Phase field models are widely employed in materials science, fluid dynamics, fracture mechanics, and image processing to describe physical processes involving complex interface evolution, with the Allen–Cahn equation serving as a classic example. Physics-informed neural networks (PINNs) have been extensively applied to various partial differential equations, but their accuracy in solving the Allen–Cahn equation is often compromised due to the sharp variations of the equation’s solution across phase interfaces. This paper proposes an enhanced PINN with spatial and temporal feature embedding strategies (STFE-PINN) for solving the Allen–Cahn equation. Specifically, for the spatial embedding, the input spatial coordinates are passed through a radial basis function (RBF) neural network, and its output is embedded as a feature into the PINN. By incorporating learnable shape parameters, the RBF embedding adaptively adjusts its basis functions, which strengthens the PINN’s ability to capture high-frequency spatial features and steep gradients. For the temporal embedding, the initial condition is embedded as an additional feature. This strategy enables the network to retain the information of the initial condition throughout training, thereby improving the learning of the phase field evolution and enhancing the accuracy of the PINN. Numerical experiments in one, two, and three dimensions validate the effectiveness and stability of the proposed method.
Wang et al. (Thu,) studied this question.