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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 <jats:italic>activation function<\/jats:italic>. The ReLU-NNs used to approximate the solutions to the BIEs depend nonlinearly on the parameters characterizing the NNs themselves. Consequently, the computation of a numerical solution to a BIE by means of ReLU-NNs boils down to a fine tuning of these parameters, in <jats:italic>network training<\/jats:italic>. We argue that ReLU-NNs of fixed depth and with a variable width allow us to recover well-known approximation rate results for the standard Galerkin Boundary Element Method (BEM). This observation hinges on existing well-known properties concerning the regularity of the solution of the BIEs on Lipschitz, polygonal boundaries, i.e.\u00a0accounting for the effect of corner singularities, and the <jats:italic>expressive<\/jats:italic> power of ReLU-NNs over different classes of functions. We prove that <jats:italic>shallow<\/jats:italic> ReLU-NNs, i.e.\u00a0networks having a fixed, moderate depth but with increasing width, can achieve optimal order algebraic convergence rates. We propose novel loss functions for NN training which are obtained using <jats:italic>computable, local residual a posteriori error estimators<\/jats:italic> with ReLU-NNs for the numerical approximation of BIEs. We find that weighted residual estimators, which are reliable without further assumptions on the quasi-uniformity of the underlying mesh, can be employed for the construction of computationally efficient loss functions for ReLU-NN training. The proposed framework allows us to leverage on state-of-the-art computational deep learning technologies such as TENSORFLOW and TPUs for the numerical solution of BIEs using ReLU-NNs. Exploratory numerical experiments validate our theoretical findings and indicate the viability of the proposed ReLU-NN Galerkin BEM approach.<\/jats:p>","DOI":"10.1007\/s10915-023-02120-w","type":"journal-article","created":{"date-parts":[[2023,3,19]],"date-time":"2023-03-19T14:02:42Z","timestamp":1679234562000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["ReLU Neural Network Galerkin BEM"],"prefix":"10.1007","volume":"95","author":[{"given":"Rub\u00e9n","family":"Aylwin","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Fernando","family":"Henr\u00edquez","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Christoph","family":"Schwab","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2023,3,19]]},"reference":[{"issue":"4","key":"2120_CR1","doi-asserted-by":"publisher","first-page":"A2474","DOI":"10.1137\/20M1366587","volume":"43","author":"M Ainsworth","year":"2021","unstructured":"Ainsworth, M., Dong, J.: Galerkin neural networks: a framework for approximating variational equations with error control. 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