Physics-informed surrogate model for mesh deformation with hard constraints

Mesh deformation technology is widely used in computational fluid dynamics simulations. Solving partial differential equations (PDEs) is a practical approach for addressing mesh deformation, and physics-informed neural networks (PINNs) have recently emerged as a promising method for solving both forward and inverse problems governed by PDEs. However, soft constraint schemes commonly adopted in PINNs often struggle to maintain mesh quality near moving boundaries. This paper proposes a novel hard constraint approach that effectively enforces both Dirichlet and Neumann boundary conditions. By incorporating the gradient auxiliary function, surrogate models based on the Laplace and biharmonic equations for mesh deformation are developed. The effectiveness of the proposed method is demonstrated through two-dimensional airfoil pitching cases and a three-dimensional wing bending case. For unsteady flow simulations where the mesh evolves over time, operator learning via DeepONet is introduced to enhance the model's generalization capability. The temporal variable is embedded as a branch network within the previous model. A mesh deformation surrogate solver is implemented based on OpenFOAM, and numerical validation is performed using a fish cruising case to investigate the underlying mechanism of fish locomotion.