In this study, we focus on transfer learning to construct an efficient surrogate model for three-dimensional fatigue crack propagation while maintaining prediction accuracy [1]. Transfer learning is applied by extending a neural network predictor initially trained on two-dimensional crack propagation simulation results to three-dimensional cases. The training datasets are generated using fatigue crack propagation analyses based on the s-version finite element method (s-FEM) [2]. The crack tip coordinates are used as input features, and the incremental crack growth is predicted to reproduce continuous crack propagation paths. Histogram-based data augmentation is employed to mitigate bias in the training parameters. In addition to conventional L2 regularization, a physically consistent partial differential equation (PDE) governing crack propagation behavior is automatically discovered from the simulation data using AI-Feynman and automatic differentiation. The discovered PDE, composed of input and output variables, is incorporated into the training process as a PDE-based loss term. By combining the prediction loss and the PDE loss in a weighted formulation, the surrogate model is effectively constrained by the discovered governing equation, leading to improved generalization performance and reduced validation loss [3]. Numerical results demonstrate that the proposed approach achieves crack tip prediction accuracy comparable to previous studies, while the training time is reduced to approximately one-third through transfer learning. Furthermore, the introduction of the discovered PDE enhances the robustness and physical consistency of the surrogate model, confirming the validity and effectiveness of the discovered equation in three-dimensional crack propagation prediction.