This study compares the Boundary Element Method (BEM) and Physics-Informed Neural Networks (PINNs) for solving the two-dimensional Helmholtz equation in wave scattering problems. The objective is to evaluate the performance of both methods under the same conditions. We solve the Helmholtz equation using BEM and PINNs for the same scattering problem. PINNs are trained by minimizing the residual of the governing equations and boundary conditions with their configuration determined through hyperparameter optimization, while BEM is applied using boundary discretization. Both methods are evaluated in terms of solution accuracy and computation time. We conducted numerical experiments by varying the number of boundary integration points for the BEM and the number of hidden layers and neurons per layer for the PINNs. We performed a hyperparameter tuning to identify an adequate PINN configuration for this problem as a network with 3 hidden layers and 25 neurons per layer, using a learning rate of $10^{-2}$ and a sine activation function. At comparable levels of accuracy, the assembly and solution of the BEM system required a computational time on the order of $10^{-2}$~s, whereas the training time of the PINN was on the order of $10^{2}$~s, corresponding to a difference of approximately four orders of magnitude. However, once trained, the PINN achieved evaluation times on the order of $10^{-2}$~s, which is about two orders of magnitude faster than the evaluation of the BEM solution at interior points. This work establishes a procedure for comparing BEM and PINNs. It also presents a direct comparison between the two methods for the scattering problem. The analysis provides quantitative data on their performance, supporting their use in future research on wave propagation problems and outlining challenges and directions for further investigation.

翻译：本研究比较了边界元法（BEM）与物理信息神经网络（PINNs）在求解波散射问题中二维亥姆霍兹方程方面的应用。目标是在相同条件下评估两种方法的性能。我们针对同一散射问题，分别使用BEM和PINNs求解亥姆霍兹方程。PINNs通过最小化控制方程和边界条件的残差进行训练，其网络结构通过超参数优化确定；而BEM则采用边界离散化方法实现。我们从解算精度和计算时间两方面对两种方法进行了评估。我们通过改变BEM的边界积分点数量以及PINNs的隐藏层数和每层神经元数量进行了数值实验。通过超参数调优，我们确定了适用于本问题的PINN配置：一个具有3个隐藏层、每层25个神经元、学习率为$10^{-2}$并使用正弦激活函数的网络。在精度相当的情况下，BEM系统的组装与求解所需的计算时间约为$10^{-2}$~s量级，而PINN的训练时间约为$10^{2}$~s量级，两者相差约四个数量级。然而，一旦训练完成，PINN的评估时间可达$10^{-2}$~s量级，这比在内部点评估BEM解的速度快约两个数量级。本研究建立了一套比较BEM与PINNs的流程，并对两种方法在散射问题中的表现进行了直接对比。该分析提供了关于二者性能的定量数据，为其在波传播问题未来研究中的应用提供了支持，并指出了进一步研究的挑战与方向。