Keywords
Python programming
Boundary value problem
Computational algorithm
Physics-informed neural networks
How to Cite
Abstract
A modern approach to solving mathematical models involving differential equations, the so-called Physics-Informed Neural Network (PINN), is based on the techniques which include the use of artificial neural networks and the method of fitting the governing differential equations at collocation points. In this paper, training of the PINN with an application of optimization techniques is performed on simple one-dimensional mechanical problems of elasticity, namely rods and beams. Different boundary conditions are considered.
Required computer algorithms are implemented using Python programming packages with the intention of creating neural networks. Numerical results are presented, and the efficiency of the proposed technique is investigated through numerical experiments with different numbers of epochs, batches, hidden layers, neurons, and collocation points.
The paper provides useful skills for using a PINN for different problems of solid mechanics. The proposed methodology is a continuation of our intention of using PINNs for problems of the theory of elasticity. The objectives are to present simply the main steps of constructing PINN and an implementation of it. A detailed explanation of the Python programming code, based on the scientific software Tensorflow, built in the Keras library and optimizers, may help compose an effective code for complicated models in mechanics.
PINNs are proposed in many recent publications to solve complicated direct and inverse problems. It seems to be a promising method that will play a central role in the development of computational mechanics in the near future. Nevertheless, the lack of educational material does not help new users to enter this scientific area. The present contribution describes the method for the solution of elementary rod and beam problems and gives computer codes that may help the reader to understand the method and to apply it to other problems.
References
Avdelas AV, Panagiotopoulos PD, Kortesis S. Neural networks for computing in the elastoplastic analysis of structures. Meccanica 1995; 30: 1–15.
Baydin A, Pearlmutter BA, Radul AA, Siskind JM. Automatic Differentiation in Machine Learning: a Survey, 2018, https://arxiv.org/pdf/1502.05767.pdf
Cande’s EJ. Harmonic Analysis of Neural Networks, Applied and Computational Harmonic Analysis 1999; 6: 197–218.
Fletcher R. Practical Methods of Optimization (2nd ed.). John Wiley & Sons, New York 1987.
Guo M, Haghighat E. An energy-based error bound of physics-informed neural network solutions in elasticity. arXiv preprint arXiv:2010.09088, 2020.
Kadeethum T, Jørgensen T, Nick H. Physics-informed neural networks for solving nonlinear diffusivity and Biot’s equations. PLoS ONE 2020, 15(5): e0232683.
Karniadakis GE, Kevrekidis IG, Lu L, et al. Physics-informed machine learning. Nat Rev Phys 2021; 3: 422–440. https://doi.org/10.1038/s42254-021-00314-5.
Kingma DP, Ba JL. Adam A. A Method for Stochastic Optimization, Computer Science, Mathematics, The International Conference on Learning Representations (ICLR) 2015.
Kovachki N, Lanthaler S, Mishra S. On Universal Approximation and Error Bounds for Fourier Neural Operator, Journal of Machine Learning Research 2021; 22: 1-76.
Kortesis S. Panagiotopoulos PD. Neural networks for computing in structural analysis: Methods and prospects of applications. International Journal for Numerical Methods in Engineering 1993; 36: 2305-2318.
Lagaris E, Likas A, Fotiadis DI. Artificial neural networks for solving ordinary and partial differential equations, IEEE Transactions on Neural Networks 1998; 9: 987–1000.
Lee Η, Kang I. Neural algorithms for solving differential equations. Journal of Computational Physics 1990; 91(1): 110-131.
Li Z, Kovachki N, Azizzadenesheli K, Liu B, Bhattacharya K, Stuart A, Anandkumar A. Fourier neural operator for parametric partial differential equations, 2020, arXiv preprint arXiv:2010.08895.
Lu L, Jin P, Karniadakis GE. DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators, 2020, https://arxiv.org/abs/1910.03193
Lu L, Jin P, Pang G, Zhang Z, Karniadakis GE. Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators. Nat Mach Intell 2021; 3: 218–229. https://doi.org/10.1038/s42256-021-00302-5.
McCulloch WS, Pitts W. A logical calculus of the ideas immanent in nervous activity. Bulletin of Mathematical Biophysics 1990; 52(1/2): 99-115.
Meade Jr AJ, Fernadez AA. The numerical solution of linear ordinary differential equations by feedforward neural networks. Math. Comput. Modelling. 1994; 19(12): 1-25.
Muradova AD, Stavroulakis GE. The projective-iterative method and neural network estimation for buckling of elastic plates in nonlinear theory. Communications in Nonlinear Science and Numerical Simulation 2007; 12: 1068-1088.
Muradova AD, Stavroulakis GE. Physics-informed neural networks for elastic plate problems with bending and Winkler-type contact effects. Journal of the Serbian Society for Computational Mechanics 2021 ; 15(2): 45-54.
Peng W, Zhang J, Zhou W, Zhao X, Yao W, Chen X. A Physics-Informed Neural Network Library 2021: https://arxiv.org/pdf/2107.04320.pdf.
Raissi M, Perdikaris P, Karniadakis GE. Physics Informed Deep Learning (Part I): Data - driven Solutions of Nonlinear Partial Differential Equations 2017: https://arxiv.org/abs/1711.10561.
Raissi M, Perdikaris P, Karniadakis GE. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 2019; 378: 686–707.
Rojas R. Neural Networks A Systematic Introduction. Springer-Verlag, Berlin, 1996.
Ruder S. An overview of gradient descent optimization algorithms 2017: https://arxiv.org/abs/1609.04747.
Russel S, Norvig P. Artificial Intelligence: A modern approach (Pearson Series in Artifical Intelligence), Pearson, 4th Edition, 2020.
Searle JR. The Rediscovery of the Mind. Cambridge, Massachusetts: MIT Press, 1992.
Shin Y, Darbon J, Karniadakis GE. On the convergence of physics informed neural networks for linear second-order elliptic and parabolic type PDEs. Commun. Comput. Phys. 2020; 28: 2042–2074.
Stavroulakis GE. Inverse and Crack Identification Problems in Engineering Mechanics. Springer, 2000.
Stavroulakis GE, Avdelas A, Abdalla KM, Panagiotopoulos PD. A neural network approach to the modelling, calculation and identification of semi-rigid connections in steel structures, Journal of Constructional Steel Research 1997; 44(1–2): 91-105.
Tartakovsky AM, Marrero CO, Perdikaris P, Tartakovsky GD, Barajas-Solano D. Learning parameters and constitutive relationships with physics informed deep neural networks 2018: arXiv preprint arXiv:1808.03398.
Waszczyszyn Z, Ziemiański L. Neural Networks in the Identification Analysis of Structural Mechanics Problems. In: Mróz Z, Stavroulakis G.E. (eds) Parameter Identification of Materials and Structures, CISM International Centre for Mechanical Sciences (Courses and Lectures), 2005; 469, Springer, Vienna.
Yagawa G, Oishi A. Computational mechanics with neural networks. Springer, 2021.
Zhang Q, Chen Y, Yang Z. Data-driven solutions and discoveries in mechanics using physics informed neural network. 2020.
Zhao X, Gong Z, Zhang Y, Yao W, Chen X. Physics-informed Convolutional Neural Networks for Temperature Field Prediction of Heat Source Layout without Labeled Data 2021, https://arxiv.org/abs/2109.12482.
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
Copyright (c) 2022 Dimitrios Katsikis, Aliki D. Muradova, Georgios E. Stavroulakis