Beyond PINNs: Boundary-Enriched Holomorphic Networks for Laplace Problems

Physics-informed neural networks (PINNs) solve partial differential equations by penalizing their residual, but this formulation can be difficult to optimize for solutions with singular behaviour. We focus specifically on the two-dimensional Laplace equation, whose strong analytical structure allows the governing equation to be enforced directly in the neural representation. We compare a hybrid harmonic--neural model with a holomorphic neural network whose output is harmonic by construction and therefore requires only boundary training. We further enrich the holomorphic model with singular functions derived from the boundary specification and allow their exponents to be learned. Across a suite of singular Laplace problems, the enriched holomorphic model consistently outperforms conventional neural approaches and accurately recovers solutions with boundary-induced singularities. Our results demonstrate how exploiting problem-specific analytical structure can provide an alternative to base and residual PINNs for challenging scientific computing problems.