A General Solution‐Based Physics‐Constrained Neural Network for Solving the One‐Dimensional Wave Equation

ABSTRACT

Wave propagation is a ubiquitous phenomenon in nature, serving as a fundamental mechanism for energy and information transfer across various fields. Traditional numerical methods for solving the wave equation, such as finite difference and spectral element methods, suffer from discretization errors that typically require a sufficiently small grid size to achieve acceptable accuracy. Physics‐informed neural networks (PINNs) offer a mesh‐free alternative by embedding physical constraints directly into the loss function to achieve accurate and efficient modeling. However, PINNs encounter convergence challenges when applied to the one‐dimensional (1D) wave equation, primarily due to the unique loss landscape associated with wave problems, particularly for long‐duration and complex wave‐input scenarios. To overcome these challenges, this paper introduces a general solution‐based physics‐constrained neural network (gs‐PCNN), which embeds the general solution of the wave equation directly into the network architecture. The gs‐PCNN consists of two subnetworks inspired by the D'Alembert general solution, so that the governing equation is satisfied by construction and only the initial and boundary conditions need to be enforced during training. Compared with the standard PINN and sinusoidal‐feature PINN (sf‐PINN) baseline models, the gs‐PCNN achieves substantially higher accuracy and shorter training times for long‐duration harmonic waves and maintains a clear accuracy advantage for broadband earthquake‐motion inputs.