ReLU Neural Network Galerkin BEM

Abstract

We introduce Neural Network (NN for short) approximation architectures for the numerical solution of Boundary Integral Equations (BIEs for short). We exemplify the proposed NN approach for the boundary reduction of the potential problem in two spatial dimensions. We adopt a Galerkin formulation-based method, in polygonal domains with a finite number of straight sides. Trial spaces used in the Galerkin discretization of the BIEs are built by using NNs that, in turn, employ the so-called Rectified Linear Units (ReLU) as the underlying activation function. The ReLU-NNs used to approximate the solutions to the BIEs depend nonlinearly on the parameters characterizing the NNs themselves.