The deep finite element method: A deep learning framework integrating the physics-informed neural networks with the finite element method

• Published in: Computer methods in applied mechanics and engineering Vol. 436; p. 117681
• Main Authors: Xiong, Wei, Long, Xiangyun, Bordas, Stéphane P.A., Jiang, Chao
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.03.2025
• Subjects: Deep energy method; Deep finite element method; Physics-informed neural networks; Solid elasticity mechanics; Three-dimensional structures; Solid elasticity mechanics; Three-dimensional structures; Physics-informed neural networks; Deep finite element method; Deep energy method
• ISSN: 0045-7825
• DOI: 10.1016/j.cma.2024.117681

•The deep finite element method integrating the physics-informed neural networks and the finite element method is proposed.•The method provides a flexible and robust framework that integrates data with physical laws.•A novel loss function is designed to eliminate high-order derivatives for efficient training.•Excellent training stability and prediction accuracy especially for complex 3D problems.•The method shows promise for extending to other partial differential equations with variational formulations. This paper proposes a deep finite element method (DFEM), which integrates physics-informed neural networks (PINNs) with the finite element method (FEM). The DFEM provides a versatile and robust framework that seamlessly combines observational data with physical laws. In this method, the outputs of the deep neural networks are utilized for approximating the displacements of grid nodes, while shape functions are used to construct trial functions within elements. A loss function is then designed by combining deep neural networks with stiffness matrices, avoiding the high-order derivative terms. Furthermore, the theoretical convergence stability of the proposed DFEM is analyzed. Finally, the training stability and predictive accuracy of the DFEM are validated through numerical examples. The results demonstrate that the proposed method greatly reduces the relative error compared to the deep energy method (DEM). Moreover, by integrating pre-training, the DFEM experiences a significant enhancement in training efficiency, reducing computation time by up to 7/8 compared to the FEM. This makes it a promising tool for fast computations and digital twin applications.