REFERENCES
1. Chen, L.; Zhao, Y. From classical thermodynamics to phase-field method. Prog. Mater. Sci. 2022, 124, 100868.
2. Zhao, Y. Understanding and design of metallic alloys guided by phase-field simulations. npj. Comput. Mater. 2023, 9, 94.
3. Mancias, J.; Vela, B.; Flórez-coronel, J.; et al. Mapping of microstructure transitions during rapid alloy solidification using Bayesian-guided phase-field simulations. Acta. Mater. 2025, 297, 121354.
4. Tang, Y.; Zhang, A.; Liu, H.; et al. Insight into scale selection of dimensionless phase-field model of alloy solidification. Mater. Des. 2025, 254, 114028.
5. Blixt, K. H.; Hallberg, H. Phase field crystal modeling of grain boundary migration: mobility, energy and structural variability. Acta. Mater. 2025, 297, 121318.
6. Jiang, L.; Liu, G.; Jin, P.; Shen, Y.; Wang, J. The driving force for twin boundary migration in phase field model coupled to crystal plasticity finite element. Int. J. Plast. 2025, 191, 104397.
7. Hu, J.; Xu, B.; Xiong, J.; Yu, C.; Kang, G. Deformation mechanism of non-textured and basal-textured polycrystalline Mg alloys: a coupled crystal plasticity-twinning phase field simulation. Int. J. Plast. 2025, 188, 104312.
8. Chen, J.; Guo, M.; Yang, M.; Cui, T.; Cui, B.; Zhang, J. Insight into element segregation mechanisms during creep in γʹ-strengthened Co-based superalloy by elastoplastic phase-field simulation. Prog. Nat. Sci. Mater. Int. 2024, 34, 873-9.
9. Zhang, H.; Zhang, Z.; Wang, S.; et al. Neutron irradiation and stress-induced creep rafting of α′ phase in Fe-Cr-Al alloy: crystal plasticity multi-phase-field simulation. J. Mater. Sci. Technol. 2026, 249, 164-73.
11. Allen, S. M.; Cahn, J. W. A correction to the ground state of FCC binary ordered alloys with first and second neighbor pairwise interactions. Scr. Metall. 1973, 7, 1261-4.
12. Doghman, J.; Bovet, C.; Ask, A. A finite element-based simulation of microstructure evolution through a 3D finite strain Cosserat phase-field model. Comput. Methods. Appl. Mech. Eng. 2025, 439, 117900.
13. Cao, W.; Yang, H.; Chen, W. An exponential time differencing Runge-Kutta method ETDRK32 for phase field models. J. Sci. Comput. 2024, 99, 6.
14. Krischok, A.; Yaraguntappa, B.; Keip, M. Fast implicit update schemes for Cahn-Hilliard-type gradient flow in the context of Fourier-spectral methods. Comput. Methods. Appl. Mech. Eng. 2024, 431, 117220.
15. Gültekin, O.; Dal, H.; Holzapfel, G. A. A phase-field approach to model fracture of arterial walls: theory and finite element analysis. Comput. Methods. Appl. Mech. Eng. 2016, 312, 542-66.
16. Valizadeh, N.; Zhuang, X.; Rabczuk, T. A monolithic finite element method for phase-field modeling of fully Eulerian fluid-structure interaction. Comput. Methods. Appl. Mech. Eng. 2025, 435, 117618.
17. Kristensen, P. K.; Martínez-pañeda, E. Phase field fracture modelling using quasi-Newton methods and a new adaptive step scheme. Theor. Appl. Fract. Mech. 2020, 107, 102446.
18. Liao, H.; Ji, B.; Zhang, L. An adaptive BDF2 implicit time-stepping method for the phase field crystal model. IMA. J. Numer. Anal. 2022, 42, 649-79.
19. Shi, X.; Huang, H.; Cao, G.; Ma, X. Accelerating large-scale phase-field simulations with GPU. AIP. Adv. 2017, 7, 105216.
20. Li, H.; Song, Z. A reduced-order finite element method based on proper orthogonal decomposition for the Allen-Cahn model. J. Math. Anal. Appl. 2021, 500, 125103.
21. Zhang, A.; Guo, Z.; Jiang, B.; Xiong, S.; Pan, F. Numerical solution to phase-field model of solidification: a review. Comput. Mater. Sci. 2023, 228, 112366.
22. Butler, K. T.; Davies, D. W.; Cartwright, H.; Isayev, O.; Walsh, A. Machine learning for molecular and materials science. Nature 2018, 559, 547-55.
23. He, Y.; De Breuck, P.; Weng, H.; Giantomassi, M.; Rignanese, G. Machine learning on multiple topological materials datasets. npj. Comput. Mater. 2025, 11, 181.
24. Li, K.; Rubungo, A. N.; Lei, X.; et al. Probing out-of-distribution generalization in machine learning for materials. Commun. Mater. 2025, 6, 9.
25. Tao, Q.; Yang, X.; Bao, L.; et al. Transforming machine learning model knowledge into material insights for multi-principal-element superalloy phase design. npj. Comput. Mater. 2025, 11, 99.
26. Barik, R. K.; Woods, L. M. Frictional properties of two-dimensional materials: data-driven machine learning predictive modeling. ACS. Appl. Mater. Interfaces. 2024, 16, 40149-59.
27. Qin, G.; Wei, Y.; Yu, L.; et al. Predicting lattice thermal conductivity from fundamental material properties using machine learning techniques. J. Mater. Chem. A. 2023, 11, 5801-10.
28. Kim, C.; Pilania, G.; Ramprasad, R. Machine learning assisted predictions of intrinsic dielectric breakdown strength of ABX3 perovskites. J. Phys. Chem. C. 2016, 120, 14575-80.
29. Dai, M.; Demirel, M. F.; Liang, Y.; Hu, J. Graph neural networks for an accurate and interpretable prediction of the properties of polycrystalline materials. npj. Comput. Mater. 2021, 7, 103.
30. Li, Z.; Hernández, F. J.; Salguero, C.; Lopez, S. A.; Crespo-otero, R.; Li, J. Machine learning photodynamics decode multiple singlet fission channels in pentacene crystal. Nat. Commun. 2025, 16, 1194.
31. Liao, W.; Yuan, R.; Xue, X.; Wang, J.; Li, J.; Lookman, T. Unsupervised learning-aided extrapolation for accelerated design of superalloys. npj. Comput. Mater. 2024, 10, 171.
32. Liu, P.; Huang, H.; Antonov, S.; et al. Machine learning assisted design of γ′-strengthened Co-base superalloys with multi-performance optimization. npj. Comput. Mater. 2020, 6, 62.
33. Deng, Y.; Zhang, Y.; Gong, X.; et al. An intelligent design for Ni-based superalloy based on machine learning and multi-objective optimization. Materials. &. Design. 2022, 221, 110935.
34. Ye, W.; Chen, C.; Wang, Z.; Chu, I.; Ong, S. P. Deep neural networks for accurate predictions of crystal stability. Nat. Commun. 2018, 9, 3800.
35. Azimi, S. M.; Britz, D.; Engstler, M.; Fritz, M.; Mücklich, F. Advanced steel microstructural classification by deep learning methods. Sci. Rep. 2018, 8, 2128.
36. Maiti, C.; Muthuswamy, S. Classification of materials in cylindrical workpieces using image processing and machine learning techniques. Int. J. Prod. Res. 2023, 62, 2566-83.
37. Cao, Y.; Zhang, C.; Tang, S.; et al. Machine learning to predict phase transformation products and their morphologies - application in design of lean high strength steel. Mater. Des. 2025, 258, 114642.
38. Lee, K.; Ayyasamy, M. V.; Delsa, P.; Hartnett, T. Q.; Balachandran, P. V. Phase classification of multi-principal element alloys via interpretable machine learning. npj. Comput. Mater. 2022, 8, 25.
39. Zhang, X.; Sun, J.; Zhang, Y.; et al. Machine learning for phase prediction of high entropy carbide ceramics from imbalanced data. npj. Comput. Mater. 2026, 12, 22.
40. Malashin, I.; Martysyuk, D. Powder particle classification with scanning electron microscopy images using machine learning techniques. Expert. Syst. Appl. 2025, 286, 128001.
41. He, J.; Zhou, Y.; Zhang, P.; Zhai, H. Machine learning assisted fast optical identification of 2D materials. Appl. Surf. Sci. 2025, 686, 162110.
42. Stan, T.; Thompson, Z. T.; Voorhees, P. W. Optimizing convolutional neural networks to perform semantic segmentation on large materials imaging datasets: X-ray tomography and serial sectioning. Mater. Charact. 2020, 160, 110119.
43. Dingreville, R.; Roberston, A. E.; Attari, V.; et al. Benchmarking machine learning strategies for phase-field problems. Modelling. Simul. Mater. Sci. Eng. 2024, 32, 065019.
44. Oommen, V.; Shukla, K.; Desai, S.; Dingreville, R.; Karniadakis, G. E. Rethinking materials simulations: blending direct numerical simulations with neural operators. npj. Comput. Mater. 2024, 10, 145.
45. Fetni, S.; Pham, T. Q. D.; Hoang, T. V.; et al. Capabilities of auto-encoders and principal component analysis of the reduction of microstructural images; application on the acceleration of phase-field simulations. Comput. Mater. Sci. 2023, 216, 111820.
46. Hu, C.; Martin, S.; Dingreville, R. Accelerating phase-field predictions via recurrent neural networks learning the microstructure evolution in latent space. Comput. Methods. Appl. Mech. Eng. 2022, 397, 115128.
47. Wu, P.; Iquebal, A. S.; Ankit, K. Emulating microstructural evolution during spinodal decomposition using a tensor decomposed convolutional and recurrent neural network. Comput. Mater. Sci. 2023, 224, 112187.
48. Ahmad, O.; Kumar, N.; Mukherjee, R.; Bhowmick, S. Accelerating microstructure modeling via machine learning: a method combining Autoencoder and ConvLSTM. Phys. Rev. Mater. 2023, 7, 083802.
49. Gesch, A.; Hu, C. Accelerating phase-field simulation of coupled microstructural evolution using autoencoder-based recurrent neural networks. J. Mater. Inf. 2025, 5.
50. Oommen, V.; Shukla, K.; Goswami, S.; Dingreville, R.; Karniadakis, G. E. Learning two-phase microstructure evolution using neural operators and autoencoder architectures. npj. Comput. Mater. 2022, 8, 190.
51. Teichert, G. H.; Garikipati, K. Machine learning materials physics: surrogate optimization and multi-fidelity algorithms predict precipitate morphology in an alternative to phase field dynamics. Comput. Methods. Appl. Mech. Eng. 2019, 344, 666-93.
52. Jokisaari, A.; Voorhees, P.; Guyer, J.; Warren, J.; Heinonen, O. Benchmark problems for numerical implementations of phase field models. Comput. Mater. Sci. 2017, 126, 139-51.
53. Wang, Y.; Gao, Z.; Long, M.; Wang, J.; Yu, P. S. PredRNN++: towards a resolution of the deep-in-time dilemma in spatiotemporal predictive learning. In International Conference on Machine Learning, Stockholm, Sweden, July 10-15, 2018; PMLR: 2018; Vol. 80, pp 5123-32. https://proceedings.mlr.press/v80/wang18b.html. (accessed 2026-05-21).
54. Wang, Y.; Long, M.; Wang, J.; Gao, Z.; Yu, P. S. PredRNN: Recurrent neural networks for predictive learning using spatiotemporal LSTMs. In Advances in Neural Information Processing Systems, Long Beach, CA, USA, December 4-9, 2017; Curran Associates, Inc.: Red Hook, NY, USA, 2017; Vol. 30, pp 879-88. https://proceedings.neurips.cc/paper/2017/hash/e. (accessed 2026-05-21).
55. Wang, Y.; Wu, H.; Zhang, J.; et al. PredRNN: a recurrent neural network for spatiotemporal predictive learning. IEEE. Trans. Pattern. Anal. Mach. Intell. 2023, 45, 2208-25.
56. Rajagukguk, R. A.; Lee, H. Sky-image-based sun-blocking index and PredRNN++ for accurate short-term solar irradiance forecasting. Build. Environ. 2025, 274, 112429.
57. Kazemzadeh Farizhandi, A. A.; Mamivand, M. Spatiotemporal prediction of microstructure evolution with predictive recurrent neural network. Comput. Mater. Sci. 2023, 223, 112110.
58. Yang, K.; Cao, Y.; Zhang, Y.; et al. Self-supervised learning and prediction of microstructure evolution with convolutional recurrent neural networks. Patterns 2021, 2, 100243.
59. Wang, S.; Wang, H.; Perdikaris, P. Learning the solution operator of parametric partial differential equations with physics-informed DeepONets. Sci. Adv. 2021, 7, eabi8605.
60. De, S.; Reynolds, M.; Hassanaly, M.; King, R. N.; Doostan, A. Bi-fidelity modeling of uncertain and partially unknown systems using DeepONets. Comput. Mech. 2023, 71, 1251-67.
