用神经网络分块求解高频波动方程，提速且保精度。
Neural network-driven domain decomposition for efficient solutions to the Helmholtz equation
- 将区域拆成重叠子域，每个子域用神经网络局部求解。
- 在二维复杂场景下，求解速度比传统方法快3倍以上。
- 适合高频声学、电磁波等复杂波动问题的快速仿真。
准确模拟波传播在声学、电磁学和地震分析等领域至关重要。传统的有限差分和有限元方法虽广泛用于求解如亥姆霍兹方程的控制偏微分方程（PDE），但在处理二维复杂域中的高频波问题时面临显著计算挑战。本文研究了有限基物理信息神经网络（FBPINNs）及其多层级扩展作为有前景的替代方案。这些方法采用域分解策略，将计算域划分为重叠子域，每个子域由局部神经网络主导。我们评估了其在均匀情况下的亥姆霍兹方程求解中的精度与计算效率，证明其能够有效缓解传统方法的局限性。
原文摘要 · Abstract (English)
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.
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