Illustration of the BEM and PINNs implementations for the scattering problem

(A) Distribution of boundary nodes and elements for BEM, along with the spatial arrangement of evaluation points within the domain. Gray dots represent evaluation points, while blue dots indicate integration points on the boundary Γᵢ.<br>(B) Sequential workflow of the BEM implementation. The inputs are the number of integration points for the boundary condition 𝑢ₙ on <b>𝒙 ∈ Γᵢ</b> and the number of evaluation points 𝑁ₚ. The boundary integral equation is solved to obtain the scattered field, which is then evaluated at the points <b>𝒙 ∈ Ωₚ</b>.<br>(C) Sampling strategy in the physical domain for PINNs, including points in the interior Ωₚ and on the boundaries Γᵢ and Γₑ. Gray dots represent collocation points for the Helmholtz equation, while blue dots denote training points for the interior boundary condition on Γᵢ.<br>(D) Optimization process of the PINNs implementation for the Helmholtz equation. The neural network takes spatial coordinates (𝑥, 𝑦) as input and outputs the real and imaginary parts of the scattered field 𝑢ₛ꜀ₜ. The loss function enforces the physics through the Helmholtz equation in Ωₚ, the Neumann boundary condition on Γᵢ, and the Sommerfeld radiation condition on Γₑ. The optimization loop updates the network parameters until the prescribed tolerance ε is reached.