A Solver Framework for Three-Dimensional Hyperelastic Problems Based on PINN and the Virtual Element Method

This work proposes a hybrid computational framework that integrates the Virtual Element Method (VEM) with neural network techniques to efficiently solve three-dimensional hyperelastic problems. The framework combines graph neural networks (GNNs), VEM, and physics-informed neural networks (PINNs), and incorporates a transfer learning strategy to improve the model generalization ability. Owing to its ability to operate on arbitrary polyhedral meshes, VEM provides a flexible and accurate discretization for hyperelastic problems involving complex geometries. Building on this property, the proposed framework employs VEM to construct high-fidelity physics constraints, utilizes GNNs to encode the topology of polyhedral meshes, and adopts PINNs to solve the governing equations under these constraints. Furthermore, the introduced transfer learning strategy enables rapid adaptation of the trained model to new material parameters and loading conditions. Numerical experiments demonstrate that the proposed framework achieves accuracy comparable to that of traditional high-fidelity numerical simulations while improving computational efficiency by one to two orders of magnitude. In addition, the method exhibits strong robustness and generalization capability under variations in parameters and loading scenarios. These results indicate that the proposed approach provides a promising and efficient solution for the rapid analysis of multi-scenario and multi-parameter three-dimensional hyperelastic problems.