Protecting Small-Coefficient Terms in PDE Discovery with a Physical-Category Magnitude-Disentangled Transformer

PDE discovery recovers the governing equation from observed data. An equation has terms and coefficients. Recovering it has two steps: choosing terms, and fitting coefficients. Terms often have coefficients of very different sizes. Existing methods compare all terms together in one step. Large-coefficient terms then dominate, and small-coefficient terms are dropped when choosing terms. When a small term is dropped, coefficient fitting gives biased values to compensate, so the recovered equation is incomplete. We introduce the Physical-Category Magnitude-Disentangled Transformer (PC-MDT) with two mechanisms. Masked local category competition compares each term only with terms of the same physical category, time, space, or nonlinear, and a learned gate exchanges information across categories only when the equation needs it. Soft-gated global assembly elects at most one term per category or none and fits coefficients only after the structure is fixed, so size plays no role in the choice. On four benchmark equations, PC-MDT recovers all exactly, while every baseline fails on at least one. A model trained on heat transfers to Burgers with two of three true terms already in place.