1. Zhang, Y. Lu, S. Tang, WK. Liu, HiDeNN-TD: Reduced-order hierarchical deep learning neural networks, Computer Methods in Applied Mechanics and Engineering, 389, (February 2022) 114414. https://doi.org/10.1016/j.cma.2021.114414.
2. Lu, Y., Li, H., Saha, S., Mojumder, S., Al Amin, A., Suarez, D., Liu, Y., Qian, D. and Liu, WK., 2021. Reduced Order Machine Learning Finite Element Methods: Concept, Implementation, and Future Applications. Computer Modeling in Engineering & Sciences,129(1).
3. Mozaffar, M., Liao, S., Xie, X., Saha, S., Park, C., Cao, J., Liu, W.K. and Gan, Z., Mechanistic artificial intelligence (Mechanistic-AI) for modeling, design, and control of advanced manufacturing processes: Current state and perspectives. Journal of Materials Processing Technology, Volume 302, April 2022, 117485.
4. Xiaoyu Xie, Wing Kam Liu, Zhengtao Gan, “Data-driven discovery of dimensionless numbers and scaling laws from experimental measurements,”
   arXiv:2111.03583[physics.flu-dyn].
5. Daoping Liu, Hang Yang, I. Elkhodary, Shan Tang, Wing Kam Liu, Xu Guo, “Mechanistically informed data-driven modeling of cyclic plasticity via artificial neural networks” Comput. Methods Appl. Mech. Engrg. 393 (2022) 114766
6. Liu, W.K., Li, S. and Park, H., 2021. Eighty Years of the Finite Element Method: Birth, Evolution, and Future. arXiv e-prints, pp.arXiv-2107.
7. Liu, W.K., Li, S. and Park, H.S., 2022. Eighty years of the finite element method: Birth, evolution, and future. Archives of Computational Methods in Engineering, pp.1-23.
8. Gao, S. Mojumder, W. Zhang, H. Li, D. Suarez, C. He, J. Cao, WK Liu, “Concurrent n-scale modeling for non-orthogonal woven composite,” Computational Mechanics, June 2022, https://doi.org/10.1007/s00466-022-02199-2.
9. Hannah Huang, Satyajit Mojumder, Derick Suarez, Abdullah Al Amin, Mark Fleming, Wing Kam Liu, “Knowledge database creation for design of polymer matrix composite,” Computational Materials Science, Volume 214, 2022,111703. https://doi.org/10.1016/j.commatsci.2022.111703.
10. D Bishara, Y Xie, WK Liu, S Li , “A State-of-the-Art Review on Machine Learning-Based Multiscale Modeling, Simulation, Homogenization and Design of Materials,” Archives of Computational Methods in Engineering, 1-32,
11. Orion L. Kafka^,, Cheng Yu, Puikei Cheng, Sarah J. Wolff, Jennifer L. Bennett, Edward J. Garboczi, Jian Cao, Xianghui Xiao, Wing Kam Liu, “X-ray computed tomography analysis of pore deformation in IN718 made with directed energy deposition via in-situ tensile testing,” International Journal of Solids and Structures, Volume 256, 1 December 2022, 111943.
12. Mahsa Tajdari, Farzam Tajdari, Pouyan Shirzadian, Aishwarya Pawar, Mirwais Wardak, Sourav Saha, Chanwook Park, Toon Huysmans, Yu Song, Yongjie Jessica Zhang, John F. Sarwark, Wing Kam Liu. “Next-Generation Prognosis Framework for Pediatric Spinal Deformities Using Bio-informed Deep Learning Networks” Special Issue of Image-Based Methods in Computational Medicine in Engineering with Computers, 2022. DOI: 10.1007/s00366-022-01742-2
13. Xiaoyu Xie, Arash Samaei, Jiachen Guo, Wing Kam Liu, Zhengtao Gan, (2022) “Data-driven discovery of dimensionless numbers and governing laws from scarce measurements, Nature Communications, 13, 7562 (2022).  https://doi.org/10.1038/s41467-022-35084-w
14. Tianju Xue, Shuheng Liao, Zhengtao Gan, Chanwook Park, Xiaoyu Xie, Wing Kam Liu, Jian Cao, “JAX-FEM: A differentiable GPU-accelerated 3D finite element solver for automatic inverse design and mechanistic data science,”
    arXiv:2212.00964,
15. Saha, S., Park, C., Knapik, S., Guo. J., Huang, O., Liu, W. K.,Deep Learning Discrete Calculus (DLDC): a family of discrete numerical methods by universal approximation for STEM education to frontier research. Comput Mech (2023). https://doi.org/10.1007/s00466-023-02292-0
16. Liu, C. Park, Y. Lu, S. Mojumder, W. K. Liu, and D. Qian, “HiDeNN-FEM: A seamless machine learning approach to nonlinear finite element analysis,” Accepted, Computational Mechanics, 2023.
17. Satyajit Mojumder, Zhengtao Gan, Yangfan Li, Abdullah Al Amin, Wing Kam Liu, “Linking process parameters with lack-of-fusion porosity for laser powder bed fusion metal additive manufacturing,” Additive Manufacturing 68 (2023) 103500.
18. Lyu, Y., Pathirage, M., Ramyar, E., Liu, W., Cusatis, G., Machine learning meta-models for fast parameter identification of the lattice discrete particle model. Comput Mech(2023). https://doi.org/10.1007/s00466-023-02320-z.
19. Huang, O., Saha, S., Guo, J. et al.An introduction to kernel and operator learning methods for homogenization by self-consistent clustering analysis. Comput Mech (2023). https://doi.org/10.1007/s00466-023-02331-w

1. Wing Kam Liu, Zhengtao Gan, Mark A. Fleming, “Mechanistic Data Science for STEM Education and Applications,” Springer, 2021.

https://www.springer.com/us/book/9783030878313#aboutBook https://www.springer.com/gp/book/9783030878313 

2. Ted Belytschko, Wing Kam Liu, Brian Moran, and Khalil Elkhodary, “Nonlinear Finite Elements for Continua and Structures,” Second Edition, John Wiley & Sons, Ltd, December, 2013.
3. Ted Belytschko, Wing Kam Liu, and Brian Moran, “Nonlinear Finite Elements for Continua and Structures,” John Wiley & Sons, Ltd, Chichester, 660pp, 2000.
4. Shaofan Li and Wing Kam Liu, “Meshfree Particle Methods,” Springer, 502pp. 2004.
5. K. Liu, E. Karpov, H. Park, Nano Mechanics and Materials: Theory, Multiscale Methods and Applications, John Wiley and Sons, 2006.

Mechanistic Data Science for STEM Education and Applications

This book introduces Mechanistic Data Science (MDS) as a structured methodology for combining data science tools with mathematical scientific principles (i.e., “mechanistic” principles) to solve intractable problems.  Traditional data science methodologies require copious quantities of data to show a reliable pattern, but the amount of required data can be greatly reduced by considering the mathematical science principles. MDS is presented here in six easy-to-follow modules: 1) Multimodal data generation and collection, 2) extraction of mechanistic features, 3) knowledge-driven dimension reduction, 4) reduced order surrogate models, 5) deep learning for regression and classification, and 6) system and design. These data science and mechanistic analysis steps are presented in an intuitive manner that emphasizes practical concepts for solving engineering problems as well as real-life problems. This book is written in a spectral style and is ideal as an entry level textbook for engineering and data science undergraduate and graduate students, practicing scientists and engineers, as well as STEM (Science, Technology, Engineering, Mathematics) high school students and teachers.

US link to the textbook: https://www.springer.com/us/book/9783030878313#aboutBook

EU link to the textbook: https://www.springer.com/gp/book/9783030878313