Patching Non-Euclidean Domains for Neural PDE Solvers

Localising a neural field by splitting its domain into patches is standard in flat domains and underlies both finite basis solvers and patch based tokenisation, but a curved surface carries no grid, so the patch family must itself be constructed. We show that overlapping decompositions of the ambient space restrict to a partition of unity on any orientable surface at no cost in generality, and that patches defined by geodesic distance can be used even though the PDE residual differentiates the windows twice while geodesic distance admits no closed form. Across four surfaces and five seeds we find that the benefit of patching is predicted not by curvature but by how far ambient proximity departs from intrinsic proximity, ranging from no benefit on a smooth torus to a factor of 4.9 on a hand scan, and that geodesic patches improve on ambient boxes on every mesh tested, by factors of 1.5 to 2.3 in the mean with p < 0.02 under a rank test.