Solving high-dimensional parametric engineering problems for inviscid flow around airfoils based on physics-informed neural networks

Engineering problems often involve solving partial differential equations (PDEs) over a range of similar problem setups, leading to high computational costs when using traditional numerical approaches to solve each setup individually. Recently developed physics-informed neural networks (PINNs) offer a novel approach to solving parametric problems, enabling the simultaneous solution of a series of similar problems. Our previous research combined PINNs with mesh transformation to learn PDE solutions in the computational space, effectively solving inviscid flow around airfoils. In this study, we expand the input dimensions of the model to include shape parameters and flow conditions, forming an input encompassing the complete state-space (i.e., all parameters determining the solution are included in the input). We spend about 18.8 h achieving the continuous solutions in a large state-space in one go, encompassing various subsonic inviscid airfoil flows encountered in engineering, thereby highlighting the model's significant advantages in addressing high-dimensional parametric problems. Once established, the model can efficiently complete airfoil flow simulation and shape inverse design tasks in about 1 s. Furthermore, we introduce a pretraining-finetuning method, enabling the fine-tuning of the model for the task of interest and quickly achieving accuracy comparable to the finite volume method.