, W., Chang, J., Xu, H.-X., Gao, J. R., Gröblacher, S., Urbach, H. P., & Adam, A. J. L. (2023). Recent advances in metasurface design and quantum optics applications with machine learning, physics-informed neural networks, and topology optimization methods. Light Science & Applications, 12, 169. https://doi.org/10.1038/s41377-023-01218-y

Jiang, J., Wu, J., Chen, Q., Chatzigeorgiou, G., & Meraghni, F. (2023). Physically informed deep homogenization neural network for unidirectional multiphase/multi-inclusion thermoconductive composites. Computer Methods in Applied Mechanics and Engineering, 409, 115972. https://doi.org/10.1016/j.cma.2023.115972

Karniadakis, G. E., Kevrekidis, I. G., Lu, L., Perdikaris, P., Wang, S., & Yang, L. (2021). Physics-informed machine learning. Nature Reviews Physics, 3, 422–440. https://doi.org/10.1038/s42254-021-00314-5

Kiran, R., Nguyen-Thanh, N., Yu, H., & Zhou, K. (2023). Adaptive isogeometric analysis–based phase-field modeling of interfacial fracture in piezoelectric composites. Engineering Fracture Mechanics, 288, 109181. https://doi.org/10.1016/j.engfracmech.2023.109181

Li, Z., Kovachki, N., Azizzadenesheli, K., Liu, B., Bhattacharya, K., Stuart, A., & Anandkumar, A. (2021). Fourier neural operator for parametric partial differential equations. https://doi.org/10.48550/arXiv.2010.08895

Li, Z., Yi, X., Yang, J., Bian, L., Yu, Z., & Dong, S. (2022). Designing artificial vibration modes of piezoelectric devices using programmable, 3D ordered structure with piezoceramic strain units. Advanced Materials, 34, 2107236. https://doi.org/10.1002/adma.202107236

Liu, C., He, Z., Lü, C., & Wang, G. (2024). Concurrent topology optimization of multiscale piezoelectric actuators. International Journal of Solids and Structures, 290, 112664. https://doi.org/10.1016/j.ijsolstr.2024.112664

Liu, Y., He, H., Cao, Y., Liang, Y., & Huang, J. (2024). Inverse design of TPMS piezoelectric metamaterial based on deep learning. Mechanics of Materials, 198, 105109. https://doi.org/10.1016/j.mechmat.2024.105109

Lu, L., Jin, P., Pang, G., Zhang, Z., & Karniadakis, G. E. (2021). Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators. Nature Machine Intelligence, 3, 218–229. https://doi.org/10.1038/s42256-021-00302-5

McClenny, L. D., & Braga-Neto, U. M. (2023). Self-adaptive physics-informed neural networks. Journal of Computational Physics, 474, 111722. https://doi.org/10.1016/j.jcp.2022.111722

Meng, C., Thrane, P. C. V., Ding, F., Gjessing, J., Thomaschewski, M., Wu, C., Dirdal, C., & Bozhevolnyi, S. I. (2021). Dynamic piezoelectric MEMS-based optical metasurfaces. Science Advances, 7, eabg5639. https://doi.org/10.1126/sciadv.abg5639

Nassar, M. E., Saeed, N. A., & Nasedkin, A. (2023). Determination of effective properties of porous piezoelectric composite with partially randomly metalized pore boundaries using finite element method. Applied Mathematical Modelling, 124, 241–256. https://doi.org/10.1016/j.apm.2023.07.025

Nguyen, C., Zhuang, X., Chamoin, L., Zhao, X., Nguyen-Xuan, H., & Rabczuk, T. (2020). Three-dimensional topology optimization of auxetic metamaterial using isogeometric analysis and model order reduction. Computer Methods in Applied Mechanics and Engineering, 371, 113306. https://doi.org/10.1016/j.cma.2020.113306

Raissi, M., Perdikaris, P., & Karniadakis, G. E. (2019). Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics, 378, 686–707. https://doi.org/10.1016/j.jcp.2018.10.045

Ravanbod, M., & Ebrahimi-Nejad, S. (2024). Perforated auxetic honeycomb booster with reentrant chirality: A new design for high-efficiency piezoelectric energy harvesting. Mechanics of Advanced Materials and Structures, 31, 9857–9872. https://doi.org/10.1080/15376494.2023.2280997

Ren, M., Wang, C., Moshrefi-Torbati, M., Yurchenko, D., Shu, Y., & Yang, K. (2025). Optimization of a comb-like beam piezoelectric energy harvester using the parallel separated multi-input neural network surrogate model. Mechanical Systems and Signal Processing, 224, 111939. https://doi.org/10.1016/j.ymssp.2024.111939

Ren, X., Das, R., Tran, P., Ngo, T. D., & Xie, Y. M. (2018). Auxetic metamaterials and structures: A review. Smart Materials and Structures, 27, 023001. https://doi.org/10.1088/1361-665X/aaa61c

Rezaei, S., Harandi, A., Moeineddin, A., Xu, B.-X., & Reese, S. (2022). A mixed formulation for physics-informed neural networks as a potential solver for engineering problems in heterogeneous domains: Comparison with finite element method. Computer Methods in Applied Mechanics and Engineering, 401, 115616. https://doi.org/10.1016/j.cma.2022.115616

Roberts, A. (2024). Embed in ensemble to rigorously and accurately homogenize quasi-periodic multi-scale heterogeneous material. ANZIAM Journal, 66, 1–34. https://doi.org/10.1017/S1446181124000099

Serrao, P. H., & Kozinov, S. (2024). Robust mixed FE for analyses of higher-order electromechanical coupling in piezoelectric solids. Computational Mechanics, 73, 1203–1217. https://doi.org/10.1007/s00466-023-02407-7

Sharma, D., & Hiremath, S. S. (2022a). Additively manufactured mechanical metamaterials based on triply periodic minimal surfaces: Performance, challenges, and application. Mechanics of Advanced Materials and Structures, 29, 5077–5107. https://doi.org/10.1080/15376494.2021.1948151

Sharma, D., & Hiremath, S. S. (2022b). Additively manufactured mechanical metamaterials based on triply periodic minimal surfaces: Performance, challenges, and application. Mechanics of Advanced Materials and Structures, 29, 5077–5107. https://doi.org/10.1080/15376494.2021.1948151

Sharma, S., Ammu, S. K., Thakolkaran, P., Jovanova, J., Masania, K., & Kumar, S. (2025). Piezoelectric truss metamaterials: Data-driven design and additive manufacturing. npj Metamaterials, 1, 9. https://doi.org/10.1038/s44455-025-00009-2

Shi, J., & Akbarzadeh, A. H. (2019). Architected cellular piezoelectric metamaterials: Thermo-electro-mechanical properties. Acta Materialia, 163, 91–121. https://doi.org/10.1016/j.actamat.2018.10.001

Shi, J., Ju, K., Chen, H., Orsat, V., Sasmito, A. P., Ahmadi, A., & Akbarzadeh, A. (2025). Ultrahigh piezoelectricity in truss‐based ferroelectric ceramics metamaterials. Advanced Functional Materials, 35, 2417618. https://doi.org/10.1002/adfm.202417618

Shin, Y., Darbon, J., & Karniadakis, G. E. (2020). On the convergence of physics informed neural networks for linear second-order elliptic and parabolic type PDEs. Communications in Computational Physics, 28, 2042–2074. https://doi.org/10.4208/cicp.OA-2020-0193

Sitzmann, V., Martel, J. N. P., Bergman, A. W., Lindell, D. B., & Wetzstein, G. (2020). Implicit neural representations with periodic activation functions. https://doi.org/10.48550/arXiv.2006.09661

Somnic, J., & Jo, B. W. (2022). Status and challenges in homogenization methods for lattice materials. Materials, 15, 605. https://doi.org/10.3390/ma15020605

Stankiewicz, G., Dev, C., Weichelt, M., Fey, T., & Steinmann, P. (2024). Towards advanced piezoelectric metamaterial design via combined topology and shape optimization. Structural and Multidisciplinary Optimization, 67, 26. https://doi.org/10.1007/s00158-024-03742-w

Sun, R., Jeong, H., Zhao, J., Gou, Y., Sauret, E., Li, Z., & Gu, Y. (2024). A physics-informed neural network framework for multi-physics coupling microfluidic problems. Computers & Fluids, 284, 106421. https://doi.org/10.1016/j.compfluid.2024.106421

Tancik, M., Srinivasan, P. P., Mildenhall, B., Fridovich-Keil, S., Raghavan, N., Singhal, U., Ramamoorthi, R., Barron, J. T., & Ng, R. (2020). Fourier features let networks learn high frequency functions in low dimensional domains. https://doi.org/10.48550/arXiv.2006.10739

Tandis, N., & Tandis, E. (2025). A physics-informed machine learning approach to piezoelectric plate modelling. Engineering Applications of Artificial Intelligence, 160, 111847. https://doi.org/10.1016/j.engappai.2025.111847

Tassi, N., Bakkali, A., Fakri, N., Azrar, L., & Aljinaidi, A. (2021). Well conditioned mathematical modeling for homogenization of thermo-electro-mechanical behaviors of piezoelectric composites. Applied Mathematical Modelling, 99, 276–293. https://doi.org/10.1016/j.apm.2021.06.019

Tran, T. V., Nanthakumar, S. S., & Zhuang, X. (2025). Deep learning-based framework for the on-demand inverse design of metamaterials with arbitrary target band gap. npj Artificial Intelligence, 1, 2. https://doi.org/10.1038/s44387-025-00001-1

Wang, B., Meng, D., Lu, C., Zhang, Q., Zhao, M., & Zhang, J. (2025). Physics-informed neural networks for analyzing size effect and identifying parameters in piezoelectric semiconductor nanowires. Journal of Applied Physics, 137, 024303. https://doi.org/10.1063/5.0248278

Wang, D., Dong, L., & Gu, G. (2023). 3D printed fractal metamaterials with tunable mechanical properties and shape reconfiguration. Advanced Functional Materials, 33, 2208849. https://doi.org/10.1002/adfm.202208849

Wu, C., Zhu, M., Tan, Q., Kartha, Y., & Lu, L. (2023). A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks. Computer Methods in Applied Mechanics and Engineering, 403, 115671. https://doi.org/10.1016/j.cma.2022.115671

Xiao, Z., Weng, Y., Yao, W., Chen, W., & Zhang, C. (2026). Nonlinear multi-field coupling analysis of piezoelectric semiconductors via PINNs. Science China Physics, Mechanics & Astronomy, 69, 214611. https://doi.org/10.1007/s11433-025-2742-6

Yang, J., Li, Z., Xin, X., Gao, X., Yuan, X., Wang, Z., Yu, Z., Wang, X., Zhou, J., & Dong, S. (2019). Designing electromechanical metamaterial with full nonzero piezoelectric coefficients. Science Advances, 5, eaax1782. https://doi.org/10.1126/sciadv.aax1782

Yang, Y., Meng, J., Fan, W., Lv, J., & Xu, B. (2025). Virtual element method for piezoelasticity. Numerical Methods in Engineering, 126, e70191. https://doi.org/10.1002/nme.70191

Yarotsky, D. (2017). Error bounds for approximations with deep ReLU networks. Neural Networks, 94, 103–114. https://doi.org/10.1016/j.neunet.2017.07.002

Yin, J., Wen, Z., Li, S., Zhang, Y., & Wang, H. (2024). Dynamically configured physics-informed neural network in topology optimization applications. Computer Methods in Applied Mechanics and Engineering, 426, 117004. https://doi.org/10.1016/j.cma.2024.117004

Zhang, Y., Huang, Y., Zhao, S., Jiao, Z., Lü, C., & Yang, J. (2025a). Nonlinear dynamic response and stability of piezoelectric shells with piezoelectric nonlinearities. International Journal of Mechanical Sciences, 304, 110731. https://doi.org/10.1016/j.ijmecsci.2025.110731

Zhang, Y., Zhu, H., Zhao, S., Ni, Z., Lü, C., & Yang, J. (2025b). Nonlinear dynamic response of functionally graded plates with piezoelectric nonlinearity. European Journal of Mechanics - A/Solids, 114, 105776. https://doi.org/10.1016/j.euromechsol.2025.105776

Zhang, Y., Guo, X., Wu, Y., Zhang, Y. Y., & Lü, C. (2025c). Active control of cables with piezoelectric actuation considering geometric and material nonlinearities. Engineering Structures, 340, 120773. https://doi.org/10.1016/j.engstruct.2025.120773

Zhang, Y., Guo, X., Wu, Y., Zhang, Y. Y., Zhang, H., & Lü, C. (2024a). Nonlinear thermo-electro-mechanical responses and active control of functionally graded piezoelectric plates subjected to strong electric fields. Thin-Walled Structures, 205, 112375. https://doi.org/10.1016/j.tws.2024.112375

Zhang, Y., Guo, X., Wu, Y., Zhang, Y. Y., Zhang, H., & Lü, C. (2024b). Vibration control of membrane structures by piezoelectric actuators considering piezoelectric nonlinearity under strong electric fields. Engineering Structures, 315, 118413. https://doi.org/10.1016/j.engstruct.2024.118413

Zhang, Z., Lee, J.-H., & Gu, G. X. (2022). Rational design of piezoelectric metamaterials with tailored electro-momentum coupling. Extreme Mechanics Letters, 55, 101785. https://doi.org/10.1016/j.eml.2022.101785

Zhao, W., Hao, R., Zhang, M., Chen, Q., Yang, Z., & Chen, X. (2025). Physically informed neural networks for homogenization and localization of composites with periodic microstructures. Composite Structures, 367, 119260. https://doi.org/10.1016/j.compstruct.2025.119260

Zhou, L., Geng, J., Chen, P., Zhu, H., Tian, H., & Liu, X. (2024). Mechanic-electric coupling cell-based smoothed isogeometric analysis for the static behaviors of piezoelectric structures. Acta Mechanica, 235, 2803–2821. https://doi.org/10.1007/s00707-024-03856-8

Zhou, Y., Wang, Z., Zhou, K., Tang, H., & Li, X. (2025). LT-PINN: Lagrangian topology-conscious physics-informed neural network for boundary-focused engineering optimization. https://doi.org/10.48550/arXiv.2506.06300