Enhanced physics‐informed neural networks for hyperelasticity

Abstract

Physics‐informed neural networks have gained growing interest. Specifically, they are used to solve partial differential equations governing several physical phenomena. However, physics‐informed neural network models suffer from several issues and can fail to provide accurate solutions in many scenarios. We discuss a few of these challenges and the techniques, such as the use of Fourier transform, that can be used to resolve these issues. This paper proposes and develops a physics‐informed neural network model that combines the residuals of the strong form and the potential energy, yielding many loss terms contributing to the definition of the loss function to be minimized. Hence, we propose using the coefficient of variation weighting scheme to dynamically and adaptively assign the weight for each loss term in the loss function. The developed PINN model is standalone and meshfree. In other words, it can accurately capture the mechanical response without requiring any labeled data. Although the framework can be used for many solid mechanics problems, we focus on three‐dimensional (3D) hyperelasticity, where we consider two hyperelastic models. Once the model is trained, the response can be obtained almost instantly at any point in the physical domain, given its spatial coordinates. We demonstrate the framework's performance by solving different problems with various boundary conditions.

PyTorch-FEA: Autograd-enabled Finite Element Analysis Methods with Applications for Biomechanical Analysis of Human Aorta

“…Physics-informed neural networks (PINNs) [52], which attracted a lot of attentions in the field of scientific machine learning, offers a direct solution to PDEs by using DNNs and physics principles. Although PINNs are intriguing, current PINNs have not led to performance improvement compared with commercial FEA methods in the solid mechanics domain and suffer from the error/instability [53] in the deformation gradient tensor calculated by differentiating a DNN with respect to the input: 𝑭𝑭 = 𝑰𝑰 + 𝜕𝜕ℎ(𝑿𝑿) 𝜕𝜕𝑿𝑿…”

“…Physics-informed neural networks (PINNs) [52], which attracted a lot of attentions in the field of scientific machine learning, offers a direct solution to PDEs by using DNNs and physics principles. Although PINNs are intriguing, current PINNs have not led to performance improvement compared with commercial FEA methods in the solid mechanics domain and suffer from the error/instability [53] in the deformation gradient tensor calculated by differentiating a DNN with respect to the input: where h ( X ) is a DNN to output displacement. The instability is not a numerical defect of autograd, and it is closely related to the well-known issue of DNN robustness [54], and it is known that [55, 56] gradients of a DNN can change dramatically if the input changes only a little or even if the same DNN is trained twice on the same data.…”

Improving the Accuracy of the Deep Energy Method

“…The modification allowed the successful resolution of stresses around the stress concentration regions. Abueidda et al 44 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 multiple loss terms.…”

Mixed formulation of physics‐informed neural networks for thermo‐mechanically coupled systems and heterogeneous domains

“…29 The idea is also further extended to a so-called mixed formulation, where both strong form and weak form are utilized simultaneously. 7,22,30,31 One important aspect of engineering applications is to consider the multi-physical characteristics. In other words, in many realistic problems, various coupled and highly nonlinear fields have to be considered simultaneously.…”