Deep learning has been widely applied to the prediction of the Remaining Useful Life (RUL) of machinery, achieving promising results. However, existing methods often focus on features extracted from single-directional signals and overlook the dynamic interactions among multi-directional signals, resulting in limited generalization. Moreover, purely data-driven models lack physical interpretability and often fail to reflect the underlying degradation dynamics. To address these issues, this paper proposes a physics-informed orthogonal dynamics network (PINN-OD) for RUL prediction of machinery. In the proposed method, an orthogonal dynamic coupling module is first designed to extract and fuse degradation features from multi-directional vibration signals using both homogeneous and heterogeneous convolutional branches. Then, a physics-informed module learns an implicit degradation partial differential equation (PDE) from data, enabling physical consistency through automatic differentiation. The proposed approach is validated on a run-to-failure bearing dataset under multiple operating conditions. Experimental results show that PINN-OD outperforms traditional data-driven methods in both prediction accuracy and generalization capability.