Learning Traveling Solitary Waves Using Separable Gaussian Neural Networks

Area of Science:

• Computational Physics
• Applied Mathematics
• Machine Learning

Background:

• Traveling solitary waves are crucial in various physical systems described by partial differential equations (PDEs).
• Traditional Physics-Informed Neural Networks (PINNs) face propagation failure issues in large computational domains due to independent spatial and temporal data treatment.

Purpose of the Study:

• To develop an interpretable neural network (NN) architecture, Separable Gaussian Neural Networks (SGNN), for learning traveling solitary waves.
• To integrate SGNN into the PINN framework to overcome limitations of traditional PINNs in handling wave propagation.
• To demonstrate the efficacy of the proposed method for various types of solitary wave solutions across different PDE families.

Main Methods:

• A novel interpretable neural network architecture, Separable Gaussian Neural Networks (SGNN), is introduced.
• Data is transformed into a co-traveling wave frame by leveraging wave characteristics, addressing spatial and temporal independence issues.
• The SGNN-integrated PINN approach is applied to (1+1)-dimensional *b*-family and (2+1)-dimensional Rosenau-Hyman PDEs, including peakon and compacton solutions.

Main Results:

• The SGNN architecture effectively approximates single-peakon, multi-peakon, and stationary ('lefton') solutions in the *b*-family of PDEs.
• The method successfully captures peakon solutions in the *ab*-family and compacton solutions in the Rosenau-Hyman family.
• Comparative analysis shows SGNN achieves comparable accuracy to multi-layer perceptrons (MLPs) using less than 10% of the neurons, highlighting significant efficiency gains.

Conclusions:

• The proposed SGNN-integrated PINN framework offers a robust and efficient solution for learning traveling solitary waves in nonlinear PDEs.
• The co-traveling wave frame transformation effectively resolves propagation failure issues in large computational domains.
• SGNN demonstrates superior efficiency and potential for broad applicability in solving complex nonlinear partial differential equations.