Residual point generation methods using Voronoi diagrams for physics-informed neural networks

Physics-Informed Neural Networks (PINNs) have become an essential method for solving various partial differential equations (PDEs) using artificial intelligence technology. In addition, The distribution of residual points significantly affects the performance of PINNs, leading many works to concentrate on optimizing the generation of these points. However, current sampling methods often focus on designing the probability density function (PDF), struggling to balance high and low residual areas, leading to instability. Hence, we introduce a series of sampling methods based on Voronoi diagrams. These include two uniform sampling methods: centroidal Voronoi tessellation (CVT) sampling and area-constrained Voronoi tessellation (AVT) sampling, as well as their corresponding resampling methods. We also propose two adaptive residual-based non-uniform sampling methods: residual-weighted centroidal Voronoi tessellation (RCVT) sampling and residual-constrained Voronoi tessellation (RVT) sampling. These Voronoi diagram-based sampling methods consider the locations of the generated points and the domain each point covers. We test the effectiveness of these six newly introduced sampling methods in six different PDE simulation experiments along with 15 existing sampling methods. These PDEs take into account both the characteristics of the equation itself, such as the linear Allen-Cahn equation and the nonlinear Wave equation, as well as the complexity of the solution and the solution domain, as seen in the Navier-Stokes equations, the Helmholtz equation and the Poisson-Boltzmann equation. The results demonstrate that our methods significantly improve prediction accuracy while keeping the computational cost on par with other sampling methods, and RCVT and RVT show superior stability across various PDE problems, initialization conditions, and resampling periods. According to our investigation, the methods involved in this work are proposed for the first time and have proven effective for two-dimensional problems. Future work will extend these methods to three-dimensional cases.