Physics-Informed Neural Networks with Complementary Soft and Hard Constraints for Solving Complex Boundary Navier-Stokes Equations

Soft- and hard-constrained Physics Informed Neural Networks (PINNs) have achieved great success in solving partial differential equations (PDEs). However, these methods still face great challenges when solving the Navier-Stokes equations (NSEs) with complex boundary conditions. To address these challenges, this paper introduces a novel complementary scheme combining soft and hard constraint PINN methods. The soft-constrained part is thus formulated to obtain the preliminary results with a lighter training burden, upon which refined results are then achieved using a more sophisticated hard-constrained mechanism with a primary network and a distance metric network. Specifically, the soft-constrained part focuses on boundary points, while the primary network emphasizes inner domain points, primarily through PDE loss. Additionally, the novel distance metric network is proposed to predict the power function of the distance from a point to the boundaries, which serves as the weighting factor for the first two components. This approach ensures accurate predictions for both boundary and inner domain areas. The effectiveness of the proposed method on the NSEs problem with complex boundary conditions is demonstrated by solving a 2D cylinder wake problem and a 2D blocked cavity flow with a segmented inlet problem, achieving significantly higher accuracy compared to traditional soft- and hard-constrained PINN approaches. Given PINN's inherent advantages in solving the inverse and the large-scale problems, which are challenging for traditional computational fluid dynamics (CFD) methods, this approach holds promise for the inverse design of required flow fields by specifically-designed boundary conditions and the reconstruction of large-scale flow fields by adding a limited number of training input points. The code for our approach will be made publicly available.