Multi-benchmark adaptive sampling physics-informed neural network for complex and coupled equations.

Area of Science:

• Computational Mathematics
• Applied Physics
• Machine Learning

Background:

• Solving complex-value and coupled partial differential equations (PDEs) presents significant computational challenges.
• Physics-Informed Neural Networks (PINNs) offer a promising approach by integrating physical laws into neural network training.
• Existing PINN methods can struggle with convergence and accuracy for intricate equation systems.

Purpose of the Study:

• To develop an advanced PINN method capable of efficiently solving complex-value and coupled PDEs.
• To enhance the accuracy and convergence speed of neural network-based PDE solvers.
• To introduce an intelligent adaptive sampling strategy for improved solution fidelity.

Main Methods:

• Introduction of the Multi-Benchmark Adaptive Sampling Physics-Informed Neural Network (MBAS-PINN) framework.
• Implementation of an adaptive sampling strategy that dynamically adjusts residual point distribution based on multiple benchmarks.
• Development of the neural tangent kernel for PINNs applied to complex PDEs.
• Utilizing two distinct training strategies to optimize focus on critical solution regions (real and imaginary parts).

Main Results:

• The MBAS-PINN method demonstrated significant improvements in accuracy and convergence for complex PDEs.
• Experimental validation on the nonlinear Schrödinger equation, Hirota equation, and Yajima-Oikawa system confirmed the method's effectiveness.
• The adaptive sampling strategy successfully guided the neural network to prioritize regions crucial for solution accuracy.

Conclusions:

• The MBAS-PINN method provides a novel and effective approach for tackling complex-value and coupled PDEs.
• This technique enhances the capability of PINNs in solving challenging mathematical physics problems.
• The adaptive sampling strategy represents a key advancement in improving the performance of physics-informed neural networks.