RF-LIN: Linear PDE Representations via Invertible Random-Feature Coordinates

Linear representations of nonlinear PDEs must preserve state information while capturing dynamics. We introduce RF-LIN, which learns invertible random-feature coordinates and a finite-dimensional linear evolution operator. Frozen nonlinear features make each appended additive-coupling readout a conditional least-squares problem. For fixed coordinates, a projected singular-value solve fits the operator with a stationarity certificate in the retained subspace. Exact reconstruction and multistep composition separate coordinate invertibility from dynamical accuracy. We derive a finite-horizon error bound linking latent residuals, matrix-power growth, and decoder sensitivity, and show that a disconnected complete fixed-point set precludes global homeomorphic linearization. In three-seed Burgers experiments at two viscosities, RF-LIN achieves mean spectral errors of 0.927% and 4.415%, reductions of 28.3% and 17.1% relative to RF-EDMD, the strongest tested spectral baseline. Reconstruction errors remain below 2.8 × 10^-14, while RF-EDMD yields lower mean rollout error. Allen–Cahn tests expose a spurious near-fixed bridge between attractors. A single Kuramoto–Sivashinsky validation window exhibits strong transient amplification and poor rollout accuracy despite a spectral radius below one. These results demonstrate spectral gains from learned invertible coordinates and identify geometric and finite-time limits on their predictive use.