Enhanced physics-informed neural networks for hyperelasticity

Abueidda, Diab W.; Korić, Seid; Guleryuz, Erman; Sobh, Nahil

doi:10.48550/arxiv.2205.14148

Cited by 4 publications

(3 citation statements)

References 27 publications

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“…Recently, physics-informed neural networks (PINNs) have gained popularity due to the novel approach for solving forward [1][2][3][4] and inverse problems [5][6][7] involving PDEs using neural networks (NNs). Unlike conventional numerical techniques for solving PDEs, PINNs are non-data-driven meshless models that satisfy the prescribed initial (IC) and Llion Evans, Michelle Tindall, and Perumal Nithiarasu have contributed equally to this work.…”

Section: Introductionmentioning

confidence: 99%

Stiff-PDEs and Physics-Informed Neural Networks

Sharma

Evans

Tindall

et al. 2023

Arch Computat Methods Eng

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In recent years, physics-informed neural networks (PINN) have been used to solve stiff-PDEs mostly in the 1D and 2D spatial domain. PINNs still experience issues solving 3D problems, especially, problems with conflicting boundary conditions at adjacent edges and corners. These problems have discontinuous solutions at edges and corners that are difficult to learn for neural networks with a continuous activation function. In this review paper, we have investigated various PINN frameworks that are designed to solve stiff-PDEs. We took two heat conduction problems (2D and 3D) with a discontinuous solution at corners as test cases. We investigated these problems with a number of PINN frameworks, discussed and analysed the results against the FEM solution. It appears that PINNs provide a more general platform for parameterisation compared to conventional solvers. Thus, we have investigated the 2D heat conduction problem with parametric conductivity and geometry separately. We also discuss the challenges associated with PINNs and identify areas for further investigation.

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Section: Introductionmentioning

confidence: 99%

Stiff-PDEs and Physics-Informed Neural Networks

Sharma

Evans

Tindall

et al. 2023

Arch Computat Methods Eng

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“…The modification allowed the successful resolution of stresses around the stress concentration regions. Abueidda et al 39 enhanced the model by developing PINN that combines residuals of the strong form and the system's potential energy. The proposed formulation yielded a loss function with many loss terms.…”

Section: Introductionmentioning

confidence: 99%

Improving the accuracy of the deep energy method

Chadha,

He,

Abueidda

et al. 2023

Acta Mech

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The deep energy method (DEM) employs the principle of minimum potential energy to train neural network models to predict displacement at a state of equilibrium under given boundary conditions. The accuracy of the model is contingent upon choosing appropriate hyperparameters. The hyperparameters have traditionally been chosen based on literature or through manual iterations. The displacements predicted using hyperparameters suggested in the literature do not ensure the minimum potential energy of the system. Additionally, they do not necessarily generalize to different load cases. Selecting hyperparameters through manual trial and error and grid search algorithms can be highly time-consuming. We propose a systematic approach using the Bayesian optimization algorithms and random search to identify optimal values for these parameters. Seven hyperparameters are optimized to obtain the minimum potential energy of the system under compression, tension, and bending loads cases. In addition to Bayesian optimization, Fourier feature mapping is also introduced to improve accuracy. The models trained using optimal hyperparameters and Fourier feature mapping could accurately predict deflections compared to finite element analysis for linear elastic materials. The deflections obtained for tension and compression load cases are found to be more sensitive to values of hyperparameters compared to bending. The approach can be easily extended to 3D and other material models.

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“…Besides, researchers have used a mixed formulation, combining both the energy method and the strong form, to capture the mechanical response with high solution gradients and stress concentrations. [24][25][26] In almost all cases, these spatial gradients are obtained utilizing automatic differentiation (AD) of the NN model. [27][28][29] This approach is widely used since AD is already applied in the backpropagation step during NN model training.…”

Section: Introductionmentioning

confidence: 99%

On the use of graph neural networks and shape‐function‐based gradient computation in the deep energy method

Abueidda

Korić

et al. 2022

Numerical Meth Engineering

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A graph convolutional network (GCN) is employed in the deep energy method (DEM) model to solve the momentum balance equation in three-dimensional space for the deformation of linear elastic and hyperelastic materials due to its ability to handle irregular domains over the traditional DEM method based on a multilayer perceptron (MLP) network. The method's accuracy and solution time are compared to the DEM model based on a MLP network. We demonstrate that the GCN-based model delivers similar accuracy while having a shorter run time through numerical examples. Two different spatial gradient computation techniques, one based on automatic differentiation (AD) and the other based on shape function (SF) gradients, are also accessed. We provide a simple example to demonstrate the strain localization instability associated with the AD-based gradient computation and show that the instability exists in more general cases by four numerical examples. The SF-based gradient computation is shown to be more robust and delivers an accurate solution even at severe deformations. Therefore, the combination of the GCN-based DEM model and SF-based gradient computation is potentially a promising candidate for solving problems involving severe material and geometric nonlinearities.

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Enhanced physics-informed neural networks for hyperelasticity

Cited by 4 publications

References 27 publications

Stiff-PDEs and Physics-Informed Neural Networks

Stiff-PDEs and Physics-Informed Neural Networks

Improving the accuracy of the deep energy method

On the use of graph neural networks and shape‐function‐based gradient computation in the deep energy method

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