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

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 $O(\text{eps})$ boundary layer in which the pointwise residual grows like ${\text{eps}}^{-1}$, 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 $\text{Lp}\gamma$ norm with $\gamma =1+1\mathrm{/}\log \text{mtil}$, a computable surrogate for the $\text{Hminusone}$ stability term, and imposing the boundary condition through a discrete $\text{HhalfBdry}$ trace norm, which treats flat and curved geometries uniformly. Under Besov regularity assumptions we prove a single a priori $\text{Hone}$ error bound, valid for all interior exponents $\gamma \in (1,2]$, with an optimal recovery rate governed by a cut-cell floor $1\mathrm{/}(2\gamma )$ specific to the curved geometry. Numerical experiments on a rectangle and a disk at $\text{eps}=2^{-s}$, $s\in \{2,4,6\}$, confirm the analysis: as the layer sharpens, the $\text{Lp}2$ interior loss becomes seed-fragile while the $\text{Lp}\gamma$ interior trains reliably, the interior norm being the decisive factor in convergence.