Energy-based PINNs using the element integral approach and their enhancement for solid mechanics problems

Abstract

Despite the growing interest in physics-informed neural networks (PINNs) for computational mechanics, significant challenges remain in their widespread application. This work proposes an energy-based PINN method rooted in the principle of virtual work, which states that the external work done on a system is equal to its strain energy. This proposed method discretizes the model into nodes and constructs elements based on these nodes. The strain energy of each element is computed through numerical integration, and the total strain energy of the model is obtained by summing these elemental contributions. Simultaneously, the external work is calculated based on the nodal forces. These calculations, combined with the principle of virtual work, allow for the definition of the model’s physical properties. A deep neural network (DNN) is then trained to map the model’s coordinates to their corresponding displacements, utilizing the defined physical properties. Furthermore, this paper proposes a method to accelerate the learning process of energy-based PINNs by using a simpler and converged model to speed up convergence and to improve the overall accuracy of more complex models. Numerical results demonstrate that the proposed approach effectively solves stress concentration and singularity problems in solid mechanics with high accuracy.