Consistent CutPINNs for Convection-Diffusion Equations on Curved Level-Set Domains

Abstract

We present an a priori error analysis of consistent-loss PINNs for stationary convection-diffusion equations on curved level-set domains. The standard mean-squared interior loss fails in the convection-dominated regime: the solution develops an $O(\eps)$ boundary layer in which the pointwise residual grows like $\eps^{-1}$, so the loss is dominated by the few collocation points inside the layer and leaves the smooth bulk unresolved. We remove this mismatch by penalising the interior residual in a discrete $\Lpγ$ norm with $γ= 1 + 1/\log\mtil$, a computable surrogate for the $\Hminusone$ stability term, and imposing the boundary condition through a discrete $\HhalfBdry$ trace norm, which treats flat and curved geometries uniformly. Under Besov regularity assumptions we prove a single a priori $\Hone$ error bound, valid for all interior exponents $γ\in (1,2]$, with an optimal recovery rate governed by a cut-cell floor $1/(2γ)$ specific to the curved geometry. Numerical experiments on a rectangle and a disk at $\eps = 2^{-s}$, $s \in \{2,4,6\}$, confirm the analysis: as the layer sharpens, the $\Lp{2}$ interior loss becomes seed-fragile while the $\Lpγ$ interior trains reliably, the interior norm being the decisive factor in convergence.