Chien-physics-informed neural networks for solving singularly perturbed boundary-layer problems

• Published in: Applied mathematics and mechanics Vol. 45; no. 9; pp. 1467 - 1480
• Main Authors: Wang, Long, Zhang, Lei, He, Guowei
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.09.2024; Springer Nature B.V; The State Key Laboratory of Nonlinear Mechanics,Institute of Mechanics,Chinese Academy of Sciences,Beijing 100190,China; School of Engineering Sciences,University of Chinese Academy of Sciences,Beijing 100049,China
• Edition: English ed.
• Subjects: Applications of Mathematics; Asymptotic methods; Asymptotic series; Classical Mechanics; Differential equations; Fluid mechanics; Fluid- and Aerodynamics; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Neural networks; Partial Differential Equations; Singular perturbation; Thin plates; 68T01; O302; singular perturbation; physics-informed neural network (PINN); composite asymptotic expansion; boundary-layer problem; 34E15; physics-informed neural network(PINN)
• ISSN: 0253-4827; 1573-2754
• DOI: 10.1007/s10483-024-3149-8

A physics-informed neural network (PINN) is a powerful tool for solving differential equations in solid and fluid mechanics. However, it suffers from singularly perturbed boundary-layer problems in which there exist sharp changes caused by a small perturbation parameter multiplying the highest-order derivatives. In this paper, we introduce Chien’s composite expansion method into PINNs, and propose a novel architecture for the PINNs, namely, the Chien-PINN (C-PINN) method. This novel PINN method is validated by singularly perturbed differential equations, and successfully solves the well-known thin plate bending problems. In particular, no cumbersome matching conditions are needed for the C-PINN method, compared with the previous studies based on matched asymptotic expansions.