Integration of machine learning with finite element analysis in materials science: a review
Bibliographic record
Abstract
No abstract is available for this record. The assistant can identify its topic from the title, but cannot infer the study’s results.
Source: Not provided · Quality: missing
Research context
OpenAlex topics: Mineral Processing and Grinding · Machine Learning in Materials Science · Advanced machining processes and optimization
SciVX discovery tags are terminology-based. OpenAlex topics are source-provided classifications.
References (100+)
Show source-provided references
1. Liu WK, Li S, Park HS (2022) Eighty years of the finite element method: birth, evolution, and future. Arch Computat Methods Eng 29:4431–4453. https://doi.org/10.1007/s11831-022-09740-910.1007/s11831-022-09740-9 ↗
2. Axelsson O, Barker VA (2001) The finite element method. In: Finite element solution of boundary value problems: theory and computation. Society for Industrial and Applied Mathematics, Philadelphia, pp 163–26710.1137/1.9780898719253.ch5 ↗
3. Chen JZ (2011) Finite flements for transient and nonlinear problens. In: Finite element method, the: its fundamentals and applications in engineering. World Scientific Publishing Company, Singapore, pp 135–15010.1142/9789814350587_0007 ↗
4. Russell SJ, Norvig P (2016) Machine learning. In: Artificial intelligence: a modern approach. Pearson Education Limited, London, pp 651–789
5. Abioye SO, Oyedele LO, Akanbi L et al (2021) Artificial intelligence in the construction industry: a review of present status, opportunities and future challenges. J Build Eng 44:103299. https://doi.org/10.1016/j.jobe.2021.10329910.1016/j.jobe.2021.103299 ↗
6. Butler KT, Davies DW, Cartwright H et al (2018) Machine learning for molecular and materials science. Nature 559(7715):547–555. https://doi.org/10.1038/s41586-018-0337-210.1038/s41586-018-0337-2 ↗
7. Deiana AMC, Tran N, Agar J et al (2022) Applications and techniques for fast machine learning in science. Fron Big Data 5:787421. https://doi.org/10.3389/fdata.2022.78742110.3389/fdata.2022.787421 ↗
8. Shailaja K, Seetharamulu B, Jabbar MA (2018) Machine learning in healthcare: a review. In: 2018 2nd International conference on electronics, communication and aerospace technology (ICECA), Coimbatore, India pp 910–914, https://doi.org/10.1109/ICECA.2018.8474918.10.1109/ICECA.2018.8474918 ↗
9. Qayyum A, Qadir J, Bilal M et al (2020) Secure and robust machine learning for healthcare: a survey. IEEE Rev Biomed Eng 14:156–180. https://doi.org/10.1109/rbme.2020.301348910.1109/rbme.2020.3013489 ↗
10. Ahmed S, Alshater MM, El Ammari A et al (2022) Artificial intelligence and machine learning in finance: a bibliometric review. Res Int Bus Finance 61:101646. https://doi.org/10.1016/j.ribaf.2022.10164610.1016/j.ribaf.2022.101646 ↗
11. Henrique BM, Sobreiro VA, Kimura H (2019) Literature review: machine learning techniques applied to financial market prediction. Expert Syst Appl 124:226–251. https://doi.org/10.1016/j.eswa.2019.01.01210.1016/j.eswa.2019.01.012 ↗
12. Liakos KG, Busato P, Moshou D et al (2018) Machine learning in agriculture: a review. Sensors 18(8):267410.3390/s18082674 ↗
13. Ghaboussi J (2010) Advances in neural networks in computational mechanics and engineering. Advances of soft computing in engineering. Springer, Vienna, pp 191–23610.1007/978-3-211-99768-0_4 ↗
14. Berg J, Nyström K (2021) Neural networks as smooth priors for inverse problems for PDEs. J Comput Math Data Sci 1:100008. https://doi.org/10.1016/j.jcmds.2021.10000810.1016/j.jcmds.2021.100008 ↗
15. Der Kiureghian A, Ke JB (1988) The stochastic finite element method in structural reliability. Prob Eng Mech 3(2):83–91. https://doi.org/10.1016/0266-8920(88)90019-710.1016/0266-8920(88)90019-7 ↗
16. Okuda H, Yoshimura S, Yagawa G et al (1998) Neural network-based parameter estimation for non-linear finite element analyses. Eng Comput 15(1):103–138. https://doi.org/10.1108/0264440981020072110.1108/02644409810200721 ↗
17. Jenkins WM (1999) A neural network for structural re-analysis. Comput Struct 72(6):687–69810.1016/S0045-7949(98)00311-3 ↗
18. Li S (2000) Global flexibility simulation and element stiffness simulation in finite element analysis with neural network. Comput Methods Appl Mech Eng 186(1):101–108. https://doi.org/10.1016/S0045-7825(99)00108-510.1016/S0045-7825(99)00108-5 ↗
19. Ziemiański L (2003) Hybrid neural network/finite element modelling of wave propagation in infinite domains. Comput Struct 81(8–11):1099–1109. https://doi.org/10.1016/S0045-7949(03)00007-510.1016/S0045-7949(03)00007-5 ↗
20. Gudur PP, Dixit US (2008) A neural network-assisted finite element analysis of cold flat rolling. Eng Appl Artif Intell 21(1):43–52. https://doi.org/10.1016/j.engappai.2006.10.00110.1016/j.engappai.2006.10.001 ↗
21. Umbrello D, Ambrogio G, Filice L et al (2008) A hybrid finite element method–artificial neural network approach for predicting residual stresses and the optimal cutting conditions during hard turning of AISI 52100 bearing steel. Mater Des 29(4):873–883. https://doi.org/10.1016/j.matdes.2007.03.00410.1016/j.matdes.2007.03.004 ↗
22. Daoheng S, Qiao H, Hao X (2000) A neurocomputing model for the elastoplasticity. Comput Methods Appl Mech Eng 182(1–2):177–186. https://doi.org/10.1016/S0045-7825(99)00091-210.1016/S0045-7825(99)00091-2 ↗
23. Li Q, Cheng F, Zhang X (2024) Numerical simulation and deformation prediction of deep pit based on PSO-BP neural network inversion of soil parameters. Sensors 24(10):2959. https://doi.org/10.3390/s2410295910.3390/s24102959 ↗
24. Zhou X, Sun X, Li Y et al (2022) Creep parameter inversion for high CFRDs based on improved BP neural network response surface method. Soft Comput 26(18):9527–9541. https://doi.org/10.1007/s00500-022-06735-310.1007/s00500-022-06735-3 ↗
25. Kang F, Liu X, Li J et al (2022) Multi-parameter inverse analysis of concrete dams using kernel extreme learning machines-based response surface model. Eng Struct 256:113999. https://doi.org/10.1016/j.engstruct.2022.11399910.1016/j.engstruct.2022.113999 ↗
26. Witczak MW, Fonseca OA (1996) Revised predictive model for dynamic (complex) modulus of asphalt mixtures. Transp Res Rec 1540(1):15–23. https://doi.org/10.1177/036119819615400010310.1177/0361198196154000103 ↗
27. Hamim A, Yusoff NIM, Omar HA et al (2020) Integrated finite element and artificial neural network methods for constructing asphalt concrete dynamic modulus master curve using deflection time-history data. Constr Build Mater 257:119549. https://doi.org/10.1016/j.conbuildmat.2020.11954910.1016/j.conbuildmat.2020.119549 ↗
28. Bastos N, Prates PA, Andrade-Campos A (2022) Material parameter identification of elastoplastic constitutive models using machine learning approaches. Key Eng Mater 926:2193–2200. https://doi.org/10.4028/p-zr575d10.4028/p-zr575d ↗
29. Davarpanah S (2023) Calibration of Drucker-Prager-Cap model parameters using particle swarm optimization method for powder compaction process. MSc Dissertation, Bishop’s University
30. Atrian A, Majzoobi GH, Markert B et al (2018) A novel approach to calibrate the Drucker-Prager Cap model for Al7075 powder. Arch Appl Mech 88:1859–1876. https://doi.org/10.1007/s00419-018-1410-x10.1007/s00419-018-1410-x ↗
31. Zirpoli A, Maier G, Novati G et al (2008) Dilatometric tests combined with computer simulations and parameter identification for in-depth diagnostic analysis of concrete dams. In: Life-Cycle Civil Engineering. CRC press, pp 279–28410.1201/9780203885307-41 ↗
32. Park S, Marimuthu KP, Han G et al (2023) Deep learning based nanoindentation method for evaluating mechanical properties of polymers. Int J Mech Sci 246:108162. https://doi.org/10.1016/j.ijmecsci.2023.10816210.1016/j.ijmecsci.2023.108162 ↗
33. Liu GR, Quek SS (2013) FEM for two-dimensional solids. In: The finite element method: a practical course. Butterworth–Heinemann, Oxford, pp 129–171
34. Bohn J, Feischl M (2021) Recurrent neural networks as optimal mesh refinement strategies. Comput Math Appl 97:61–76. https://doi.org/10.1016/j.camwa.2021.05.01810.1016/j.camwa.2021.05.018 ↗
35. Zhang Z, Wang Y, Jimack P K et al (2020) MeshingNet: a new mesh generation method based on deep learning. In: International conference on computational science Springer, Cham, pp 186–19810.1007/978-3-030-50420-5_14 ↗
36. Sprave J, Drescher C (2021) Evaluating the quality of finite element meshes with machine learning. arXiv preprint https://doi.org/10.48550/arXiv.2107.1050710.48550/arXiv.2107.10507 ↗
37. Chan CL, Scholz F, Takacs T (2022) Locally refined quad meshing for linear elasticity problems based on convolutional neural networks. Eng Comput 38(5):4631–4652. https://doi.org/10.1007/s00366-022-01677-810.1007/s00366-022-01677-8 ↗
38. Kirchdoerfer T, Ortiz M (2016) Data-driven computational mechanics. Comput Methods Appl Mech Eng 304:81–101. https://doi.org/10.1016/j.cma.2016.02.00110.1016/j.cma.2016.02.001 ↗
39. Kirchdoerfer T, Ortiz M (2017) Data driven computing with noisy material data sets. Comput Methods Appl Mech Eng 326:622–641. https://doi.org/10.1016/j.cma.2017.07.03910.1016/j.cma.2017.07.039 ↗
40. Korzeniowski TF, Weinberg K (2021) A multi-level method for data-driven finite element computations. Comput Methods Appl Mech Eng 379:113740. https://doi.org/10.1016/j.cma.2021.11374010.1016/j.cma.2021.113740 ↗
41. Böhringer P, Sommer D, Haase T et al (2023) A strategy to train machine learning material models for finite element simulations on data acquirable from physical experiments. Comput Methods Appl Mech Eng 406:115894. https://doi.org/10.1016/j.cma.2023.11589410.1016/j.cma.2023.115894 ↗
42. Carneiro AMC, Alves AFC, Coelho RPC et al (2023) A simple machine learning-based framework for faster multi-scale simulations of path-independent materials at large strains. Finite Elem Anal Des 222:103956. https://doi.org/10.1016/j.finel.2023.10395610.1016/j.finel.2023.103956 ↗
43. Mai HT, Kang J, Lee J (2021) A machine learning-based surrogate model for optimization of truss structures with geometrically nonlinear behavior. Finite Elem Anal Des 196:103572. https://doi.org/10.1016/j.finel.2021.10357210.1016/j.finel.2021.103572 ↗
44. Kim S, Shin H (2023) Deep learning framework for multiscale finite element analysis based on data-driven mechanics and data augmentation. Comput Methods Appl Mech Eng 414:116131. https://doi.org/10.1016/j.cma.2023.11613110.1016/j.cma.2023.116131 ↗
45. Ghavamian F, Simone A (2019) Accelerating multiscale finite element simulations of history-dependent materials using a recurrent neural network. Comput Methods Appl Mech Eng 357:112594. https://doi.org/10.1016/j.cma.2019.11259410.1016/j.cma.2019.112594 ↗
46. Vaswani A, Shazeer N, Parmar N et al (2017) Attention is all you need. In: Proc 31st Int conf on neural information processing systems, pp 6000–6010
47. Kim S, Shin H (2025) Accelerating the data-driven multiscale finite element analysis for elastoplastic materials by using proper orthogonal decomposition and transformer architecture. Comput Methods Appl Mech Eng 437:117827. https://doi.org/10.1016/j.cma.2025.11782710.1016/j.cma.2025.117827 ↗
48. Lu Y, Li H, Saha S et al (2021) Reduced order machine learning finite element methods: concept, implementation, and future applications. Comput Model Eng Sci 129(1):1351–1371. https://doi.org/10.32604/cmes.2021.01771910.32604/cmes.2021.017719 ↗
49. Liu Z, Wu CT, Koishi M (2019) A deep material network for multiscale topology learning and accelerated nonlinear modeling of heterogeneous materials. Comput Methods Appl Mech Eng 345:1138–1168. https://doi.org/10.1016/j.cma.2018.09.02010.1016/j.cma.2018.09.020 ↗
50. Chen G, Li T, Chen Q et al (2019) Application of deep learning neural network to identify collision load conditions based on permanent plastic deformation of shell structures. Comput Mech 64:435–449. https://doi.org/10.1007/s00466-019-01706-210.1007/s00466-019-01706-2 ↗
51. Bruski D, Pachocki L, Sciegaj A et al (2023) Speed estimation of a car at impact with a W-beam guardrail using numerical simulations and machine learning. Adv Eng Softw 184:103502. https://doi.org/10.1016/j.advengsoft.2023.10350210.1016/j.advengsoft.2023.103502 ↗
52. Yang C, Meng K, Yang L et al (2023) Transfer learning-based crashworthiness prediction for the composite structure of a subway vehicle. Int J Mech Sci 248:108244. https://doi.org/10.1016/j.ijmecsci.2023.10824410.1016/j.ijmecsci.2023.108244 ↗
53. Wriggers P (2003) Computational contact mechanics. Comput Mech 32(1–2):141–141. https://doi.org/10.1016/j.ijmecsci.2023.10824410.1016/j.ijmecsci.2023.108244 ↗
54. Oishi A, Yoshimura S (2007) A new local contact search method using a multi-layer neural network. Comput Model Eng Sci 21(2):93. https://doi.org/10.3970/cmes.2007.021.09310.3970/cmes.2007.021.093 ↗
55. Oishi A, Yagawa G (2020) A surface-to-surface contact search method enhanced by deep learning. Comput Mech 65(4):1125–1147. https://doi.org/10.1007/s00466-019-01811-210.1007/s00466-019-01811-2 ↗
56. Polat A (2023) Estimation of contact lengths using deep neural network. Gumushane Univ J Sci 13(2):458–470. https://doi.org/10.17714/gumusfenbil.112222510.17714/gumusfenbil.1122225 ↗
57. Hattori G, Serpa AL (2015) Contact stiffness estimation in ANSYS using simplified models and artificial neural networks. Finite Elem Anal Des 97:43–53. https://doi.org/10.1016/j.finel.2015.01.00310.1016/j.finel.2015.01.003 ↗
58. Zhou JM, Dong L, Guan W et al (2019) Impact load identification of nonlinear structures using deep recurrent neural network. Mech Syst Signal Process 133:106292. https://doi.org/10.1016/j.ymssp.2019.10629210.1016/j.ymssp.2019.106292 ↗
59. Kalliorinne K, Larsson R, Pérez-Ràfols F et al (2021) Artificial neural network architecture for prediction of contact mechanical response. Front Mech Eng 6:579825. https://doi.org/10.3389/fmech.2020.57982510.3389/fmech.2020.579825 ↗
60. Furlan M, Mavros G (2022) A neural network approach for roughness-dependent update of tyre friction. Simul Model Pract Theory 116:102484. https://doi.org/10.1016/j.simpat.2021.10248410.1016/j.simpat.2021.102484 ↗
61. Bathe KJ (2006) Vectors, matrices, and tensors. In: Finite element procedures. Pearson Education, Watertown, pp 17–72
62. Jung J, Yoon K, Lee PS (2020) Deep learned finite elements. Comput Methods Appl Mech Eng 372:113401. https://doi.org/10.1016/j.cma.2020.11340110.1016/j.cma.2020.113401 ↗
63. Han X, Sun X, Chen X (2023) Locally assembled stiffness matrix: a novel method to obtain global stiffness matrix. Acta Mech 234(6):2461–2480. https://doi.org/10.1007/s00707-023-03507-410.1007/s00707-023-03507-4 ↗
64. Zienkiewicz OC, Taylor RL, Zhu JZ (2005) Plate bending approximation: thin and thick plates. In: The finite element method: its basis and fundamentals. Elsevier, Amsterdam, pp 407–466
65. Jung J, Jun H, Lee PS (2022) Self-updated four-node finite element using deep learning. Comput Mech 69(1):23–44. https://doi.org/10.1007/s00466-021-02081-710.1007/s00466-021-02081-7 ↗
66. Tandale SB, Markert B, Stoffel M (2022) Smart stiffness computation of one-dimensional finite elements. Mech Res Commun 119:103817. https://doi.org/10.1016/j.mechrescom.2021.10381710.1016/j.mechrescom.2021.103817 ↗
67. Balkrishna Tandale S, Markert B, Stoffel M (2022) Intelligent stiffness computation for plate and beam structures by neural network enhanced finite element analysis. Int J Numer Methods Eng 123(17):4001–4031. https://doi.org/10.1002/nme.699610.1002/nme.6996 ↗
68. Capuano G, Rimoli JJ (2019) Smart finite elements: a novel machine learning application. Comput Methods Appl Mech Eng 345:363–381. https://doi.org/10.1016/j.cma.2018.10.04610.1016/j.cma.2018.10.046 ↗
69. Raissi M (2018) Deep hidden physics models: deep learning of nonlinear partial differential equations. J Mach Learn Res 19(25):1–24. https://doi.org/10.48550/arXiv.1801.0663710.48550/arXiv.1801.06637 ↗
70. Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J Comput Phys 378:686–707. https://doi.org/10.1016/j.jcp.2018.10.04510.1016/j.jcp.2018.10.045 ↗
71. Roy AM, Guha S, Sundararaghavan V et al (2024) Physics-infused deep neural network for solution of non-associative Drucker-Prager elastoplastic constitutive model. J Mech Phys Solids 185:105570. https://doi.org/10.1016/j.jmps.2024.10557010.1016/j.jmps.2024.105570 ↗
72. Guo J, Yao Y, Wang H et al (2023) Pre-training strategy for solving evolution equations based on physics-informed neural networks. J Comput Phys 489:112258. https://doi.org/10.1016/j.jcp.2023.11225810.1016/j.jcp.2023.112258 ↗
73. Wang J, Mo YL, Izzuddin B et al (2023) Exact Dirichlet boundary physics-informed neural network EPINN for solid mechanics. Comput Methods Appl Mech Eng 414:116184. https://doi.org/10.1016/j.cma.2023.11618410.1016/j.cma.2023.116184 ↗
74. Choi J, Kim N, Hong Y (2023) Unsupervised Legendre-Galerkin neural network for solving partial differential equations. IEEE Access 11:23433–23446. https://doi.org/10.1109/ACCESS.2023.324468110.1109/ACCESS.2023.3244681 ↗
75. Baharlouei S, Mokhtari R, Mostajeran F (2023) DNN-HDG: a deep learning hybridized discontinuous Galerkin method for solving some elliptic problems. Eng Anal Bound Elem 151:656–669. https://doi.org/10.1016/j.enganabound.2023.03.03910.1016/j.enganabound.2023.03.039 ↗
76. Abueidda DW, Lu Q, Koric S (2021) Meshless physics-informed deep learning method for three-dimensional solid mechanics. Int J Numer Methods Eng 122(23):7182–7201. https://doi.org/10.1002/nme.682810.1002/nme.6828 ↗
77. Liu M, Cai Z, Chen J (2022) Adaptive two-layer ReLU neural network: I. Best least-squares approximation. Comput Math Appl 113:34–44. https://doi.org/10.1016/j.camwa.2022.03.00510.1016/j.camwa.2022.03.005 ↗
78. Liu M, Cai Z (2022) Adaptive two-layer ReLU neural network: II. Ritz approximation to elliptic PDEs. Comput Math Appl 113:103–116. https://doi.org/10.1016/j.camwa.2022.03.01010.1016/j.camwa.2022.03.010 ↗
79. Zuo Y, You H, Zou X et al (2024) Digital twin enhanced quality prediction method of powder compaction process. Robotics Comput Integr Manuf 89:102762. https://doi.org/10.1016/j.rcim.2024.10276210.1016/j.rcim.2024.102762 ↗
80. Jafar-Salehi E, Ghasempoor A, Eslamian M (2013) Experimental study and predictive modelling of cold compaction green density in powder metallurgy of stainless steel components. Powder Metall 56(3):208–515. https://doi.org/10.1179/1743290112Y.000000004710.1179/1743290112Y.0000000047 ↗
81. Merayo D, Rodríguez-Prieto A, Camacho AM (2020) Prediction of physical and mechanical properties for metallic materials selection using big data and artificial neural networks. IEEE access 8:13444–13456. https://doi.org/10.1109/ACCESS.2020.296576910.1109/ACCESS.2020.2965769 ↗
82. Trower M, Emerson J, Yu M et al (2023) Reduced-order hybrid modelling for powder compaction: predicting density and classifying diametrical hardness. Powder Technol 427:118745. https://doi.org/10.1016/j.powtec.2023.11874510.1016/j.powtec.2023.118745 ↗
83. Chen Q, Kong L, Dugast F et al (2023) Using the transformer model for physical simulation: an application on transient thermal analysis for 3D printing process simulation. In: NeurIPS 2023 AI for science workshop
84. Ahmadi M, Ansari R, Hassanzadeh-Aghdam MK (2023) Micromechanical finite element analysis of Young’s modulus, yield strength and thermal expansion coefficient of nano-sized ceramic particle/metal matrix nanocomposites. J Braz Soc Mech Sci Eng 45(9):478. https://doi.org/10.1007/s40430-023-04385-610.1007/s40430-023-04385-6 ↗
85. Yang Z, Yu CH, Buehler MJ (2021) Deep learning model to predict complex stress and strain fields in hierarchical composites. Sci Adv 7(15):eabde416. https://doi.org/10.1126/sciadv.abd74110.1126/sciadv.abd741 ↗
86. Yang Z, Yu CH, Guo K, Buehler MJ (2021) End-to-end deep learning method to predict complete strain and stress tensors for complex hierarchical composite microstructures. J Mech PhysSolids 154:104506. https://doi.org/10.1016/j.jmps.2021.10450610.1016/j.jmps.2021.104506 ↗
87. Cheng J, Ling Y, De Waele W (2024) An ANN hardness prediction tool based on a finite element implementation of a thermal-metallurgical model for mild steel produced by WAAM. Metals 14(5):556. https://doi.org/10.3390/met1405055610.3390/met14050556 ↗
88. Hadhri M, El Ouafi A, Barka N (2017) Prediction of the hardness profile of an AISI 4340 steel cylinder heat-treated by laser-3D and artificial neural networks modelling and experimental validation. J Mech Sci Technol 31:615–623. https://doi.org/10.1007/s12206-017-0114-410.1007/s12206-017-0114-4 ↗
89. Quarto M, Bocchi S, D’Urso G et al (2022) Hybrid finite elements method-artificial neural network approach for hardness prediction of AA6082 friction stir welded joints. Int J Mechatron Manuf Syst 15(2–3):149–166. https://doi.org/10.1504/IJMMS.2022.12491910.1504/IJMMS.2022.124919 ↗
90. Karina CN, Chun PJ, Okubo K (2017) Tensile strength prediction of corroded steel plates by using machine learning approach. Steel Compos Struct 24(5):635–641. https://doi.org/10.12989/scs.2017.24.5.63510.12989/scs.2017.24.5.635 ↗
91. Huang DJ, Li H (2021) A machine learning guided investigation of quality repeatability in metal laser powder bed fusion additive manufacturing. Mater Des 203:109606. https://doi.org/10.1016/j.matdes.2021.10960610.1016/j.matdes.2021.109606 ↗
92. Silva GC, Beber VC, Pitz DB (2021) Machine learning and finite element analysis: an integrated approach for fatigue lifetime prediction of adhesively bonded joints. Fatigue Fract Eng Mater Struct 44(12):3334–3348. https://doi.org/10.1111/ffe.1355910.1111/ffe.13559 ↗
93. He L, Wang ZL, Akebono H et al (2021) Machine learning-based predictions of fatigue life and fatigue limit for steels. J Mater Sci Technol 90:9–19. https://doi.org/10.1016/j.jmst.2021.02.02110.1016/j.jmst.2021.02.021 ↗
94. Lee JW, Park C, Do Lee B et al (2021) A machine-learning-based alloy design platform that enables both forward and inverse predictions for thermo-mechanically controlled processed (TMCP) steel alloys. Sci Rep 11(1):11012. https://doi.org/10.1038/s41598-021-90237-z10.1038/s41598-021-90237-z ↗
95. Yang F, Li Z, Wang Q et al (2020) Cluster-formula-embedded machine learning for design of multicomponent β-Ti alloys with low Young’s modulus. npj Comput Mater 6(1):101. https://doi.org/10.1038/s41524-020-00372-w10.1038/s41524-020-00372-w ↗
96. Shen SC, Buehler MJ (2022) Nature-inspired architected materials using unsupervised deep learning. Commun Eng 1(1):37. https://doi.org/10.1038/s44172-022-00037-010.1038/s44172-022-00037-0 ↗
97. Liu W, Xu G, Fan W et al (2024) Machine-learning-based characterization and inverse design of metamaterials. Materials 17(14):3512. https://doi.org/10.3390/ma1714351210.3390/ma17143512 ↗
98. Hamel CM, Roach DJ, Long KN et al (2019) Machine-learning based design of active composite structures for 4D printing. Smart Mater Struct 28(6):065005. https://doi.org/10.1088/1361-665X/ab143910.1088/1361-665X/ab1439 ↗
99. Bendsøe MP, Kikuchi N (1988) Generating optimal topologies in structural design using a homogenization method. Comput Methods Appl Mech Eng 71(2):197–224. https://doi.org/10.1016/0045-7825(88)90086-210.1016/0045-7825(88)90086-2 ↗
100. Shin S, Shin D, Kang N (2023) Topology optimization via machine learning and deep learning: a review. J Comput Design Eng 10(4):1736–1766. https://doi.org/10.1093/jcde/qwad07210.1093/jcde/qwad072 ↗