Moving sampling physics-informed neural networks induced by moving mesh PDE.

Area of Science:

• Computational Science
• Numerical Analysis
• Machine Learning

Background:

• Traditional numerical methods often struggle with adaptive sampling for complex problems.
• Improving the quality and control of sampling points is crucial for simulation accuracy.
• Deep learning and mesh-based methods offer potential for advanced sampling strategies.

Purpose of the Study:

• To develop an end-to-end adaptive sampling framework using deep neural networks and the moving mesh method.
• To enhance the precision and controllability of sampling point distribution.
• To integrate the adaptive sampling framework with physics-informed neural networks (PINN) for improved performance.

Main Methods:

• Proposing the Moving Mesh Partial Differential Equation Network (MMPDE-Net) for adaptive sampling point generation.
• Developing an iterative algorithm based on MMPDE-Net for precise sampling point distribution.
• Combining MMPDE-Net with Physics-Informed Neural Networks (PINN) to create Moving Sampling PINN (MS-PINN).

Main Results:

• MMPDE-Net adaptively generates high-quality sampling points by solving moving mesh PDEs.
• The iterative algorithm ensures more precise and controllable sampling point distribution.
• MS-PINN demonstrates significant performance improvements over standard PINN in numerical experiments.

Conclusions:

• The proposed MMPDE-Net framework effectively enhances sampling point generation quality.
• MS-PINN offers a robust and accurate approach for numerical simulations, outperforming traditional PINN.
• The method provides a novel way to improve the efficiency and reliability of scientific computing.