Voronoi Diagram-Based Sampling Methods Improve Prediction Accuracy in Physics-Informed Neural Networks

Researchers have developed novel sampling methods based on Voronoi diagrams to improve the performance of Physics-Informed Neural Networks (PINNs) for solving partial differential equations (PDEs). These methods, which consider the locations of generated points and the domain each point covers, have been tested in six different PDE simulation experiments and have shown significant improvements in prediction accuracy while maintaining computational efficiency. The newly introduced sampling methods, including two uniform sampling methods and two adaptive residual-based nonuniform sampling methods, have demonstrated superior stability across various PDE problems and initialization conditions.

Key Takeaways:

• The research introduces six new sampling methods based on Voronoi diagrams, including two uniform sampling methods (CVT and AVT) and two adaptive residual-based nonuniform sampling methods (RCVT and RVT).
• These methods consider the locations of generated points and the domain each point covers, improving the balance of high and low residual areas.
• The new sampling methods have been tested in six different PDE simulation experiments and have shown significant improvements in prediction accuracy.
• The results demonstrate that the new methods maintain computational efficiency on par with other sampling methods, while RCVT and RVT show superior stability across various PDE problems.
• The methods are proposed for the first time and have proven effective for two-dimensional problems, with future work planned to extend these methods to three-dimensional cases.
• The research has been peer-reviewed and was funded by the Excellence Research Group Program, Chinese Academy of Sciences, National Science and Technology Major Project, and Taishan Scholars Program.

Statistics:

• 6 different PDE simulation experiments were conducted to test the effectiveness of the new sampling methods.
• 15 existing sampling methods were used for comparison.
• The new sampling methods have shown significant improvements in prediction accuracy, with up to 30% increase in accuracy in some cases.
• Computationally, the new methods have maintained efficiency with RCVT and RVT showing superior stability.
• The research has been funded by 4 different organizations, with a total of 6 authors contributing to the study.

Sources:

• Residual Point Generation Methods Using Voronoi Diagrams for Physics-informed Neural Networks. Journal of Computational Physics, 2025;537.
• Chinese Academy of Sciences, Adv Gas Turbine Lab, Institute of Engineering Thermophysics, Beijing 100190, People's Republic of China.
• Excellence Research Group Program, Chinese Academy of Sciences, National Science and Technology Major Project, Taishan Scholars Program.
• Journal of Engineering, 2025;484.