A Fourier physics-informed graph neural network for Allen–Cahn and Conservative Allen–Cahn equations

In this paper, a Fourier physics-informed graph neural network is proposed to simulate complex interfacial evolution within the phase-field framework. The approach employs an encoder–processor–decoder architecture, representing the physical system as an unstructured graph to ensure strong adaptability to complex geometric domains. Within the encoder, a Fourier feature mapping layer preprocesses spatial coordinates to mitigate spectral bias and improve the resolution of sharp phase boundaries. To capture spatial interactions robustly, the processor utilizes an edge-feature-free message-passing mechanism. The governing dynamics of the Allen–Cahn and conservative Allen–Cahn equations are reformulated into a discrete variational optimization problem based on the minimizing movement scheme. For conservative Allen–Cahn dynamics, the global mass constraint is incorporated into the variational formulation through a Lagrange multiplier. A deterministic projection step eliminates residual mass errors and enforces strict discrete mass conservation. By minimizing the graph-discretized energy functional at each time step, this formulation effectively integrates physical principles, driving the macroscopic system toward energy dissipative evolution. Finally, several numerical simulations in two- and three-dimensional spaces are presented to demonstrate the accuracy of the proposed method.