Physics-Informed Neural Networks (PINNs) provide a mesh-free, optimization-based framework for the numerical solution of partial differential equations by enforcing the governing equations and boundary conditions through a unified loss functional. In parallel, classical numerical methods such as the Galerkin Boundary Element Method (BEM) offer a well-established discretization strategy based on boundary integral formulations and boundary-only representations.

In this work, we present a systematic comparison between PINNs and a Galerkin BEM for the numerical solution of the Laplace equation posed on Koch prefractal domains. These geometries form a sequence of polygonal approximations converging to a fractal limit, with increasing geometric complexity and decreasing boundary regularity as the prefractal level increases. Both approaches are implemented independently and tested on identical geometries and discretizations in order to ensure a fair and consistent comparison.

The numerical experiments investigate the influence of boundary refinement and prefractal level on the performance of each method. The comparison focuses on solution accuracy, computational cost, sensitivity to the prescribed boundary data, and robustness with respect to increasing geometric irregularity.

The goal of this study is to provide a balanced assessment of the capabilities and limitations of PINNs and classical boundary integral techniques when applied to non-smooth and highly structured computational domains, and to highlight their respective advantages in the context of fractal-like geometries.