Ensemble of physics-informed neural networks for solving plane elasticity problems with examples

Two-dimensional (plane) elasticity equations in solid mechanics are solved numerically with the use of an ensemble of physics-informed neural networks (PINNs). The system of equations consists of the kinematic definitions, i.e. the strain–displacement relations, the equilibrium equations connecting...

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Published in:	Acta mechanica Vol. 235; no. 11; pp. 6703 - 6722
Main Authors:	Mouratidou, Aliki D., Drosopoulos, Georgios A., Stavroulakis, Georgios E.
Format:	Journal Article
Language:	English
Published:	Vienna Springer Vienna 01.11.2024 Springer Springer Nature B.V
Subjects:	Analysis Application programming interface Applications programming Artificial intelligence Boundary conditions Classical and Continuum Physics Commercial aircraft Computer programs Constitutive equations Constitutive relationships Control Deep learning Dynamical Systems Elasticity Engineering Engineering Fluid Dynamics Engineering Thermodynamics Equilibrium equations Exact solutions Finite element method Heat and Mass Transfer Kinematics Machine learning Material properties Mathematical analysis Mathematical models Neural networks Original Paper Physics Plane stress Python Science Software Solid Mechanics Strain Tensors Theoretical and Applied Mechanics Vibration
ISSN:	0001-5970, 1619-6937
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Abstract	Two-dimensional (plane) elasticity equations in solid mechanics are solved numerically with the use of an ensemble of physics-informed neural networks (PINNs). The system of equations consists of the kinematic definitions, i.e. the strain–displacement relations, the equilibrium equations connecting a stress tensor with external loading forces and the isotropic constitutive relations for stress and strain tensors. Different boundary conditions for the strain tensor and displacements are considered. The proposed computational approach is based on principles of artificial intelligence and uses a developed open-source machine learning platform, scientific software Tensorflow, written in Python and Keras library, an application programming interface, intended for a deep learning. A deep learning is performed through training the physics-informed neural network model in order to fit the plain elasticity equations and given boundary conditions at collocation points. The numerical technique is tested on an example, where the exact solution is given. Two examples with plane stress problems are calculated with the proposed multi-PINN model. The numerical solution is compared with results obtained after using commercial finite element software. The numerical results have shown that an application of a multi-network approach is more beneficial in comparison with using a single PINN with many outputs. The derived results confirmed the efficiency of the introduced methodology. The proposed technique can be extended and applied to the structures with nonlinear material properties.
AbstractList	Two-dimensional (plane) elasticity equations in solid mechanics are solved numerically with the use of an ensemble of physics-informed neural networks (PINNs). The system of equations consists of the kinematic definitions, i.e. the strain–displacement relations, the equilibrium equations connecting a stress tensor with external loading forces and the isotropic constitutive relations for stress and strain tensors. Different boundary conditions for the strain tensor and displacements are considered. The proposed computational approach is based on principles of artificial intelligence and uses a developed open-source machine learning platform, scientific software Tensorflow, written in Python and Keras library, an application programming interface, intended for a deep learning. A deep learning is performed through training the physics-informed neural network model in order to fit the plain elasticity equations and given boundary conditions at collocation points. The numerical technique is tested on an example, where the exact solution is given. Two examples with plane stress problems are calculated with the proposed multi-PINN model. The numerical solution is compared with results obtained after using commercial finite element software. The numerical results have shown that an application of a multi-network approach is more beneficial in comparison with using a single PINN with many outputs. The derived results confirmed the efficiency of the introduced methodology. The proposed technique can be extended and applied to the structures with nonlinear material properties.
Audience	Academic
Author	Drosopoulos, Georgios A. Stavroulakis, Georgios E. Mouratidou, Aliki D.
Author_xml	– sequence: 1 givenname: Aliki D. orcidid: 0000-0002-8382-1263 surname: Mouratidou fullname: Mouratidou, Aliki D. email: amouratidou@tuc.gr organization: School of Production Engineering and Management, Institute of Computational Mechanics and Optimization, Technical University of Crete – sequence: 2 givenname: Georgios A. surname: Drosopoulos fullname: Drosopoulos, Georgios A. organization: Discipline of Civil Engineering, School of Engineering and Computing, University of Central Lancashire, Discipline of Civil Engineering, School of Engineering, University of KwaZulu-Natal – sequence: 3 givenname: Georgios E. surname: Stavroulakis fullname: Stavroulakis, Georgios E. organization: School of Production Engineering and Management, Institute of Computational Mechanics and Optimization, Technical University of Crete
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SubjectTerms	Analysis Application programming interface Applications programming Artificial intelligence Boundary conditions Classical and Continuum Physics Commercial aircraft Computer programs Constitutive equations Constitutive relationships Control Deep learning Dynamical Systems Elasticity Engineering Engineering Fluid Dynamics Engineering Thermodynamics Equilibrium equations Exact solutions Finite element method Heat and Mass Transfer Kinematics Machine learning Material properties Mathematical analysis Mathematical models Neural networks Original Paper Physics Plane stress Python Science Software Solid Mechanics Strain Tensors Theoretical and Applied Mechanics Vibration
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Title	Ensemble of physics-informed neural networks for solving plane elasticity problems with examples
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