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

doi:10.1007/s10483-024-3149-8

Applied Mathematics and Mechanics (English Edition) ›› 2024, Vol. 45 ›› Issue (9): 1467-1480.doi: https://doi.org/10.1007/s10483-024-3149-8

• • 下一篇

收稿日期:2024-07-31 出版日期:2024-09-01 发布日期:2024-08-27

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

Long WANG¹^,², Lei ZHANG¹^,²^,*(), Guowei HE¹^,²

¹ The State Key Laboratory of Nonlinear Mechanics, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100190, China
² School of Engineering Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

Received:2024-07-31 Online:2024-09-01 Published:2024-08-27
Contact: Lei ZHANG E-mail:zhanglei@imech.ac.cn
Supported by:
the National Natural Science Foundation of China Basic Science Center Program for “Multiscale Problems in Nonlinear Mechanics”(11988102);the National Natural Science Foundation of China(12202451);Project supported by the National Natural Science Foundation of China Basic Science Center Program for “Multiscale Problems in Nonlinear Mechanics” (No. 11988102) and the National Natural Science Foundation of China (No. 12202451)

摘要/Abstract

Abstract:

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.

Key words: physics-informed neural network (PINN), singular perturbation, boundary-layer problem, composite asymptotic expansion

中图分类号:

34E15
68T01

. [J]. Applied Mathematics and Mechanics (English Edition), 2024, 45(9): 1467-1480.

Long WANG, Lei ZHANG, Guowei HE. Chien-physics-informed neural networks for solving singularly perturbed boundary-layer problems[J]. Applied Mathematics and Mechanics (English Edition), 2024, 45(9): 1467-1480.

图/表 8

参考文献 35

1	NAYFEH, A. H. Perturbation Methods, John Wiley & Sons, New York 35- 37 (2000)
2	ANDERSON, J. D., JR. Fundamentals of Aerodynamics, 6th ed. McGraw-Hill, New York 997- 1012 (2017)
3	ANDERSON, J. D., JR. Hypersonic and High-Temperature Gas Dynamics, 2nd ed. AIAA Education, Reston 261- 374 (2006)
4	WHITE, F. M. Fluid Mechanics, 8th ed. McGraw-Hill Education, New York 449- 520 (1979)
5	CHIEN, W. Z. Large deflection of a circular clamped plate under uniform pressure. Chinese Journal of Physics, 7 (2), 102- 113 (1947)
6	CHIEN, W. Z., and YEH, K. Y. On the large deflection of circular plate. Acta Physica Sinica, 10 (3), 209- 238 (1954)
7	RAISSI, M., PERDIKARIS, P., and KARNIADAKIS, G. E. Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics, 378, 686- 707 (2019)
8	KARNIADAKIS, G. E., KEVREKIDIS, I. G., LU, L., PERDIKARIS, P., WANG, S., and YANG, L. Physics-informed machine learning. Nature Reviews Physics, 3 (6), 422- 440 (2021)
9	LU, L., MENG, X., MAO, Z., and KARNIADAKIS, G. E. DeepXDE: a deep learning library for solving differential equations. SIAM Review, 63 (1), 208- 228 (2021)
10	CHIEN, W. Z. Asymptotic behavior of a thin clamped circular plate under uniform normal pressure at very large deflection. Science Reports of the National Tsing Hua University, 5, 193- 208 (1948)
11	VAN DYKE, M. Higher approximations in boundary-layer theory, part 1, general analysis. Journal of Fluid Mechanics, 14 (2), 161- 177 (1962)
12	VAN DYKE, M. Higher approximations in boundary-layer theory, part 2, application to leading edges. Journal of Fluid Mechanics, 14 (4), 481- 495 (1962)
13	VAN DYKE, M. Higher approximations in boundary-layer theory, part 3, parabola in uniform stream. Journal of Fluid Mechanics, 19 (1), 145- 159 (1964)
14	MILES, J. W. Fluid mechanics and singular perturbations: a collection of papers by Saul Kaplun. Journal of Fluid Mechanics, 36 (1), 207- 208 (1969)
15	LATTA, G. E. Singular Perturbation Problems, Ph. D. dissertation, California Institute of Technology (1951)
16	BROMBERG, E., and STOKER, J. J. Nonlinear theory of curved elastic sheets. Quarterly of Applied Mathematics, 3 (3), 246- 265 (1945)
17	VISHIK, M. I., and LYUSTERNIK, L. A. Regular degeneration and boundary layer for linear differential equations with small parameter. Uspekhi Matematicheskikh Nauk, 12 (5), 3- 122 (1957)
18	RAISSI, M., YAZDANI, A., and KARNIADAKIS, G. E. Hidden fluid mechanics: learning velocity and pressure fields from flow visualizations. Science, 367 (6481), 1026- 1030 (2020)
19	WANG, H., LIU, Y., and WANG, S. Dense velocity reconstruction from particle image velocimetry/particle tracking velocimetry using a physics-informed neural network. Physics of Fluids, 34 (1), 017116 (2022)
20	REYES, B., HOWARD, A. A., PERDIKARIS, P., and TARTAKOVSKY, A. M. Learning unknown physics of non-Newtonian fluids. Physical Review Fluids, 6 (7), 073301 (2021)
21	HAGHIGHAT, E., RAISSI, M., MOURE, A., GOMEZ, H., and JUANES, R. A physics-informed deep learning framework for inversion and surrogate modeling in solid mechanics. Computer Methods in Applied Mechanics and Engineering, 379, 113741 (2021)
22	HORNIK, K., STINCHCOMBE, M., and WHITE, H. Multilayer feedforward networks are universal approximators. Neural Networks, 2 (5), 359- 366 (1989)
23	CYBENKO, G. Approximation by superpositions of a sigmoidal function. Mathematics of Control, Signals and Systems, 2 (4), 303- 314 (1989)
24	ZHANG, L., CHENG, L., LI, H., GAO, J., YU, C., DOMEL, R., YANG, Y., and LIU, W. K. Hierarchical deep-learning neural networks: finite elements and beyond. Computational Mechanics, 67, 207- 230 (2021)
25	TANG, S., and YANG, Y. Why neural networks apply to scientific computing?. Theoretical and Applied Mechanics Letters, 11 (3), 100242 (2021)
26	ZHANG, L., and HE, G. Multi-scale-matching neural networks for thin plate bending problem. Theoretical and Applied Mechanics Letters, 14 (1), 94 (1004)
27	ARZANI, A., CASSEL, K. W., and D'SOUZA, R. M. Theory-guided physics-informed neural networks for boundary layer problems with singular perturbation. Journal of Computational Physics, 473, 111768 (2023)
28	HUANG, J., QIU, R., WANG, J., and WANG, Y. Multi-scale physics-informed neural networks for solving high Reynolds number boundary layer flows based on matched asymptotic expansions. Theoretical and Applied Mechanics Letters, 14 (2), 96 (1004)
29	BAYDIN, A. G., PEARLMUTTER, B. A., RADUL, A. A., and SISKIND, J. M. Automatic differentiation in machine learning: a survey. Journal of Machine Learning Research, 18, 153 (2018)
30	PASZKE, A., GROSS, S., CHINTALA, S., CHANAN, G., YANG, E., DEVITO, Z., LIN, Z., DESMAISON, A., ANTIGA, L., and LERER, A. Automatic differentiation in PyTorch. Proceedings of the 31st Conference on Neural Information Processing Systems, Long Beach, CA, U. S. A. (2017)
31	NOCEDAL, J. Updating quasi-Newton matrices with limited storage. Mathematics of Computation, 35 (151), 773- 782 (1980)
32	KINGMA, D. and BA, J. Adam: a method for stochastic optimization. arXiv Preprint, arXiv: 1412.6980 (2014) https://doi.org/10.48550/arXiv.1412.6980
33	GLOROT, X., and BENGIO, Y. Understanding the difficulty of training deep feedforward neural networks. Proceedings of the 13th International Conference on Artificial Intelligence and Statistics, 9, 249- 256 (2010)
34	ALZHEIMER, W. E., and DAVIS, R. T. Unsymmetrical bending of prestressed annular plates. Journal of the Engineering Mechanics Division, 94 (4), 905- 918 (1968)
35	TIMOSHENKO, S., and WOINOWSKY-KRIEGER, S. Theory of Plates and Shells, 2nd ed. McGraw-Hill, New York 415- 419 (1959)

[1]	莫嘉琪温朝辉. A class of boundary value problems for third-order differential equation with a turning point[J]. Applied Mathematics and Mechanics (English Edition), 2010, 31(8): 1027-1032.
[2]	莫嘉琪. A class of singular perturbation solutions to semilinear equations of fourth order[J]. Applied Mathematics and Mechanics (English Edition), 2009, 30(11): 1455-1460.
[3]	陈丽华;莫嘉琪. Singularly perturbed solution for weakly nonlinear equations with two parameters[J]. Applied Mathematics and Mechanics (English Edition), 2007, 28(10): 1343-1348 .
[4]	黄蔚章;陈育森. INITIAL LAYER PHENOMENA FOR A CLASS OF SINGULAR PERTURBED NONLINEAR SYSTEM WITH SLOW VARIABLES[J]. Applied Mathematics and Mechanics (English Edition), 2004, 25(7): 836-844.

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

RichHTML

PDF

可视化

摘要/Abstract

引用本文

使用本文

图/表 8

参考文献 35

相关文章 4

编辑推荐

Metrics

本文评价