Energy-minimizing generalized node network with machine learning optimization for complex fracture problems

Pengfei Yan, Bangke Ren, Yaocai Ma,
Energy-minimizing generalized node network with machine learning optimization for complex fracture problems,
Engineering Applications of Artificial Intelligence,
Volume 162, Part E,
2025,
112717,
ISSN 0952-1976,
https://doi.org/10.1016/j.engappai.2025.112717.
(https://www.sciencedirect.com/science/article/pii/S0952197625027484)
Abstract: Physics-informed neural networks (PINNs) are increasingly employed in scientific computing for solving partial differential equations. However, their inherent smoothness can limit their performance in complex fracture problems characterized by non-smooth solutions. This paper presents a novel energy-minimizing generalized node network (EMGNN) that integrates the partition-of-unity generalized node method (PUGNM) with modern machine learning optimization. The EMGNN treats the generalized node system as a differentiable physical network where the PU approximations are defined on isolated blocks. During forward propagation, the displacements and their derivatives at integral points are evaluated to compute the system's total potential energy. This energy, consisting of the strain energy of all isolated blocks and the work done by external forces, serves as the loss function. Displacement boundary conditions are enforced by incorporating the strain energy of virtual thin-layer elements into the loss function. The solution is then found through backpropagation by optimizing the generalized degrees of freedom (DOFs) as trainable parameters. By incorporating discontinuity features characterized by the PUGNM, it can effectively capture displacement jumps across crack interfaces and stress singularities near crack tips. This offers a potential pathway for applying machine learning to fracture problems. Compared to plain PINN, the EMGNN attains higher accuracy with reduced training time and improved training stability. Several numerical experiments demonstrate the effectiveness and accuracy of the proposed EMGNN.
Keywords: Physics-informed neural network; Machine learning; Partition-of-unity generalized node method; Fracture problems; Complex discontinuities