A structure-preserving PINN with embedded periodic boundary layer and adaptively enforced initial conditions for geometric flows

Geometric flow models, driven by mechanisms such as surface tension, chemical potential, or curvature energy, are widely applied in fields including materials science, biological membrane dynamics, and image processing. In this paper, we propose a structure-preserving physics-informed neural network (PINN) method that incorporates an embedded periodic boundary layer and adaptively enforced initial conditions. This method, referred to as sp-epai-PINN, is specifically designed to solve a range of representative geometric flow problems, including mean curvature flow, surface diffusion flow and elastic flow. The PINN is a mesh-free approach that is particularly well-suited for solving geometric flow problems, as traditional numerical methods often encounter difficulties related to mesh quality during the evolution process. The embedded periodic boundary layer plays a key role in accurately handling flows involving closed curves, while the adaptively enforced initial conditions significantly enhance the model's capability to address complex curve evolution. Pretraining enables the network to inherit the stability of the initial topology, providing a reliable foundation for the subsequent fine-tuning stage. Furthermore, incorporating energy and/or area constraints into the loss function results in a structure-preserving algorithm that maintains the intrinsic geometric properties of the continuous model. These design elements collectively form the main innovations of the sp-epai-PINN framework, which proves to be highly effective in solving the geometric flows. Numerical experiments were extensively conducted to validate the proposed methods, revealing that sp-epai-PINN consistently surpasses traditional PINN approaches in long-term stability, geometric structure preservation, and convergence efficiency.