Dual Physics-Informed Neural Networks Based on Traveling- Wave Decomposition for Boundary Reflection Solutions of Wave Equations

We propose a dual physics-informed neural network (Dual-PINN) for wave equations by incorporating the characteristic structure of the one-dimensional d’Alembert solution. Two neural branches are associated with the characteristic coordinates x−ct and x+ct to represent counter-propagating wave components, and their outputs are jointly constrained by the governing equation and auxiliary conditions. Homogeneous Dirichlet boundaries can be imposed exactly through a trial function. For one-dimensional problems, an approximation result shows that the dual-branch representation can approximate solutions of the form u(x,t)=F(x−ct)+G(x+ct) arbitrarily well under standard universal-approximation assumptions, without implying convergence of the non-convex training procedure. Numerical examples involving periodic propagation, boundary reflection, and a two-dimensional wave problem show that the proposed representation captures the tested wave dynamics effectively. The two-dimensional case is treated as an empirical extension, suggesting the potential of characteristic-informed neural representations for wave-equation modeling.