Correcting State-Dependent Propagation Defects for Operator Learning

Neural operators provide efficient surrogate models for the time evolution of parameterized partial differential equations (PDEs). Existing spectral neural operators reuse a shared response across input states, while nonlinear dynamics exhibit state-varying local responses. To address this structural discrepancy, we introduce State-Dependent Propagation Defect Correction (SDC), which enables an observable one-step propagation defect between a numerical reference map and a frozen neural propagator. We establish the connection between state-varying local responses and the propagation defect and show that standard prediction-space calibration is equivalent to regression of the observable defect. SDC-based operator learning is designed by a low-rank complex spectral update that adapts global mode-wise channel responses and a bias-free bottleneck that provides nonlinear state-conditioned correction from globally mixed hidden features. Both correction paths are zero-initialized and the pretrained propagator remains frozen during calibration, preserving the original propagation map at initialization. Experiments on time-dependent PDEs and weather forecasting demonstrate improved autoregressive prediction with minimal parameters and computational overhead.