Consistent CutPINNs for Elliptic PDEs on Curved Level-Set Domains

Abstract

We propose \emph{Consistent CutPINN}, a framework for partial differential equations posed on bounded curved domains defined implicitly by a $\C^2$ level-set function, $\Omega = \{\varphi < 0\}$. In this paper we develop the framework for second-order elliptic problems in two dimensions. The standard PINN loss penalises the boundary mismatch in $L^2(\partial\Omega)$, but $L^2(\partial\Omega)$ does not control the $H^{1/2}(\partial\Omega)$ trace norm that appears in the $H^1(\Omega)$ energy estimate. The consistent PINN framework of Bonito et al.~\cite{bonito2025} fixes this on the unit cube $(0,1)^d$ via a Kuhn--Tucker simplicial decomposition of the flat boundary faces, but the construction relies on the affine structure of the faces and does not carry over to smooth curved boundaries. We address this gap. Specifically, (i) we introduce a discrete $H^{1/2}(\partial\Omega)$ surrogate built directly from collocation points on a $\C^2$ curve, (ii) we prove a \textit{Chord-arc} norm equivalence between this surrogate and the continuous trace norm, (iii) we establish an \emph{a priori} $H^1$ error bound on cut domains, and (iv) we derive convergence rates under Besov regularity using optimal recovery theory. Numerical experiments on a disk and a non-convex flower domain confirm that the consistent loss is much more accurate than the standard PINN loss and far more robust to cut-cell configurations.