A new arXiv preprint by Nathanael Tepakbong, Jun Fan, Xiang Zhou, and Ding-Xuan Zhou analyzes physics-informed neural networks (PINNs) for uniformly elliptic Dirichlet boundary value problems. The authors study networks that satisfy boundary conditions exactly by multiplying the output by a user-defined distance-to-boundary approximation, $\rho$, rather than adding a boundary penalty term.
The paper shows that an $L^2(\partial\Omega)$ boundary penalty cannot generally guarantee $H^2(\Omega)$ error control, but exact enforcement can, provided $\rho$ is chosen appropriately. The authors identify a sharp sufficient condition on $\rho$ that preserves the regularity needed for $H^2(\Omega)$ error control, give practical constructions for general smooth domains, and derive explicit high-probability convergence rates for boundary-adapted ReQU and tanh networks.
As part of the proof, the authors obtain new VC-dimension bounds for hypothesis spaces of higher-order derivatives of ReQU and tanh networks, along with new approximation bounds for shallow ReQU networks in higher-order Sobolev norms. Numerical experiments support the theoretical results: appropriate choices of $\rho$ improve accuracy and convergence, while poorly chosen distance functions can substantially degrade the solution.