Physics-Encoded Graph Deep Learning for Physical Diffusion Systems

Physics-Encoded Graph Deep Learning for Physical Diffusion Systems

Diffusion phenomena are fundamental in physics, engineering, and biology, yet their simulation via traditional numerical methods is computationally intensive for complex systems. While deep learning offers promise, prevailing approaches like the physics-informed neural network (PINN) suffer 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 the graph message-passing mechanism with discrete differential geometry principles. By explicitly encoding differential operators such as the Laplacian and gradient operators into the network architecture for spatial discretization, PeMN reduces nonlinearity in function approximation while enforcing strict adherence to physics.

Diffusion equations are fundamental mathematical models that describe the spatial and temporal evolution of physical quantities such as heat, mass, and momentum. These partial differential equations (PDEs) are widely applied in a variety of disciplines, including physics, engineering, and biology, and they are used to characterize complex spatiotemporal dynamic processes such as chemical reactions, fluid dynamics, and climate change. Accurate simulation of diffusion processes is therefore crucial in optimizing designs, conducting risk assessment, and facilitating decision-making.

However, the simulation of diffusion systems using traditional numerical computation methods, such as the finite element method (FEM) and the finite volume method, faces a series of challenges. The primary issue is the high computational cost associated with handling large-scale, high-dimensional problems, especially in scenarios requiring numerous repetitive simulations, such as optimization design, inverse problems, and uncertainty quantification. Moreover, for diffusion problems with complex boundary conditions and nonlinear characteristics, the stability and convergence of numerical methods are particularly pronounced, necessitating careful algorithm design and parameter tuning. With the advancement of computing power, deep learning methods offer a new possibility for addressing these challenges, as deep learning models are capable of learning complex spatiotemporal patterns from data and enabling rapid predictions at a significantly lower computational cost.