A novel number-theoretic sampling method for neural network solutions of partial differential equations

Area of Science:

• Computational mathematics
• Numerical analysis
• Deep learning

Background:

• Traditional Monte Carlo integration struggles with low-regularity or high-dimensional problems.
• Uniform random sampling in numerical methods leads to degraded efficiency for complex equations.

Purpose of the Study:

• To develop a novel deep learning framework for efficient numerical integration of partial differential equations (PDEs).
• To address the limitations of traditional methods in handling low-regularity and high-dimensional PDEs.
• To provide rigorous mathematical guarantees for improved error bounds.

Main Methods:

• Utilizing deterministic number-theoretic sampling points generated by a generating vector for minimal discrepancy.
• Integrating Physics-Informed Neural Networks (PINNs) within the deep learning architecture.
• Employing numerical validation on diverse PDE problems, including Poisson, inverse Helmholtz, and high-dimensional PDEs.

Main Results:

• The proposed framework demonstrates superior performance compared to conventional uniform random sampling.
• Achieved lower error bounds due to deterministic sampling and PINN integration.
• Showcased robust generalization capabilities across various complex PDE scenarios.

Conclusions:

• The deep learning framework with deterministic sampling offers a robust and efficient solution for integrating PDEs.
• This approach significantly enhances accuracy and performance for problems with rough solutions or in high dimensions.
• The method provides a promising advancement in numerical analysis for scientific computing.