NPINN+: an enhanced physics-informed neural network for solving wave equations with nonlocal boundary conditions

Wave equations with nonlocal conditions appear in many scientific and engineering applications, such as, the population dynamics, the mathematical biology, and the materials science. The numerical challenge mainly stems from nonlocal terms, whose global coupling degrades the efficiency and stability of classical methods. In recent years, physics-informed neural networks (PINN) have achieved notable success in solving partial differential equations. In this paper, we propose an enhanced physics-informed neural network for wave equations subject to nonlocal conditions, termed NPINN+. By exploiting an equivalent transformation of the nonlocal condition, the original problem is reformulated into a wave equation satisfying Neumann boundary conditions with an additional integral-form source term. NPINN+ employs a single neural network to provide a unified representation of the spatiotemporal solution, while incorporating the governing equation, derivative information, initial and boundary conditions, and nonlocal constraints into a unified physics-informed loss function, enabling effective capture of the underlying physical features. Furthermore, a residual-based dynamic sampling strategy and a SoftAdapt-driven adaptive loss weighting mechanism are introduced to enhance accuracy and training robustness. Numerical experiments on regular domains demonstrate the effectiveness of the proposed method, and its extension to star-shaped domains is achieved via a polar coordinate transformation. Comparative results with PINN, APINN, and RAR-PINN show that NPINN+ consistently achieves superior accuracy and stability.