Physics-Encoded Graph Deep Learning for Physical Diffusion Systems

Diffusion phenomena are fundamental in physics, engineering, biology, and etc., yet their simulation via traditional numerical methods is computationally intensive for complex systems. While deep learning offers promise, prevailing approaches like physics-informed neural network (PINN) suffers from limited generalizability and error accumulation, especially with sparse data or irregular geometries. Here, we propose a physics-encoded graph neural network (PeMN) that integrates graph message-passing mechanism with discrete differential geometry principles. By explicitly encoding differential operators such as Laplacian and gradient operators into network architecture for spatial discretization, PeMN reduces nonlinearity in function approximation while enforcing strict adherence to physics. Meanwhile, combining with recurrent temporal modeling it achieves global spatiotemporal optimization, avoiding error propagation inherent in numerical iterative methods. Benchmark study demonstrates very significant improvements in accuracy over other models. Notably, PeMN remains robust and enables generalization across various initial conditions. Our framework establishes a new paradigm for embedding PDEs into network architecture.