PT-symmetric PINN for integrable nonlocal equations: Forward and inverse problems.

Area of Science:

• Computational Physics
• Applied Mathematics
• Nonlinear Dynamics

Background:

• Solving PT-symmetric nonlocal equations presents challenges for traditional Physics-Informed Neural Networks (PINNs).
• Nonlocal terms in these equations complicate direct numerical differentiation within neural network architectures.

Purpose of the Study:

• To introduce a novel method, PT-symmetric semi-supervised neural networks (PTS-PINN), for solving PT-symmetric nonlocal equations.
• To enhance the accuracy and applicability of neural network-based solutions for complex nonlinear systems.

Main Methods:

• PTS-PINN reformulates nonlocal terms as coupled local components, avoiding direct differentiation.
• Physical information of PT-symmetry is integrated into the neural network's loss function to improve accuracy.
• The method is tested on various nonlocal equations, including NLS and three-wave interaction systems.

Main Results:

• PTS-PINN demonstrates robust performance in solving forward and inverse problems for diverse PT-symmetric nonlocal equations.
• The method shows exceptional capability in learning large space-time scale rogue waves governed by nonlocal dynamics.
• Numerical experiments confirm the efficacy and improved accuracy of PTS-PINN over standard approaches.

Conclusions:

• PTS-PINN offers a powerful and accurate framework for addressing PT-symmetric nonlocal equations.
• The integration of physical symmetry information into the loss function is a key factor in the method's success.
• This approach opens new avenues for simulating complex phenomena in nonlinear physics and applied mathematics.