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. On a smooth torus patching brings no benefit. On three meshes, with up to ten seeds each, the gain of geodesic patches over a tuned global network grows with how far ambient proximity departs from intrinsic proximity, from a factor of 1.7 to 4.6 on a hand scan. Geodesic patches improve on ambient boxes on every mesh tested, by factors of 1.4 to 2.1 in the mean with p < 0.001 under a rank test, and at low distortion the whole gain comes from going intrinsic, since ambient boxes alone do not beat a tuned global network. On a family of progressively squashed tori the advantage of geodesic patches over ambient boxes grows from parity to a factor of 5.5.