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Benchmarking Physics-Informed Neural Networks and Boundary Elements Methods for Wave Scattering

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.

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