Law-based and data-based methods: stability issues revisited with PINNs

Gui-Rong Liu

University of Cincinnati

Abstract: Machine learning methods such as Artificial Neural Networks (ANNs) [1][2] and Physics-Informed Neural Networks (PINNs)[3], have been successfully applied to a broad range of problems in science and engineering. Recently, the author proposed TrumpetNets and TubeNets [4][5] two novel classes of two-way deep networks trained using the standard Finite Element Method (FEM) [6],  the Smoothed Finite Element Method (S-FEM) (S-FEM) [7], and PINN formulations based on partial differential equations (PDEs). These approaches have been applied to static, dynamic, linear, and nonlinear mechanics problems [3]. 

This talk explores the fundamental differences between law-based and data-based methods, focusing on their underlying principles, computational procedures, predictive capabilities, and key properties across various problem categories. Insights into the stability characteristics of these two classes of methods are presented, highlighting that techniques effective for one may not necessarily perform well for the other.

[1] Liu GR, Machine Learning with Python: Theory and Applications,  World Scientific, in-printing, 2021.
[2] Liu GR, A Neural Element Method, IJCM 17 (07), 2050021, 2020.
[3] Liu GR, PINN with Python: An Introduction, ScienTech Publisher, 2025.
[4] Liu GR, S.Y. Duan, Z.M. Zhang and X. Han, TubeNet: A Special TrumpetNet for Explicit Solutions to Inverse Problems, International Journal of Computational Methods (IJCM), in printing (2020).
[5] Liu, GR, FEA-AI and AI-AI: Two-Way Deepnets for Real-Time Computations for Both Forward and Inverse Mechanics Problems, IJCM, Vol. 16, No. 08, 1950045 (2019).
[6] Liu G.R., and Quek, S.S., Finite Element Method: a practical course, BH, Burlington, MA, 2003.
[7] Liu G.R. and Nguyen-Thoi T, Smoothed Finite Element Methods, CRC Press: Boca Raton, 2010.

Bio: GR Liu received his Ph.D. from Tohoku University, Japan, in 1991. He was a Postdoctoral Fellow at Northwestern University, USA, from 1991–1993. He was a Professor at the National University of Singapore until 2010. He is currently a Professor at the Department of Aerospace Engineering and Engineering Mechanics, University of Cincinnati, USA. He was the Founder of the Association for Computational Mechanics (Singapore) (SACM) and served as the President of SACM until 2010. He served as the President of the Asia-Pacific Association for Computational Mechanics (APACM) (2010–2013) and an Executive Council Member of the International Association for Computational Mechanics (IACM) (2005–2010; 2020–2026). He authored over 600 journal papers and more than 15 books, including two bestsellers, Mesh Free Method: Moving Beyond the Finite Element Method and Smoothed Particle Hydrodynamics: A Meshfree Particle Methods. He is the Editor-in-Chief of the International Journal of Computational Methods and served as an Associate Editor for IPSE and MANO. He is the recipient of numerous awards, including the Singapore Defence Technology Prize, NUS Outstanding University Researcher Award, NUS Best Teacher Award, APACM Computational Mechanics Award, JSME Computational Mechanics Award, ASME Ted Belytschko Applied Mechanics Award, Zienkiewicz Medal from APACM,  AJCM Computational Mechanics Award, Humboldt Research Award, and SACM Medal from the Association of Computational Mechanics (Singapore). He has been listed as one among the world’s top 1% most influential scientists (Highly Cited Researchers) by Thomson Reuters for a number of years. Google citations: ~65000, h-index: 118.