arXiv:2511.11679v1 Announce Type: cross Abstract: Free-boundary diffeomorphism optimization is a core ingredient in the surface mapping problem but remains notoriously difficult because the boundary is unconstrained and local bijectivity must be preserved under large deformation. Numerical Least-Squares Quasiconformal (LSQC) theory, with its provable existence, uniqueness, similarity-invariance and resolution-independence, offers an elegant mathematical remedy. However, the conventional numerical algorithm requires landmark conditioning, and cannot be applied into gradient-based optimization. We propose a neural surrogate, the Spectral Beltrami Network (SBN), that embeds LSQC energy into a multiscale mesh-spectral architecture. Next, we propose the SBN guided optimization framework SBN-Opt which optimizes free-boundary diffeomorphism for the problem, with local geometric distortion explicitly controllable. Extensive experiments on density-equalizing maps and inconsistent surface registration demonstrate our SBN-Opt's superiority over traditional numerical algorithms.