Kirchhoff plate bending and Winkler-type contact problems with different boundary conditions are solved with the use of physics-informed neural networks (PINN). The PINN is built on the base of mechanics laws and deep learning. The idea of the technique includes fitting the governing partial differential equations at collocation points and then training the neural network with the use of optimization techniques. Training of the neural network is performed by numerical optimization using Adam’s method and the L-BFGS (Limited- Broyden–Fletcher–Goldfarb–Shanno) algorithm. The error loss function and the computational error of the approximate solution (output of the neural network) of the bending problem and contact problem with Winkler type elastic foundation are shown on examples. The predictions of the NN are investigated for different values of the foundation’s constants. The effectiveness of the proposed framework is demonstrated through numerical experiments with different numbers of epochs, hidden layers, neurons and numbers of collocation points. The Tensorflow deep learning and scientific computing package of Python is used through a Jupyter Notebook.
Al-Aradi A, Correia A, Naiff D, Jardim G, Saporito Y. Solving nonlinear and highdimensional partial differential equations via deep Learning. 2018;
2.
Avdelas A, Panagiotopoulos P, Kortesis S. Neural networks for computing in the elastoplastic analysis of structures. Meccanica. 1995;1–15.
3.
Baydin A, Pearlmutter B, Radul A, Siskind J. Automatic Differentiation in Machine Learning: a Survey. Journal of Machine Learning Research. 2018;1–43.
4.
Fletcher R. Practical Methods of Optimization . John Wiley & Sons, New York. 1987;(2).
5.
Kharazmi E, Zhang Z, Karniadakis G. Variational physics-informed neural networks for solving partial differential equations. 2019;
6.
Karniadakis GE, Kevrekidis IG, Lu L, Perdikaris P, Wang S, Yang L. Physics-informed machine learning. Nature Reviews Physics. 2021;3(6):422–40.
7.
Katsikadelis J, Yiotis A. The BEM for plates of variable thickness on nonlinear biparametric elastic foundation. An analog equation solution. Journal of Engineering Mathematics. 2003;313–30.
8.
Kingma D, Ba J. Adam: A Method for Stochastic Optimization. 2015;
9.
Kortesis S, Panagiotopoulos P. Neural networks for computing in structural analysis: Methods and prospects of applications. International Journal for Numerical Methods in Engineering. 1993;2305–18.
10.
Lagaris E, Likas A, Fotiadis D. Artificial neural networks for solving ordinary and partial differential equations. IEEE Transactions on Neural Networks. 1998;987–1000.
11.
Mcculloch W, Pitts W. A logical calculus of the ideas immanent in nervous activity. The Bulletin of Mathematical Biophysics. 1943;(4):115–33.
12.
Muradova A. The spectral method and numerical continuation algorithm for the von Kármán problem with postbuckling behaviour of solution. 2008;(2):179–206.
13.
Muradova A, Stavroulakis G. The projective-iterative method and neural network estimation for buckling of elastic plates in nonlinear theory. Communications in Nonlinear Science and Numerical Simulation. 2007;1068–88.
14.
Muradova A, Stavroulakis G. Buckling and postbuckling analysis of rectangular plates resting on elastic foundations with the use of the spectral method. Computer Methods in Applied Mechanics and Engineering. 2012;(208):213–20.
15.
Muradova A, Stavroulakis G, Tairidis G. A spectral collocation method for vibration suppression of smart elastic plates. 2018;352–60.
16.
Muradova A, Stavroulakis G. Mathematical Models with Buckling and Contact Phenomena for Elastic Plates: A Review Mathematics. 2020;(4):566.
17.
Raissi M, Perdikaris P, Karniadakis G. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics. 2019;686–707.
18.
Hui-Shen S. Postbuckling analysis of orthotropic rectangular plates on nonlinear elastic foundations, Engineering Structures. 1995;(6):407–12.
19.
Shin Y, Darbon J, Karniadakis G. On the convergence of physics informed neural networks for linear second-order elliptic and parabolic type PDEs. Commun Comput Phys. 2020;2042–74.
20.
Stavroulakis G. Inverse and Crack Identification Problems in Engineering Mechanics. 2000;
21.
Stavroulakis G, Avdelas A, Abdalla K, Panagiotopoulos P. A neural network approach to the modelling, calculation and identification of semi-rigid connections in steel structures. Journal of Constructional Steel Research. 1997;(1–2):91–105.
22.
Stavroulakis G, Bolzon G, Waszczyszyn Z, Ziemianski L. Inverse analysis. Numerical and computational methods. 2003;685–718.
23.
Waszczyszyn Z, Ziemiański L. Neural Networks in the Identification Analysis of Structural Mechanics Problems. CISM International Centre for Mechanical Sciences (Courses and Lectures). 2005;
24.
Yagawa G, Oishi. Computational mechanics with neural networks. 2021;
The statements, opinions and data contained in the journal are solely those of the individual authors and contributors and not of the publisher and the editor(s). We stay neutral with regard to jurisdictional claims in published maps and institutional affiliations.
