Accurately simulating wave propagation is crucial in fields such as acoustics, electromagnetism, and seismic analysis. Traditional numerical methods, like finite difference and finite element approaches, are widely used to solve governing partial differential equations (PDEs) such as the Helmholtz equation. However, these methods face significant computational challenges when applied to high-frequency wave problems in complex two-dimensional domains. This work investigates Finite Basis Physics-Informed Neural Networks (FBPINNs) and their multilevel extensions as a promising alternative. These methods leverage domain decomposition, partitioning the computational domain into overlapping sub-domains, each governed by a local neural network. We assess their accuracy and computational efficiency in solving the Helmholtz equation for the homogeneous case, demonstrating their potential to mitigate the limitations of traditional approaches.

翻译：精确模拟波传播在声学、电磁学和地震分析等领域至关重要。传统数值方法，如有限差分法和有限元法，被广泛用于求解诸如亥姆霍兹方程等控制性偏微分方程。然而，当这些方法应用于复杂二维域中的高频波问题时，面临着显著的计算挑战。本研究探讨了有限基物理信息神经网络及其多级扩展作为一种有前景的替代方案。这些方法利用域分解技术，将计算域划分为重叠的子域，每个子域由一个局部神经网络控制。我们评估了它们在求解齐次情况亥姆霍兹方程时的精度和计算效率，证明了其缓解传统方法局限性的潜力。