Four-dimensional printed structures transform shape under external stimuli, and designing these structures to achieve desired deformation outcomes is crucial for 4-D printing technology. Current machine-learning (ML)-assisted methods can design 4-D-printed structures rapidly and accurately, but often overlook the E(3) equivariance required for the model, leading to inconsistent prediction structure between different coordinate representations of the same structure. Consequently, many ML-assisted design approaches rely on data augmentation techniques such as rotation to improve the model's generalization across different coordinate representations, but the method through data augmentation is inefficient. We observe that equivariant graph neural networks (GNNs), due to their equivariant properties, can address the inefficiencies associated with the aforementioned training augmentation. Therefore, we propose utilizing an equivariant GNN to improve these efficiencies. Nonetheless, existing equivariant GNNs struggle to balance computational efficiency with expressive power, which will impact their effectiveness in 4-D printing structure design. In response, we introduce a novel equivariant GNN, the graph rotation network (GRN), which approximates arbitrary E(3) equivariant functions by constructing a 3-D equivariant basis through rotations. This approach enhances the model's performance with trainable rotational angles. We theoretically analyze the E(3) equivariance of our model as well as its computational complexity and empirically demonstrate the improved training efficiency introduced by equivariance. Finally, we apply GRN to both forward prediction and inverse design of 4-D-printed meshes. Our equivariant GNN significantly outperforms nonequivariant GNNs and other equivariant GNNs in forward prediction tasks. For inverse design, our method achieves an accuracy of 98.8% (relative to the mesh size), providing an effective tool for the design of 4-D-printed structures.
Wang et al. (Thu,) studied this question.