Train on the Sphere, Deploy on the Hill: Closed-Form-Anchored Surrogates for Real-Terrain Boundary-Integral Equations

The analysis of electrostatic induction effects on transmission lines in real-world power distribution systems requires inferring induced surface charge density $\sigma$ on overhead-line corridors at deployment scale, where dense $\mathcal{O}(N^3)$ boundary-element solvers for the underlying Poisson boundary integral equation (BIE) are infeasible. We present a recipe for training cheap PDE surrogates from canonical-case closed forms (flat ground, grounded sphere) plus sparse dense-solver fine-tuning where the canonical case breaks like in our case study. The recipe also avoids a structural degeneracy between training operator $A_{\theta }$ and solution $\sigma$ in self-supervised operator learning that tends to lead to internally consistent but non-physical solutions. The recipe combines a closed-form local-tangent-plane baseline plus a small neural correction; a frozen analytic kernel; an $SE(3)$- and scale-invariant dimensionless feature stack derived from a $2$-term identity $\sigma =-(\nabla \varphi \cdot \hat{n})/(2\pi )+H\varphi /(4\pi )$, exact on flat ground and the grounded sphere and supervised training against this identity. An $8$-parameter linear head recovers the analytic curvature coefficient $+1/(4\pi )$ from data to within $2\%$; holds to $1\%$ relative RMS in $\sigma$ across $12$ orders of magnitude in source-charge scale, where DeepONet, Set Transformer, and FNO baselines miss the analytic answer by orders of magnitude out-of-distribution; and matches direct-collocation boundary-element ground truth on a synthetic curved patch to $3.8\%$ relative RMS. A regime-of-validity result delineates when the canonical $2$-term identity applies; outside it (terrain curvature radius ${\kappa }^{-1}$ much greater than source clearance height $|h|$, the corridor regime), we extend the recipe with sparse dense-BEM-target supervision. On several industrial transmission-line corridors with native triangulated meshes, the extended recipe attains $9.2\%$ median relative RMS to dense BEM on held-out corridors, $20\times$ tighter than the analytic 2-term plug-in. Deployment: a $1.7$kB MLP, seconds per corridor on a single CPU core.