LCKoop: Koopman-Inspired Smooth Autoregressive Neural Operators in PDE Forecasting

Modern autoregressive neural operators evolve high-dimensional states through expressive nonlinear transition maps. However, their local first-order behavior varies rapidly along generated trajectories, leading to unstable error propagation. In contrast, Koopman-based models and dynamic mode decomposition (DMD) impose a globally or locally static linear transition, providing more stable representations of state evolution, but are inherently limited in their ability to capture high-dimensional strongly nonlinear dynamics. In this work, we introduce **LCKoop**: **L**ow-**C**urvature-is-**Koop**man, an easy-to-use, *plug-in* regularizer that bridges these two frameworks. LCKoop instantiates the model-implied first-order affine approximation and encourages it to remain consistent along generated trajectories, without modifying the forecasting architecture or inference path. We show that this regularization controls a trajectory-conditioned higher-order remainder in autoregressive error propagation, complementing conventional first-order sensitivity analysis. Across four PDE forecasting tasks and three neural-operator families, LCKoop reduces MSE by an average of 8.9% at the short forecasting horizon and 14.6% at the extended horizon, with maximum reductions of 53.7% and 42.4%, respectively.