arXiv:2602.18801v1 Announce Type: new Abstract: Neural operators provide fast PDE surrogates and often generalize across parameters and resolutions. However, in the short train long test setting, autoregressive rollouts can become unstable. This typically happens for two reasons: one step errors accumulate over time, and high frequency components feed back and grow. We introduce the Spectral Generator Neural Operator (SGNO), a residual time stepper that targets both effects. For the linear part, SGNO uses an exponential time differencing update in Fourier space with a learned diagonal generator. We constrain the real part of this generator to be nonpositive, so iterating the step does not amplify the linear dynamics. For nonlinear dynamics, SGNO adds a gated forcing term with channel mixing within each Fourier mode, which keeps the nonlinear update controlled. To further limit high frequency feedback, SGNO applies spectral truncation and an optional smooth mask on the forcing pathway. We derive a one step amplification bound and a finite horizon rollout error bound. The bound separates generator approximation error from nonlinear mismatch and gives sufficient conditions under which the latent $L^2$ norm does not grow across rollout steps. On APEBench spanning 1D, 2D, and 3D PDE families, SGNO achieves lower long horizon error and longer stable rollout lengths than strong neural operator baselines. Ablations confirm the roles of the generator constraint, gating, and filtering.The code is available at https://github.com/lijy32123-cloud/SGNO.