Abstract
Ferroelectric ceramics are promising energy-storage candidates for miniaturizing high-power electronic systems, yet synergistically enhancing energy density and efficiency remains constrained by intricate coupling between chemical compositions and polarization configurations. Achieving high-throughput compositional exploration while solving real-time polarization dynamics is nearly impossible with traditional simulations due to prohibitive computational costs. Here, we propose an inverse design framework integrating a variational generative model with active learning optimization to accelerate the development of ferroelectrics with enhanced energy-storage performance under limited electric fields. By formulating the time-dependent Ginzburg-Landau equation governing domain structure evolution as conditional sampling within model latent space, achieving synergistic optimization of chemistry and polarization configurations. Through four-round closed-loop synthesis, we successfully obtain Bi0.5Na0.5TiO3-based relaxor-ferroelectrics exhibiting exceptional energy density of ~2.3 J cm-3 and ~80% efficiency at a low field of 200 kV cm-1. This work establishes an efficient, generalizable route for the inverse design of next-generation energy-storage dielectric materials.
Similar content being viewed by others
Data availability
The data that support the findings of this study are available on request from the corresponding authors.
Code availability
The source code and pre-trained model of this study is openly available in Github repository https://github.com/Zhaochen-Xi/FEs-Inverse-Design.
References
Kim, J. et al. Ultrahigh capacitive energy density in ion-bombarded relaxor ferroelectric films. Science 369, 81 (2020).
Yang, B. B. et al. Engineering relaxors by entropy for high energy storage performance. Nat. Energy 8, 956–964 (2023).
Han, S. et al. High energy density in artificial heterostructures through relaxation time modulation. Science 384, 312–317 (2024).
Liu, Y. et al. Ultrahigh capacitive energy storage through dendritic nanopolar design. Science 388, 211–216 (2025).
Pan, H. et al. Ultrahigh energy storage in superparaelectric relaxor ferroelectrics. Science 374, 100–104 (2021).
Yang, L. T. et al. Perovskite lead-free dielectrics for energy storage applications. PROGRESS MATERIALS SCIENCE 102, 72–108 (2019).
Pan, H. et al. Ultrahigh-energy density lead-free dielectric films via polymorphic nanodomain design. Science 365, 578–582 (2019).
Liu, Y. et al. Harnessing local inhomogeneity for enhanced dielectric energy storage. Nat. Commun. 16, 6236 (2025).
Kamboj, V. et al. Grain morphological engineering for optimization of energy storage in (1-x)(Na0.5Bi0.5)0.75Sr0.25TiO3-xBaTiO3 ferroelectric ceramics: A microwave sintering approach. J. Energy Storage 114, 115874 (2025).
Yang, B. et al. Enhanced energy storage in antiferroelectrics via antipolar frustration. Nature 637, 1104–1110 (2025).
Shu, L. et al. Partitioning polar-slush strategy in relaxors leads to large energy-storage capability. Science 385, 204–209 (2024).
Yan, F. et al. Superior energy storage properties and excellent stability achieved in environment-friendly ferroelectrics via composition design strategy. Nano Energy 75, 105012 (2020).
Yan, F. et al. Significantly enhanced energy storage density and efficiency of BNT-based perovskite ceramics via A-site defect engineering. Energy Storage Mater. 30, 392–400 (2020).
Yuan, R. H. et al. Accelerated discovery of large electrostrains in BaTiO3-based piezoelectrics using active learning. Adv. Mater. 30, 1702884 (2018).
Gurnani, R. et al. AI-assisted discovery of high-temperature dielectrics for energy storage. Nat. Commun. 15, 6107 (2024).
Liu, Y. et al. Experimental discovery of structure–property relationships in ferroelectric materials via active learning. Nat. Mach. Intell. 4, 341–350 (2022).
Li, H. et al. Machine learning-accelerated discovery of heat-resistant polysulfates for electrostatic energy storage. Nat. Energy 10, 90–100 (2025).
Yang, M. et al. High-temperature polymer composite capacitors with high energy density designed via machine learning. Nat. Energy 10, 1323–1333 (2025).
He, J. et al. Accelerated discovery of high-performance piezocatalyst in BaTiO3-based ceramics via machine learning. Nano Energy 97, 107218 (2022).
Duan, X. et al. Machine learning accelerated discovery of entropy-stabilized oxide catalysts for catalytic oxidation. J. Am. Chem. Soc. 147, 651–661 (2025).
Yuan, R. H. et al. Accelerated search for BaTiO3-based ceramics with large energy storage at low fields using machine learning and experimental design. Adv. Sci. 6, 1901395 (2019).
Li, W. et al. Generative learning facilitated discovery of high-entropy ceramic dielectrics for capacitive energy storage. Nat. Commun. 15, 4940 (2024).
Zhu, B. et al. Designing Pb-free high-entropy relaxor ferroelectrics with machine learning assistance for high energy storage. J. Am. Chem. Soc. 147, 27912–27921 (2025).
Wang, X. et al. Machine learning assisted composition design of high-entropy Pb-free relaxors with giant energy-storage. Nat. Commun. 16, 1254 (2025).
Shen, Z.-H. et al. Machine learning in energy storage materials. Interdiscip. Mater. 1, 175–195 (2022).
Devonshire, A. F. XCVI. Theory of barium titanate. Lond., Edinb., Dublin Philos. Mag. J. Sci. 40, 1040–1063 (1949).
Devonshire, A. F. CIX. Theory of barium titanate—Part II. Lond., Edinb., Dublin Philos. Mag. J. Sci. 42, 1065–1079 (1951).
Chen, L. Q. Phase-field models for microstructure evolution. Ann. Rev. Mater. Res. 32, 113–140 (2002).
Wang, J. J., Wu, P. P., Ma, X. Q. & Chen, L. Q. Temperature-pressure phase diagram and ferroelectric properties of BaTiO3 single crystal based on a modified Landau potential. J. Appl. Phys. 108, 114105 (2010).
Kingma, D. P. et al Improved variational inference with inverse autoregressive flow. In: Proceedings of the 30th International Conference on Neural Information Processing Systems). Curran Associates Inc. (2016).
Huang, Y. H. et al. A thermodynamic potential, energy storage performances, and electrocaloric effects of Ba1-xSrxTiO3 single crystals. Appl. Phys. Lett. 112, 102901 (2018).
Yin, J. et al. Deciphering the atomic-scale structural origin for large dynamic electromechanical response in lead-free Bi0.5Na0.5TiO3-based relaxor ferroelectrics. Nat. Commun. 13, 6333 (2022).
Marlton, F., Standard, O., Kimpton, J. A. & Daniels, J. E. Phase boundaries in the ternary (Bi0.5Na0.5TiO3)x(BaTiO3)y(SrTiO3)1−x−y system. Appl. Phys. Lett. 111, 202903 (2017).
Deb, K., Pratap, A., Agarwal, S. & Meyarivan, T. A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans. Evolut. Comput. 6, 182–197 (2002).
Li, D. et al. Lead-free relaxor ferroelectric ceramics with ultrahigh energy storage densities via polymorphic polar nanoregions design. Small 19, 2206958 (2023).
Zhou, X., Xue, G., Luo, H., Bowen, C. R. & Zhang, D. Phase structure and properties of sodium bismuth titanate lead-free piezoelectric ceramics. Prog. Mater. Sci. 122, 100836 (2021).
Hiruma, Y., Imai, Y., Watanabe, Y., Nagata, H. & Takenaka, T. Large electrostrain near the phase transition temperature of (Bi0.5Na0.5)TiO3–SrTiO3 ferroelectric ceramics. Appl. Phys. Lett. 92, 262904 (2008).
Li, W. et al. Structural modification and piezoelectric properties in Bi0.5Na0.5TiO3–BaTiO3–SrTiO3 thin films. J. Mater. Sci. Mater. Electron. 27, 215–220 (2016).
Xie, Y. et al. The energy-storage performance and dielectric properties of (0.94-x)BNT-0.06BT-xST thin films prepared by sol–gel method. J. Alloy. Compd. 860, 158164 (2021).
Takenaka, T., Kei-ichi Maruyama, K. -iM. & Koichiro Sakata, K. S. (Bi1/2Na1/2)TiO3-BaTiO3 system for lead-free piezoelectric ceramics. Jpn. J. Appl. Phys. 30, 2236 (1991).
Li, J. et al. Grain-orientation-engineered multilayer ceramic capacitors for energy storage applications. Nat. Mater. 19, 999–1005 (2020).
Acknowledgements
The work is sponsored by the National Key Research and Development Program of China (2025YFF0521300), National Natural Science Foundation of China (52472136, 92463306, 52372100, 525B2020), the Hong Kong Scholars Program (XJ2025025), and the Fundamental Research Funds for the Central University, China. We acknowledge the computational resources provided by the High-Performance Computing platform of Xi’an Jiaotong University. The authors also thank F. Yang and X. D. Zhang from the Network Information Center of Xi’an Jiaotong University for their support of the HPC platform.
Author information
Authors and Affiliations
Contributions
The work was conceived and designed by Z.X., H.H., and D.Z.; Z.X. built the machine learning approaches, wrote the source code, and processed related data assisted by C.G., Z.L.; Z.X. synthesized the samples and performed tests on energy-storage performances, dielectric properties, structural stability, assisted by Z.W., J.B., W.Z., H.Z., Z.C.; Z.X., C.G., K.X., and H.H. conducted the phase-field simulations. Z.X. wrote the initial draft of the manuscript; H.H., C.G., and D.Z. revised the manuscript.
Corresponding authors
Ethics declarations
Competing interests
The authors declare no competing interests.
Peer review
Peer review information
Nature Communications thanks Varun Kamboj and the other anonymous reviewer(s) for their contribution to the peer review of this work. A peer review file is available.
Additional information
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary information
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/.
About this article
Cite this article
Xi, Z., Wang, Z., Guo, C. et al. Active learning in latent spaces enables rapid inverse design of ferroelectric ceramics for energy storage. Nat Commun (2026). https://doi.org/10.1038/s41467-026-70792-7
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/s41467-026-70792-7
