Typed Frame Search: Reference-Free Coordinate Selection for PDEs

Many partial differential equation (PDE) solutions become stationary profiles in moving or rescaled coordinates, yet an accurate field fit need not identify a unique coordinate representation. We introduce typed frame search, which selects an explicit frame–profile program from the PDE and prescribed initial and boundary data, without observing future solution trajectories. Candidates from a finite transformation grammar are fitted to physics and data constraints, ranked on independent validation designs, and penalized for frame complexity to favor canonical representations. We establish conditions for frame identifiability and reliable selection within the fitted bank. Across 28 in-bank configurations from 14 PDE families and ten seeds, the method recovers 96.8% of canonical frames (271/280), with median relative L2 error of 1.43 × 10^-6. Compared with a degree-four continuous-clock baseline using the same number of optimization starts, it achieves approximately 90-fold lower median error and 66.6-fold lower median search time. Tests with unseen profiles and two-dimensional affine flows recover all 50 and 60 frames, respectively. Ablations distinguish the constraints that determine the field from the complexity penalty that resolves equivalent representations. These results support interpretable coordinate recovery within a finite grammar, with limitations from initial-data noise and unsupported transformations.