| Abstract: |
| We derive quadrature error estimates for the evaluation of boundary integral potentials with logarithmic singularities, considering standard discretization techniques such as Gauss--Legendre and trapezoidal rules on general geometries. The analysis provides explicit asymptotic expressions for the quadrature error, highlighting its dependence on the distance of the evaluation point from the boundary and on the number of quadrature points. These estimates offer a practical tool for predicting accuracy in nearly singular regimes and for guiding adaptive strategies. As a complementary study, we investigate the applicability of these error estimates when the density is not known analytically but is approximated via physics-informed neural networks (PINNs). The learned density is used to reconstruct the potential through standard quadrature, and the resulting error is compared with the theoretical predictions. We examine whether the derived estimates remain valid in this data-driven setting and identify conditions under which they accurately capture the observed behavior. In particular, we show that the consistency of the estimates depends on the smoothness and local accuracy of the PINN density, especially near the boundary |
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