Deep learning-based inverse design and forward prediction of bi-material 4D-printed facial shells

Abstract

The programmable properties of polylactic acids and shape memory polymers in 4D printing enable time-dependent shape transformations, allowing the fabrication of 3D shells with zero material waste. However, achieving target geometries requires inverse design, often constrained by slow evolutionary algorithms or complex analytical models. Herein, we present a 2D curve matrix, tuned by material ratios and arc angles, to enable contraction or elongation and thereby reproduce protruding features such as noses. A fully convolutional network (FCN) directly generates design patterns with high accuracy from depth images in a single step, with multi-task learning predicting rib composition and curvature. In parallel, we refine the inverse design of the line matrix and utilize transfer learning to accurately reconstruct human facial geometries, while the FCN also performs well in forward prediction to bypass computational costs. Furthermore, the fabricated 3D shells closely match target facial features in both scale and geometry, with minimal deviation between simulations and experiments, demonstrating the method’s potential for scalable, customizable 4D-printed applications.

Keywords

4D printing / shape morphing / deep learning / inverse design / fully convolutional network / human face

References

[1] Bischoff M,Wall WA.Models and finite elements for thin-walled structures. In: Stein E, Borst R, Hughes TJR, editors. Encyclopedia of Computational Mechanics. Wiley; 2004.

[2] Loughlan J.Thin-walled structures: advances in research, design and manufacturing technology. 1st edition. CRC Press; 2018.

[3] Chen D,Yang J.Functionally graded porous structures: analyses, performances, and applications - a review.Thin Walled Struct2023;191:111046

[4] Ren R,Yue H,Lu Y.Stiffness enhancement methods for thin-walled aircraft structures: a review.Thin Walled Struct2024;201:111995

[5] Azanaw GM.Thin-walled structures in structural engineering: a comprehensive review of design innovations, stability challenges, and sustainable frontiers.Am J Mater Synth Process2025;10:18-26

[6] Wan X,Tian Y.Recent advances in 4D printing of advanced materials and structures for functional applications.Adv Mater2024;36:e2312263

[7] Yarali E,Ghalayaniesfahani A,Diaz-Payno PJ.4D printing for biomedical applications.Adv Mater2024;36:e2402301

[8] Kuang X,Wu J.Advances in 4D printing: materials and applications.Adv Funct Mater2019;29:1805290

[9] Khoo ZX,Liu Y.3D printing of smart materials: a review on recent progresses in 4D printing.Virt Phys Prototyping2015;10:103-22

[10] Jin B,Shi Y,Zhao Q.Solvent-assisted 4D programming and reprogramming of liquid crystalline organogels.Adv Mater2022;34:e2107855

[11] Yang Y,Lin M,Huang C.Bio-inspired facile strategy for programmable osmosis-driven shape-morphing elastomer composite structures.Mater Horiz2024;11:2180-90

[12] Khalid MY,Mamoun OS,Bodaghi M.Transformative 4D printed SMPs into soft electronics and adaptive structures: innovations and practical insights.Adv Mater Technol2025;10:e00309

[13] Liu B,Meng F.4D printed hydrogel scaffold with swelling-stiffening properties and programmable deformation for minimally invasive implantation.Nat Commun2024;15:1587 PMCID:PMC10881973

[14] Zhang L,Cole T.3D-printed liquid metal polymer composites as NIR-responsive 4D printing soft robot.Nat Commun2023;14:7815 PMCID:PMC10684855

[15] Soleimanzadeh H,Jamalabadi M,Zolfagharian A.Rotary 4D printing of programmable metamaterials on sustainable 4D mandrel. Adv. Mater. Technol. 2025, e01581.

[16] Yan Z,Wang J.Controlled mechanical buckling for origami-inspired construction of 3D microstructures in advanced materials.Adv Funct Mater2016;26:2629-39 PMCID:PMC4972027

[17] Yang X,Zhang B.Hierarchical tessellation enables programmable morphing matter.Matter2024;7:603-19

[18] Nojoomi A,Lee K.Bioinspired 3D structures with programmable morphologies and motions.Nat Commun2018;9:3705 PMCID:PMC6135848

[19] Arslan H,Jeon J.3D printing of anisotropic hydrogels with bioinspired motion.Adv Sci2019;6:1800703 PMCID:PMC6343088

[20] Boley JW,Lissandrello C.Shape-shifting structured lattices via multimaterial 4D printing.Proc Natl Acad Sci U S A2019;116:20856-62 PMCID:PMC6800333

[21] Sun X,Yu L.Machine learning-enabled forward prediction and inverse design of 4D-printed active plates.Nat Commun2024;15:5509 PMCID:PMC11217466

[22] Chiu YH,Juang JY.Designing bioinspired composite structures via genetic algorithm and conditional variational autoencoder.Polymers2023;15:281 PMCID:PMC9860612

[23] Jin L,Cheng J.Machine learning driven forward prediction and inverse design for 4D printed hierarchical architecture with arbitrary shapes.Appl Mater Today2024;40:102373

[24] Sun X,Yu L.Machine learning-evolutionary algorithm enabled design for 4D-printed active composite structures.Adv Funct Mater2022;32:2109805

[25] Athinarayanarao D,Chamoret D.Computational design for 4D printing of topology optimized multi-material active composites.npj Comput Mater2023;9:962

[26] Bai Y,Xue Y.A dynamically reprogrammable surface with self-evolving shape morphing.Nature2022;609:701-8

[27] Yang X,Zhao H.Morphing matter: from mechanical principles to robotic applications.Soft Sci2023;3:38

[28] Ni X,Kim JT.Soft shape-programmable surfaces by fast electromagnetic actuation of liquid metal networks.Nat Commun2022;13:5576 PMCID:PMC9508113

[29] Wang X,Ye J.Freestanding 3D mesostructures, functional devices, and shape-programmable systems based on mechanically induced assembly with shape memory polymers.Adv Mater2019;31:e1805615

[30] Tong D,Liu M.Inverse design of snap-actuated jumping robots powered by mechanics-aided machine learning.IEEE Robot Autom Lett2025;10:1720-7

[31] Bodaghi M,Yousefi A,Demoly F.Metamaterial boat fenders with supreme shape recovery and energy absorption/dissipation via FFF 4D printing.Smart Mater Struct2023;32:095028

[32] Jolaiy S,Hosseini M,Demoly F.Limpet-inspired design and 3D/4D printing of sustainable sandwich panels: pioneering supreme resiliency, recoverability and repairability.Appl Mater Today2024;38:102243

[33] Zolfagharian A,Xavier MS,Bodaghi M.Multimaterial 4D printing with a tunable bending model.Smart Mater Struct2023;32:065001

[34] Nojoomi A,Yum K.2D material programming for 3D shaping.Nat Commun2021;12:603 PMCID:PMC7841157

[35] Ahn SJ,Joo HJ,Lee DY.4D printing of continuous shape representation.Adv Mater Technol2021;6:2100133

[36] Kansara H,He Y.Inverse design and additive manufacturing of shape-morphing structures based on functionally graded composites.J Mech Phys Solids2023;180:105382

[37] Yun S,Kim S.Tailoring elastomeric meshes with desired 1D tensile behavior using an inverse design algorithm and material extrusion printing.Addit Manuf2022;60:103254

[38] Zheng X,Chen TT.Deep learning in mechanical metamaterials: from prediction and generation to inverse design.Adv Mater2023;35:e2302530

[39] Chen CT.Generative deep neural networks for inverse materials design using backpropagation and active learning.Adv Sci2020;7:1902607 PMCID:PMC7055566

[40] Chen CT.Physics-informed deep-learning for elasticity: forward, inverse, and mixed problems.Adv Sci2023;10:e2300439 PMCID:PMC10288249

[41] Wang R,Wang X,Liang X.DM net: a multiple nonlinear regression net for the inverse design of disordered metamaterials.Addit Manuf2024;96:104577

[42] Mohammadi M,Bodaghi M.Sustainable robotic joints 4D printing with variable stiffness using reinforcement learning.Robotics Comput Integr Manuf2024;85:102636

[43] Mohammadi M,Bodaghi M.Inverse design of adaptive flexible structures using physical-enhanced neural network.Virtual Phys Prototyp2025;20:e2530732 PMCID:PMC12502832

[44] Cheng X,Yao S.Programming 3D curved mesosurfaces using microlattice designs.Science2023;379:1225-32

[45] Chiu Y,Chen M,Yen Y.Inverse design of face-like 3D surfaces via bi-material 4D printing and shape morphing.Virt Phys Prototyping2025;20:e2507099

[46] Bodaghi M,Liao WH.Triple shape memory polymers by 4D printing.Smart Mater Struct2018;27:065010

[47] Ma DS,Wittenbrink B.The Chicago face database: a free stimulus set of faces and norming data.Behav Res Methods2015;47:1122-35

[48] Ma DS,Wittenbrink B.Chicago face database: multiracial expansion.Behav Res Methods2021;53:1289-300 PMCID:PMC8219557

[49] Lakshmi A,Correll J.The India face set: international and cultural boundaries impact face impressions and perceptions of category membership.Front Psychol2021;12:627678 PMCID:PMC7905305

[50] Thomaz CE.A new ranking method for principal components analysis and its application to face image analysis.Image Vis Comput2010;28:902-13

[51] DeBruine L. Face research lab london set. 2021. https://figshare.com/articles/dataset/Face_Research_Lab_London_Set/5047666. (accessed 2025-11-11)

[52] Daugman J.Iris recognition.Am Sci2001;89:326

[53] Ma L,Wang Y.Efficient iris recognition by characterizing key local variations.IEEE Trans Image Process2004;13:739-50

[54] Bashkatov AN,Bakutkin VV,Savchenko EP.Estimation of melanin content in iris of human eye. In Ophthalmic Technologies XV. SPIE; 2005. pp. 302-11.

[55] Iyamu E.Age, gender, corneal diameter, corneal curvature and central corneal thickness in Nigerians with normal intra ocular pressure.J Optomm2012;5:87-97 PMCID:PMC3861109

[56] Aguirre GK.A model of the entrance pupil of the human eye.Sci Rep2019;9:9360 PMCID:PMC6597540

[57] Mohammed S,Feuerpfeil M.The effects of data quality on machine learning performance on tabular data.Inf Syst2025;132:102549

[58] Vaswani A,Parmar N. Attention is all you need. arXiv 2017, arXiv:1706.03762. Available online: https://doi.org/10.48550/arXiv.1706.03762. (accessed 11 Nov 2025)

[59] Lin K,Zhou T.Applying machine learning to fine classify construction and demolition waste based on deep residual network and knowledge transfer.Environ Dev Sustain2023;25:8819-36

[60] Nakano T.You trust a face like yours.Humanit Soc Sci Commun2022;9:1248

[61] Mitteroecker P,Windhager S.A brief review of shape, form, and allometry in geometric morphometrics, with applications to human facial morphology.Hystrix2013;24:59-66

[62] He K,Ren S.Deep residual learning for image recognition. In 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), Las Vegas, USA. June 27-30, 2016. IEEE; 2016. pp. 770-8.

[63] Chen Z,Lee CY. GradNorm: gradient normalization for adaptive loss balancing in deep multitask networks. arXiv 2017, arXiv:1711.02257. Available online: https://doi.org/10.48550/arXiv.1711.02257. (accessed 11 Nov 2025)

[64] Wang Z,Sheikh HR.Image quality assessment: from error visibility to structural similarity.IEEE Trans Image Process2004;13:600-12

[65] Yoo J.Fast normalized cross-correlation.Circuits Syst Signal Process2009;28:819-43

[66] Radford A,Chintala S. Unsupervised representation learning with deep convolutional generative adversarial networks. arXiv 2015, arXiv:1511.06434. Available online: https://doi.org/10.48550/arXiv.1511.06434. (accessed 11 Nov 2025)

[67] Kaelbling LP,Moore AW. Reinforcement learning: a survey. arXiv 1996, arXiv:cs/9605103. Available online: https://doi.org/10.48550/arXiv.cs/9605103. (accessed 11 Nov 2025)

[68] Wiering M.Reinforcement learning: state-of-art. 1st edition. Springer Berlin, 2012.

[69] Li Y. Deep reinforcement learning: an overview. arXiv 2017, arXiv:1701.07274. Available online: https://doi.org/10.48550/arXiv.1701.07274. (accessed 11 Nov 2025)

[70] Srivastava N,Krizhevsky A,Salakhutdinov R. Dropout: a simple way to prevent neural networks from overfitting. J. Mach. Learn. Res. 2014, 15, 1929-58. https://www.jmlr.org/papers/volume15/srivastava14a/srivastava14a.pdf?utm_content=buffer79b4. (accessed 11 Nov 2025)

[71] Aung SC,Lee ST.Three dimensional laser scan assessment of the Oriental nose with a new classification of Oriental nasal types.Br J Plast Surg2000;53:109-16

[72] Uzun A,Bilgic S.The average values of the nasal anthropometric measurements in 108 young Turkish males.Auris Nasus Larynx2006;33:31-5

[73] Farkas LG,Forrest CR.International anthropometric study of facial morphology in various ethnic groups/races.J Craniofac Surg2005;16:615-46

[74] Chen Y.Object modelling by registration of multiple range images.Image Vis Comput1992;10:145-55

[75] Besl P.A method for registration of 3-D shapes.IEEE Trans Pattern Anal Mach Intell1992;14:239-56