PINN Based on Domain Adaptation for Solving Fornberg–Whitham-Type Equations

The Fornberg–Whitham equation, which contains high-order nonlinear derivatives, is widely recognized as a prominent model for describing shallow water dynamics. We explore physics-informed neural networks (PINsN) in conjunction with transfer learning technology to investigate numerical solutions for the FW equation. The proposed domain adaptation in transfer learning for PINN transforms the complex problem defined over the entire spatiotemporal domain into simpler problems defined over smaller subdomains. Training a neural network on these subdomains provides extra supervised learning data, effectively addressing optimization challenges associated with PINNs. Consequently, it enables the resolution of anisotropic and long-term predictive issues in these types of equations. Moreover, it enhances prediction accuracy and accelerates loss convergence by circumventing local optima in the specified scenarios. The method efficiently handles both forward and inverse FW equation problems, excelling in cost-effective accurate predictions for inverse problems. The efficiency and accuracy of our proposed approaches are demonstrated through examples and results.