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Engineering with Computers
Published in:

20-04-2024 | Original Article

A novel normalized reduced-order physics-informed neural network for solving inverse problems

Authors: Khang A. Luong, Thang Le-Duc, Seunghye Lee, Jaehong Lee

Published in: Engineering with Computers | Issue 5/2024

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Abstract

The utilization of Physics-informed Neural Networks (PINNs) in deciphering inverse problems has gained significant attention in recent years. However, the PINN training process for inverse problems is notably restricted due to gradient failures provoked by magnitudes of partial differential equations (PDEs) parameters or source functions. To address these matters, normalized reduced-order physics-informed neural network (nr-PINN) is developed in this study. The goal of the nr-PINN is to reconfigure the original PDE into a system of normalized lower-order PDEs through two sequential steps. To start with, self-homeomorphisms of the PDEs are implemented via scaling factors determined based on measured data. Afterward, each normalized PDE is transformed into a system of lower-order PDEs by primary and secondary variables. Besides, a technique to exactly impose many types of boundary conditions (BCs) by redefining NNs outputs is developed in the context of reduced-order method. The advantages of the nr-PINN model over the original one regarding solution accuracy and training cost are demonstrated through several inverse problems in solid mechanics with different types of PDEs and BCs.
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Literature
1.
LeVeque RJ (2007) Finite difference methods for ordinary and partial differential equations: steady-state and time-dependent problems. SIAM, PhiladelphiaCrossRef
2.
Reddy JN (2019) Introduction to the finite element method. McGraw-Hill Education, New York
3.
Logan DL (2022) First course in the finite element method, Enhanced. Cengage Learning, SI Version, Chennai
4.
Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J Comput Phys 378:686–707MathSciNetCrossRef
5.
Zhuang X, Guo H, Alajlan N, Zhu H, Rabczuk T (2021) Deep autoencoder based energy method for the bending, vibration, and buckling analysis of Kirchhoff plates with transfer learning. Eur J Mech A/Solids 87:104225MathSciNetCrossRef
6.
Samaniego E, Anitescu C, Goswami S, Nguyen-Thanh VM, Guo H, Hamdia K, Zhuang X, Rabczuk T (2020) An energy approach to the solution of partial differential equations in computational mechanics via machine learning: Concepts, implementation and applications. Comput Methods Appl Mech Eng 362:112790MathSciNetCrossRef
7.
Zhu Q, Zhao Z, Yan J (2023) Physics-informed machine learning for surrogate modeling of wind pressure and optimization of pressure sensor placement. Comput Mech 71(3):481–491MathSciNetCrossRef
8.
Cai S, Mao Z, Wang Z, Yin M, Karniadakis GE (2021) Physics-informed neural networks (PINNS) for fluid mechanics: A review. Acta Mech Sin 37(12):1727–1738MathSciNetCrossRef
9.
Li A, Zhang YJ (2023) Isogeometric analysis-based physics-informed graph neural network for studying traffic jam in neurons. Comput Methods Appl Mech Eng 403:115757MathSciNetCrossRef
10.
Jeong H, Bai J, Batuwatta-Gamage CP, Rathnayaka C, Zhou Y, Gu Y (2023) A physics-informed neural network-based topology optimization (PINNTO) framework for structural optimization. Eng Struct 278:115484CrossRef
11.
Wilt JK, Yang C, Gu GX (2020) Accelerating auxetic metamaterial design with deep learning. Adv Eng Mater 22(5):1901266CrossRef
12.
Chen Y, Lu L, Karniadakis GE, Dal Negro L (2020) Physics-informed neural networks for inverse problems in nano-optics and metamaterials. Opt Exp 28(8):11618–11633CrossRef
13.
Xu C, Cao BT, Yuan Y, Meschke G (2023) Transfer learning based physics-informed neural networks for solving inverse problems in engineering structures under different loading scenarios. Comput Methods Appl Mech Eng 405:115852MathSciNetCrossRef
14.
Yuan L, Ni Y-Q, Deng X-Y, Hao S (2022) A-PINN: Auxiliary physics informed neural networks for forward and inverse problems of nonlinear integro-differential equations. J Comput Phys 462:111260MathSciNetCrossRef
15.
Yu J, Lu L, Meng X, Karniadakis GE (2022) Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems. Comput Methods Appl Mech Eng 393:114823MathSciNetCrossRef
16.
Gao H, Zahr MJ, Wang J-X (2022) Physics-informed graph neural Galerkin networks: a unified framework for solving PDE-governed forward and inverse problems. Comput Methods Appl Mech Eng 390:114502MathSciNetCrossRef
17.
Kissas G, Yang Y, Hwuang E, Witschey WR, Detre JA, Perdikaris P (2020) Machine learning in cardiovascular flows modeling: predicting arterial blood pressure from non-invasive 4D flow MRI data using physics-informed neural networks. Comput Methods Appl Mech Eng 358:112623MathSciNetCrossRef
18.
Lu L, Meng X, Mao Z, Karniadakis GE (2021) Deepxde: a deep learning library for solving differential equations. SIAM Rev 63(1):208–228MathSciNetCrossRef
19.
Glorot X, Bengio Y (2010) Understanding the difficulty of training deep feedforward neural networks. In: Proceedings of the 13th international conference on artificial intelligence and statistics, JMLR workshop and conference proceedings, pp 249–256
20.
Zong Y, He Q, Tartakovsky AM (2023) Improved training of physics-informed neural networks for parabolic differential equations with sharply perturbed initial conditions. Comput Methods Appl Mech Eng 414:116125MathSciNetCrossRef
21.
Laubscher R, Rousseau P (2022) Application of a mixed variable physics-informed neural network to solve the incompressible steady-state and transient mass, momentum, and energy conservation equations for flow over in-line heated tubes. Appl Soft Comput 114:108050CrossRef
22.
Laubscher R (2021) Simulation of multi-species flow and heat transfer using physics-informed neural networks. Phys Fluids 33(8):12CrossRef
23.
Li W, Bazant MZ, Zhu J (2021) A physics-guided neural network framework for elastic plates: comparison of governing equations-based and energy-based approaches. Comput Methods Appl Mech Eng 383:113933MathSciNetCrossRef
24.
Yu B et al (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Commun Math Stat 6(1):1–12MathSciNetCrossRef
25.
Luong KA, Le-Duc T, Lee J (2023) Deep reduced-order least-square method—a parallel neural network structure for solving beam problems. Thin-Walled Struct 191:111044CrossRef
26.
Lu L, Pestourie R, Yao W, Wang Z, Verdugo F, Johnson SG (2021) Physics-informed neural networks with hard constraints for inverse design. SIAM J Sci Comput 43(6):B1105–B1132MathSciNetCrossRef
27.
Nguyen-Thanh VM, Zhuang X, Rabczuk T (2020) A deep energy method for finite deformation hyperelasticity. Eur J Mech A/Solids 80:103874MathSciNetCrossRef
28.
Nguyen-Thanh VM, Anitescu C, Alajlan N, Rabczuk T, Zhuang X (2021) Parametric deep energy approach for elasticity accounting for strain gradient effects. Comput Methods Appl Mech Eng 386:114096MathSciNetCrossRef
29.
Liu Z, Yang Y, Cai Q-D (2019) Solving differential equation with constrained multilayer feedforward network. arXiv preprint arXiv:​1904.​06619
30.
Sukumar N, Srivastava A (2022) Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks. Comput Methods Appl Mech Eng 389:114333MathSciNetCrossRef
31.
Luong KA, Le-Duc T, Lee J (2023) Automatically imposing boundary conditions for boundary value problems by unified physics-informed neural network. Eng Comput 2:1–23
32.
Baydin AG, Pearlmutter BA, Radul AA, Siskind JM (2018) Automatic differentiation in machine learning: a survey. J Mach Learn Res 18:1–43MathSciNet
33.
Krylov VI, Stroud AH (2006) Approximate calculation of integrals. Courier Corporation, North Chelmsford
34.
Stoer J, Bulirsch R, Bartels R, Gautschi W, Witzgall C (1980) Introduction to numerical analysis, vol 1993. Springer, New YorkCrossRef
35.
36.
Abadi M, Barham P, Chen J, Chen Z, Davis A, Dean J, Devin M, Ghemawat S, Irving G, Isard M et al (2016) Tensorflow: a system for large-scale machine learning. In: Osdi, vol 16. Savannah, pp 265–283
37.
38.
Liu DC, Nocedal J (1989) On the limited memory BFGS method for large scale optimization. Math Program 45(1–3):503–528MathSciNetCrossRef
39.
Chen W, Wang Q, Hesthaven JS, Zhang C (2021) Physics-informed machine learning for reduced-order modeling of nonlinear problems. J Comput Phys 446:110666MathSciNetCrossRef
40.
Timoshenko S, Woinowsky-Krieger S et al (1959) Theory of plates and shells, vol 2. McGraw-Hill, New York
41.
Huang Y, Ouyang Z-Y (2020) Exact solution for bending analysis of two-directional functionally graded Timoshenko beams. Arch Appl Mech 90(5):1005–1023CrossRef
Metadata
Title
A novel normalized reduced-order physics-informed neural network for solving inverse problems
Authors
Khang A. Luong
Thang Le-Duc
Seunghye Lee
Jaehong Lee
Publication date
20-04-2024
Publisher
Springer London
Published in
Engineering with Computers / Issue 5/2024
Print ISSN: 0177-0667
Electronic ISSN: 1435-5663
DOI
https://doi.org/10.1007/s00366-024-01971-7

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