arXiv:2611.13205v1 Announce Type: new Abstract: Localized interfaces, layers, and shocks can require high spatial resolution even when most of a partial differential equation (PDE) solution remains smooth. In global spectral methods, resolving these features requires increasing the approximation order across the entire domain. LegONet builds PDE solvers from reusable mechanism blocks, but its global coefficient representation faces the same limitation. We introduce Patch LegONet, which reuses pretrained mechanism blocks on smaller physical patches to resolve localized fine scales. The solution consists of a global background and local corrections. Mapping a fixed reference representation onto smaller patches increases local resolution without enlarging the blocks. The corrections join smoothly to the global background at patch boundaries. Differential operators are rescaled analytically to account for each patch's physical size. Nonlinear mechanisms are evaluated on the combined solution to include interactions between the background and local corrections. Patches can move, merge, or be removed during rollout while the pretrained patch blocks remain frozen. Six experiments on Allen-Cahn and viscous Burgers equations assess static, multiple, and moving patches in one to three dimensions. At comparable numbers of degrees of freedom, Patch LegONet consistently lowers the final relative solution error compared with global refinement. The reduction is approximately one order of magnitude in most tests. These results extend mechanism reuse across spatial scales and provide a practical route to local refinement in learned PDE solvers.
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